29.48/15.63 YES 31.93/16.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 31.93/16.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 31.93/16.19 31.93/16.19 31.93/16.19 H-Termination with start terms of the given HASKELL could be proven: 31.93/16.19 31.93/16.19 (0) HASKELL 31.93/16.19 (1) BR [EQUIVALENT, 0 ms] 31.93/16.19 (2) HASKELL 31.93/16.19 (3) COR [EQUIVALENT, 0 ms] 31.93/16.19 (4) HASKELL 31.93/16.19 (5) Narrow [SOUND, 0 ms] 31.93/16.19 (6) QDP 31.93/16.19 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 31.93/16.19 (8) AND 31.93/16.19 (9) QDP 31.93/16.19 (10) MRRProof [EQUIVALENT, 88 ms] 31.93/16.19 (11) QDP 31.93/16.19 (12) TransformationProof [EQUIVALENT, 0 ms] 31.93/16.19 (13) QDP 31.93/16.19 (14) QReductionProof [EQUIVALENT, 0 ms] 31.93/16.19 (15) QDP 31.93/16.19 (16) TransformationProof [EQUIVALENT, 0 ms] 31.93/16.19 (17) QDP 31.93/16.19 (18) TransformationProof [EQUIVALENT, 0 ms] 31.93/16.19 (19) QDP 31.93/16.19 (20) UsableRulesProof [EQUIVALENT, 0 ms] 31.93/16.19 (21) QDP 31.93/16.19 (22) QReductionProof [EQUIVALENT, 0 ms] 31.93/16.19 (23) QDP 31.93/16.19 (24) TransformationProof [EQUIVALENT, 0 ms] 31.93/16.19 (25) QDP 31.93/16.19 (26) UsableRulesProof [EQUIVALENT, 0 ms] 31.93/16.19 (27) QDP 31.93/16.19 (28) QReductionProof [EQUIVALENT, 0 ms] 31.93/16.19 (29) QDP 31.93/16.19 (30) QDPOrderProof [EQUIVALENT, 20 ms] 31.93/16.19 (31) QDP 31.93/16.19 (32) TransformationProof [EQUIVALENT, 0 ms] 31.93/16.19 (33) QDP 31.93/16.19 (34) DependencyGraphProof [EQUIVALENT, 0 ms] 31.93/16.19 (35) TRUE 31.93/16.19 (36) QDP 31.93/16.19 (37) QDPPairToRuleProof [EQUIVALENT, 0 ms] 31.93/16.19 (38) AND 31.93/16.19 (39) QDP 31.93/16.19 (40) TransformationProof [EQUIVALENT, 1 ms] 31.93/16.19 (41) QDP 31.93/16.19 (42) DependencyGraphProof [EQUIVALENT, 0 ms] 31.93/16.19 (43) QDP 31.93/16.19 (44) UsableRulesProof [EQUIVALENT, 0 ms] 31.93/16.19 (45) QDP 31.93/16.19 (46) QReductionProof [EQUIVALENT, 0 ms] 31.93/16.19 (47) QDP 31.93/16.19 (48) InductionCalculusProof [EQUIVALENT, 0 ms] 31.93/16.19 (49) QDP 31.93/16.19 (50) NonInfProof [EQUIVALENT, 115 ms] 31.93/16.19 (51) QDP 31.93/16.19 (52) DependencyGraphProof [EQUIVALENT, 0 ms] 31.93/16.19 (53) TRUE 31.93/16.19 (54) QDP 31.93/16.19 (55) QDPSizeChangeProof [EQUIVALENT, 0 ms] 31.93/16.19 (56) YES 31.93/16.19 (57) QDP 31.93/16.19 (58) QDPSizeChangeProof [EQUIVALENT, 0 ms] 31.93/16.19 (59) YES 31.93/16.19 (60) QDP 31.93/16.19 (61) QDPSizeChangeProof [EQUIVALENT, 0 ms] 31.93/16.19 (62) YES 31.93/16.19 31.93/16.19 31.93/16.19 ---------------------------------------- 31.93/16.19 31.93/16.19 (0) 31.93/16.19 Obligation: 31.93/16.19 mainModule Main 31.93/16.19 module Main where { 31.93/16.19 import qualified Prelude; 31.93/16.19 data List a = Cons a (List a) | Nil ; 31.93/16.19 31.93/16.19 data MyBool = MyTrue | MyFalse ; 31.93/16.19 31.93/16.19 data MyInt = Pos Main.Nat | Neg Main.Nat ; 31.93/16.19 31.93/16.19 data Main.Nat = Succ Main.Nat | Zero ; 31.93/16.19 31.93/16.19 data Ordering = LT | EQ | GT ; 31.93/16.19 31.93/16.19 data Tup2 a b = Tup2 a b ; 31.93/16.19 31.93/16.19 data Main.WHNF a = WHNF a ; 31.93/16.19 31.93/16.19 compareMyInt :: MyInt -> MyInt -> Ordering; 31.93/16.19 compareMyInt = primCmpInt; 31.93/16.19 31.93/16.19 dsEm :: (b -> a) -> b -> a; 31.93/16.19 dsEm f x = Main.seq x (f x); 31.93/16.19 31.93/16.19 enforceWHNF :: Main.WHNF a -> b -> b; 31.93/16.19 enforceWHNF (Main.WHNF x) y = y; 31.93/16.19 31.93/16.19 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 31.93/16.19 enumFromToMyInt = numericEnumFromTo; 31.93/16.19 31.93/16.19 esEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.19 esEsOrdering LT LT = MyTrue; 31.93/16.19 esEsOrdering LT EQ = MyFalse; 31.93/16.19 esEsOrdering LT GT = MyFalse; 31.93/16.19 esEsOrdering EQ LT = MyFalse; 31.93/16.19 esEsOrdering EQ EQ = MyTrue; 31.93/16.19 esEsOrdering EQ GT = MyFalse; 31.93/16.19 esEsOrdering GT LT = MyFalse; 31.93/16.19 esEsOrdering GT EQ = MyFalse; 31.93/16.19 esEsOrdering GT GT = MyTrue; 31.93/16.19 31.93/16.19 flip :: (b -> c -> a) -> c -> b -> a; 31.93/16.19 flip f x y = f y x; 31.93/16.19 31.93/16.19 fromIntMyInt :: MyInt -> MyInt; 31.93/16.19 fromIntMyInt x = x; 31.93/16.19 31.93/16.19 fsEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.19 fsEsOrdering x y = not (esEsOrdering x y); 31.93/16.19 31.93/16.19 ltEsMyInt :: MyInt -> MyInt -> MyBool; 31.93/16.19 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 31.93/16.19 31.93/16.19 not :: MyBool -> MyBool; 31.93/16.19 not MyTrue = MyFalse; 31.93/16.19 not MyFalse = MyTrue; 31.93/16.19 31.93/16.19 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 31.93/16.19 31.93/16.19 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 31.93/16.19 31.93/16.19 otherwise :: MyBool; 31.93/16.19 otherwise = MyTrue; 31.93/16.19 31.93/16.19 primCmpInt :: MyInt -> MyInt -> Ordering; 31.93/16.19 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.19 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.19 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.19 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 31.93/16.20 31.93/16.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 31.93/16.20 primCmpNat Main.Zero Main.Zero = EQ; 31.93/16.20 primCmpNat Main.Zero (Main.Succ y) = LT; 31.93/16.20 primCmpNat (Main.Succ x) Main.Zero = GT; 31.93/16.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 31.93/16.20 31.93/16.20 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 31.93/16.20 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 31.93/16.20 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 31.93/16.20 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 31.93/16.20 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 31.93/16.20 31.93/16.20 primPlusInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 31.93/16.20 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 31.93/16.20 31.93/16.20 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 31.93/16.20 primPlusNat Main.Zero Main.Zero = Main.Zero; 31.93/16.20 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 31.93/16.20 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 31.93/16.20 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 31.93/16.20 31.93/16.20 psMyInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 psMyInt = primPlusInt; 31.93/16.20 31.93/16.20 rangeMyInt :: Tup2 MyInt MyInt -> List MyInt; 31.93/16.20 rangeMyInt (Tup2 m n) = enumFromToMyInt m n; 31.93/16.20 31.93/16.20 seq :: a -> b -> b; 31.93/16.20 seq x y = Main.enforceWHNF (Main.WHNF x) y; 31.93/16.20 31.93/16.20 takeWhile :: (a -> MyBool) -> List a -> List a; 31.93/16.20 takeWhile p Nil = takeWhile3 p Nil; 31.93/16.20 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 31.93/16.20 31.93/16.20 takeWhile0 p x xs MyTrue = Nil; 31.93/16.20 31.93/16.20 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 31.93/16.20 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 31.93/16.20 31.93/16.20 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 31.93/16.20 31.93/16.20 takeWhile3 p Nil = Nil; 31.93/16.20 takeWhile3 vx vy = takeWhile2 vx vy; 31.93/16.20 31.93/16.20 } 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (1) BR (EQUIVALENT) 31.93/16.20 Replaced joker patterns by fresh variables and removed binding patterns. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (2) 31.93/16.20 Obligation: 31.93/16.20 mainModule Main 31.93/16.20 module Main where { 31.93/16.20 import qualified Prelude; 31.93/16.20 data List a = Cons a (List a) | Nil ; 31.93/16.20 31.93/16.20 data MyBool = MyTrue | MyFalse ; 31.93/16.20 31.93/16.20 data MyInt = Pos Main.Nat | Neg Main.Nat ; 31.93/16.20 31.93/16.20 data Main.Nat = Succ Main.Nat | Zero ; 31.93/16.20 31.93/16.20 data Ordering = LT | EQ | GT ; 31.93/16.20 31.93/16.20 data Tup2 b a = Tup2 b a ; 31.93/16.20 31.93/16.20 data Main.WHNF a = WHNF a ; 31.93/16.20 31.93/16.20 compareMyInt :: MyInt -> MyInt -> Ordering; 31.93/16.20 compareMyInt = primCmpInt; 31.93/16.20 31.93/16.20 dsEm :: (b -> a) -> b -> a; 31.93/16.20 dsEm f x = Main.seq x (f x); 31.93/16.20 31.93/16.20 enforceWHNF :: Main.WHNF b -> a -> a; 31.93/16.20 enforceWHNF (Main.WHNF x) y = y; 31.93/16.20 31.93/16.20 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 31.93/16.20 enumFromToMyInt = numericEnumFromTo; 31.93/16.20 31.93/16.20 esEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.20 esEsOrdering LT LT = MyTrue; 31.93/16.20 esEsOrdering LT EQ = MyFalse; 31.93/16.20 esEsOrdering LT GT = MyFalse; 31.93/16.20 esEsOrdering EQ LT = MyFalse; 31.93/16.20 esEsOrdering EQ EQ = MyTrue; 31.93/16.20 esEsOrdering EQ GT = MyFalse; 31.93/16.20 esEsOrdering GT LT = MyFalse; 31.93/16.20 esEsOrdering GT EQ = MyFalse; 31.93/16.20 esEsOrdering GT GT = MyTrue; 31.93/16.20 31.93/16.20 flip :: (b -> c -> a) -> c -> b -> a; 31.93/16.20 flip f x y = f y x; 31.93/16.20 31.93/16.20 fromIntMyInt :: MyInt -> MyInt; 31.93/16.20 fromIntMyInt x = x; 31.93/16.20 31.93/16.20 fsEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.20 fsEsOrdering x y = not (esEsOrdering x y); 31.93/16.20 31.93/16.20 ltEsMyInt :: MyInt -> MyInt -> MyBool; 31.93/16.20 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 31.93/16.20 31.93/16.20 not :: MyBool -> MyBool; 31.93/16.20 not MyTrue = MyFalse; 31.93/16.20 not MyFalse = MyTrue; 31.93/16.20 31.93/16.20 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 31.93/16.20 31.93/16.20 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 31.93/16.20 31.93/16.20 otherwise :: MyBool; 31.93/16.20 otherwise = MyTrue; 31.93/16.20 31.93/16.20 primCmpInt :: MyInt -> MyInt -> Ordering; 31.93/16.20 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 31.93/16.20 31.93/16.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 31.93/16.20 primCmpNat Main.Zero Main.Zero = EQ; 31.93/16.20 primCmpNat Main.Zero (Main.Succ y) = LT; 31.93/16.20 primCmpNat (Main.Succ x) Main.Zero = GT; 31.93/16.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 31.93/16.20 31.93/16.20 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 31.93/16.20 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 31.93/16.20 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 31.93/16.20 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 31.93/16.20 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 31.93/16.20 31.93/16.20 primPlusInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 31.93/16.20 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 31.93/16.20 31.93/16.20 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 31.93/16.20 primPlusNat Main.Zero Main.Zero = Main.Zero; 31.93/16.20 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 31.93/16.20 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 31.93/16.20 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 31.93/16.20 31.93/16.20 psMyInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 psMyInt = primPlusInt; 31.93/16.20 31.93/16.20 rangeMyInt :: Tup2 MyInt MyInt -> List MyInt; 31.93/16.20 rangeMyInt (Tup2 m n) = enumFromToMyInt m n; 31.93/16.20 31.93/16.20 seq :: a -> b -> b; 31.93/16.20 seq x y = Main.enforceWHNF (Main.WHNF x) y; 31.93/16.20 31.93/16.20 takeWhile :: (a -> MyBool) -> List a -> List a; 31.93/16.20 takeWhile p Nil = takeWhile3 p Nil; 31.93/16.20 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 31.93/16.20 31.93/16.20 takeWhile0 p x xs MyTrue = Nil; 31.93/16.20 31.93/16.20 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 31.93/16.20 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 31.93/16.20 31.93/16.20 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 31.93/16.20 31.93/16.20 takeWhile3 p Nil = Nil; 31.93/16.20 takeWhile3 vx vy = takeWhile2 vx vy; 31.93/16.20 31.93/16.20 } 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (3) COR (EQUIVALENT) 31.93/16.20 Cond Reductions: 31.93/16.20 The following Function with conditions 31.93/16.20 "undefined |Falseundefined; 31.93/16.20 " 31.93/16.20 is transformed to 31.93/16.20 "undefined = undefined1; 31.93/16.20 " 31.93/16.20 "undefined0 True = undefined; 31.93/16.20 " 31.93/16.20 "undefined1 = undefined0 False; 31.93/16.20 " 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (4) 31.93/16.20 Obligation: 31.93/16.20 mainModule Main 31.93/16.20 module Main where { 31.93/16.20 import qualified Prelude; 31.93/16.20 data List a = Cons a (List a) | Nil ; 31.93/16.20 31.93/16.20 data MyBool = MyTrue | MyFalse ; 31.93/16.20 31.93/16.20 data MyInt = Pos Main.Nat | Neg Main.Nat ; 31.93/16.20 31.93/16.20 data Main.Nat = Succ Main.Nat | Zero ; 31.93/16.20 31.93/16.20 data Ordering = LT | EQ | GT ; 31.93/16.20 31.93/16.20 data Tup2 b a = Tup2 b a ; 31.93/16.20 31.93/16.20 data Main.WHNF a = WHNF a ; 31.93/16.20 31.93/16.20 compareMyInt :: MyInt -> MyInt -> Ordering; 31.93/16.20 compareMyInt = primCmpInt; 31.93/16.20 31.93/16.20 dsEm :: (a -> b) -> a -> b; 31.93/16.20 dsEm f x = Main.seq x (f x); 31.93/16.20 31.93/16.20 enforceWHNF :: Main.WHNF b -> a -> a; 31.93/16.20 enforceWHNF (Main.WHNF x) y = y; 31.93/16.20 31.93/16.20 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 31.93/16.20 enumFromToMyInt = numericEnumFromTo; 31.93/16.20 31.93/16.20 esEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.20 esEsOrdering LT LT = MyTrue; 31.93/16.20 esEsOrdering LT EQ = MyFalse; 31.93/16.20 esEsOrdering LT GT = MyFalse; 31.93/16.20 esEsOrdering EQ LT = MyFalse; 31.93/16.20 esEsOrdering EQ EQ = MyTrue; 31.93/16.20 esEsOrdering EQ GT = MyFalse; 31.93/16.20 esEsOrdering GT LT = MyFalse; 31.93/16.20 esEsOrdering GT EQ = MyFalse; 31.93/16.20 esEsOrdering GT GT = MyTrue; 31.93/16.20 31.93/16.20 flip :: (b -> a -> c) -> a -> b -> c; 31.93/16.20 flip f x y = f y x; 31.93/16.20 31.93/16.20 fromIntMyInt :: MyInt -> MyInt; 31.93/16.20 fromIntMyInt x = x; 31.93/16.20 31.93/16.20 fsEsOrdering :: Ordering -> Ordering -> MyBool; 31.93/16.20 fsEsOrdering x y = not (esEsOrdering x y); 31.93/16.20 31.93/16.20 ltEsMyInt :: MyInt -> MyInt -> MyBool; 31.93/16.20 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 31.93/16.20 31.93/16.20 not :: MyBool -> MyBool; 31.93/16.20 not MyTrue = MyFalse; 31.93/16.20 not MyFalse = MyTrue; 31.93/16.20 31.93/16.20 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 31.93/16.20 31.93/16.20 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 31.93/16.20 31.93/16.20 otherwise :: MyBool; 31.93/16.20 otherwise = MyTrue; 31.93/16.20 31.93/16.20 primCmpInt :: MyInt -> MyInt -> Ordering; 31.93/16.20 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 31.93/16.20 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 31.93/16.20 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 31.93/16.20 31.93/16.20 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 31.93/16.20 primCmpNat Main.Zero Main.Zero = EQ; 31.93/16.20 primCmpNat Main.Zero (Main.Succ y) = LT; 31.93/16.20 primCmpNat (Main.Succ x) Main.Zero = GT; 31.93/16.20 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 31.93/16.20 31.93/16.20 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 31.93/16.20 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 31.93/16.20 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 31.93/16.20 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 31.93/16.20 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 31.93/16.20 31.93/16.20 primPlusInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 31.93/16.20 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 31.93/16.20 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 31.93/16.20 31.93/16.20 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 31.93/16.20 primPlusNat Main.Zero Main.Zero = Main.Zero; 31.93/16.20 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 31.93/16.20 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 31.93/16.20 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 31.93/16.20 31.93/16.20 psMyInt :: MyInt -> MyInt -> MyInt; 31.93/16.20 psMyInt = primPlusInt; 31.93/16.20 31.93/16.20 rangeMyInt :: Tup2 MyInt MyInt -> List MyInt; 31.93/16.20 rangeMyInt (Tup2 m n) = enumFromToMyInt m n; 31.93/16.20 31.93/16.20 seq :: a -> b -> b; 31.93/16.20 seq x y = Main.enforceWHNF (Main.WHNF x) y; 31.93/16.20 31.93/16.20 takeWhile :: (a -> MyBool) -> List a -> List a; 31.93/16.20 takeWhile p Nil = takeWhile3 p Nil; 31.93/16.20 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 31.93/16.20 31.93/16.20 takeWhile0 p x xs MyTrue = Nil; 31.93/16.20 31.93/16.20 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 31.93/16.20 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 31.93/16.20 31.93/16.20 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 31.93/16.20 31.93/16.20 takeWhile3 p Nil = Nil; 31.93/16.20 takeWhile3 vx vy = takeWhile2 vx vy; 31.93/16.20 31.93/16.20 } 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (5) Narrow (SOUND) 31.93/16.20 Haskell To QDPs 31.93/16.20 31.93/16.20 digraph dp_graph { 31.93/16.20 node [outthreshold=100, inthreshold=100];1[label="rangeMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 31.93/16.20 3[label="rangeMyInt vz3",fontsize=16,color="burlywood",shape="triangle"];811[label="vz3/Tup2 vz30 vz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 811[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 811 -> 4[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 4[label="rangeMyInt (Tup2 vz30 vz31)",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 31.93/16.20 5[label="enumFromToMyInt vz30 vz31",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 31.93/16.20 6[label="numericEnumFromTo vz30 vz31",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 31.93/16.20 7[label="takeWhile (flip ltEsMyInt vz31) (numericEnumFrom vz30)",fontsize=16,color="black",shape="triangle"];7 -> 8[label="",style="solid", color="black", weight=3]; 31.93/16.20 8[label="takeWhile (flip ltEsMyInt vz31) (Cons vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 31.93/16.20 9[label="takeWhile2 (flip ltEsMyInt vz31) (Cons vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 31.93/16.20 10[label="takeWhile1 (flip ltEsMyInt vz31) vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))) (flip ltEsMyInt vz31 vz30)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 31.93/16.20 11[label="takeWhile1 (flip ltEsMyInt vz31) vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))) (ltEsMyInt vz30 vz31)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 31.93/16.20 12[label="takeWhile1 (flip ltEsMyInt vz31) vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))) (fsEsOrdering (compareMyInt vz30 vz31) GT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 31.93/16.20 13[label="takeWhile1 (flip ltEsMyInt vz31) vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (compareMyInt vz30 vz31) GT))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 31.93/16.20 14[label="takeWhile1 (flip ltEsMyInt vz31) vz30 (dsEm numericEnumFrom (psMyInt vz30 (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt vz30 vz31) GT))",fontsize=16,color="burlywood",shape="box"];812[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 812[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 812 -> 15[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 813[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 813[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 813 -> 16[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 15[label="takeWhile1 (flip ltEsMyInt vz31) (Pos vz300) (dsEm numericEnumFrom (psMyInt (Pos vz300) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos vz300) vz31) GT))",fontsize=16,color="burlywood",shape="box"];814[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];15 -> 814[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 814 -> 17[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 815[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 815[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 815 -> 18[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 16[label="takeWhile1 (flip ltEsMyInt vz31) (Neg vz300) (dsEm numericEnumFrom (psMyInt (Neg vz300) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg vz300) vz31) GT))",fontsize=16,color="burlywood",shape="box"];816[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];16 -> 816[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 816 -> 19[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 817[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 817[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 817 -> 20[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 17[label="takeWhile1 (flip ltEsMyInt vz31) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz3000)) vz31) GT))",fontsize=16,color="burlywood",shape="box"];818[label="vz31/Pos vz310",fontsize=10,color="white",style="solid",shape="box"];17 -> 818[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 818 -> 21[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 819[label="vz31/Neg vz310",fontsize=10,color="white",style="solid",shape="box"];17 -> 819[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 819 -> 22[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 18[label="takeWhile1 (flip ltEsMyInt vz31) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) vz31) GT))",fontsize=16,color="burlywood",shape="box"];820[label="vz31/Pos vz310",fontsize=10,color="white",style="solid",shape="box"];18 -> 820[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 820 -> 23[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 821[label="vz31/Neg vz310",fontsize=10,color="white",style="solid",shape="box"];18 -> 821[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 821 -> 24[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 19[label="takeWhile1 (flip ltEsMyInt vz31) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz3000)) vz31) GT))",fontsize=16,color="burlywood",shape="box"];822[label="vz31/Pos vz310",fontsize=10,color="white",style="solid",shape="box"];19 -> 822[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 822 -> 25[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 823[label="vz31/Neg vz310",fontsize=10,color="white",style="solid",shape="box"];19 -> 823[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 823 -> 26[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 20[label="takeWhile1 (flip ltEsMyInt vz31) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) vz31) GT))",fontsize=16,color="burlywood",shape="box"];824[label="vz31/Pos vz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 824[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 824 -> 27[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 825[label="vz31/Neg vz310",fontsize=10,color="white",style="solid",shape="box"];20 -> 825[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 825 -> 28[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 21[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz3000)) (Pos vz310)) GT))",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3]; 31.93/16.20 22[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz3000)) (Neg vz310)) GT))",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3]; 31.93/16.20 23[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos vz310)) GT))",fontsize=16,color="burlywood",shape="box"];826[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];23 -> 826[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 826 -> 31[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 827[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 827[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 827 -> 32[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 24[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Neg vz310)) GT))",fontsize=16,color="burlywood",shape="box"];828[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];24 -> 828[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 828 -> 33[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 829[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 829[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 829 -> 34[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 25[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz3000)) (Pos vz310)) GT))",fontsize=16,color="black",shape="box"];25 -> 35[label="",style="solid", color="black", weight=3]; 31.93/16.20 26[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz3000)) (Neg vz310)) GT))",fontsize=16,color="black",shape="box"];26 -> 36[label="",style="solid", color="black", weight=3]; 31.93/16.20 27[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos vz310)) GT))",fontsize=16,color="burlywood",shape="box"];830[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];27 -> 830[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 830 -> 37[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 831[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 831[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 831 -> 38[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 28[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Neg vz310)) GT))",fontsize=16,color="burlywood",shape="box"];832[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];28 -> 832[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 832 -> 39[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 833[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 833[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 833 -> 40[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 29[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz3000) vz310) GT))",fontsize=16,color="burlywood",shape="box"];834[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];29 -> 834[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 834 -> 41[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 835[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 835[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 835 -> 42[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 30[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3]; 31.93/16.20 31[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vz3100))) GT))",fontsize=16,color="black",shape="box"];31 -> 44[label="",style="solid", color="black", weight=3]; 31.93/16.20 32[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 31.93/16.20 33[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vz3100))) GT))",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3]; 31.93/16.20 34[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Neg Zero)) GT))",fontsize=16,color="black",shape="box"];34 -> 47[label="",style="solid", color="black", weight=3]; 31.93/16.20 35[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];35 -> 48[label="",style="solid", color="black", weight=3]; 31.93/16.20 36[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz310 (Succ vz3000)) GT))",fontsize=16,color="burlywood",shape="box"];836[label="vz310/Succ vz3100",fontsize=10,color="white",style="solid",shape="box"];36 -> 836[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 836 -> 49[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 837[label="vz310/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 837[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 837 -> 50[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 37[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vz3100))) GT))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3]; 31.93/16.20 38[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 31.93/16.20 39[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vz3100))) GT))",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3]; 31.93/16.20 40[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Neg Zero)) GT))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3]; 31.93/16.20 41[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz3000) (Succ vz3100)) GT))",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3]; 31.93/16.20 42[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz3000) Zero) GT))",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3]; 31.93/16.20 43[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3]; 31.93/16.20 44[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz3100)) GT))",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3]; 31.93/16.20 45[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3]; 31.93/16.20 46[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3]; 31.93/16.20 47[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3]; 31.93/16.20 48[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3]; 31.93/16.20 49[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz3100) (Succ vz3000)) GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3]; 31.93/16.20 50[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz3000)) GT))",fontsize=16,color="black",shape="box"];50 -> 64[label="",style="solid", color="black", weight=3]; 31.93/16.20 51[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];51 -> 65[label="",style="solid", color="black", weight=3]; 31.93/16.20 52[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];52 -> 66[label="",style="solid", color="black", weight=3]; 31.93/16.20 53[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz3100) Zero) GT))",fontsize=16,color="black",shape="box"];53 -> 67[label="",style="solid", color="black", weight=3]; 31.93/16.20 54[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];54 -> 68[label="",style="solid", color="black", weight=3]; 31.93/16.20 55 -> 447[label="",style="dashed", color="red", weight=0]; 31.93/16.20 55[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz3000 vz3100) GT))",fontsize=16,color="magenta"];55 -> 448[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 55 -> 449[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 55 -> 450[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 55 -> 451[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 56[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];56 -> 71[label="",style="solid", color="black", weight=3]; 31.93/16.20 57[label="takeWhile1 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];57 -> 72[label="",style="solid", color="black", weight=3]; 31.93/16.20 58[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];58 -> 73[label="",style="solid", color="black", weight=3]; 31.93/16.20 59[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];59 -> 74[label="",style="solid", color="black", weight=3]; 31.93/16.20 60[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];60 -> 75[label="",style="solid", color="black", weight=3]; 31.93/16.20 61[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];61 -> 76[label="",style="solid", color="black", weight=3]; 31.93/16.20 62[label="takeWhile1 (flip ltEsMyInt (Pos vz310)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];62 -> 77[label="",style="solid", color="black", weight=3]; 31.93/16.20 63 -> 723[label="",style="dashed", color="red", weight=0]; 31.93/16.20 63[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz3100 vz3000) GT))",fontsize=16,color="magenta"];63 -> 724[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 63 -> 725[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 63 -> 726[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 63 -> 727[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 63 -> 728[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 64[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];64 -> 80[label="",style="solid", color="black", weight=3]; 31.93/16.20 65[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];65 -> 81[label="",style="solid", color="black", weight=3]; 31.93/16.20 66[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];66 -> 82[label="",style="solid", color="black", weight=3]; 31.93/16.20 67[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];67 -> 83[label="",style="solid", color="black", weight=3]; 31.93/16.20 68[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];68 -> 84[label="",style="solid", color="black", weight=3]; 31.93/16.20 448[label="vz3100",fontsize=16,color="green",shape="box"];449[label="vz3000",fontsize=16,color="green",shape="box"];450[label="vz3100",fontsize=16,color="green",shape="box"];451[label="vz3000",fontsize=16,color="green",shape="box"];447[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz23 vz24) GT))",fontsize=16,color="burlywood",shape="triangle"];838[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];447 -> 838[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 838 -> 476[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 839[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];447 -> 839[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 839 -> 477[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 71[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];71 -> 89[label="",style="solid", color="black", weight=3]; 31.93/16.20 72[label="takeWhile0 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];72 -> 90[label="",style="solid", color="black", weight=3]; 31.93/16.20 73[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];73 -> 91[label="",style="solid", color="black", weight=3]; 31.93/16.20 74[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];74 -> 92[label="",style="solid", color="black", weight=3]; 31.93/16.20 75[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];75 -> 93[label="",style="solid", color="black", weight=3]; 31.93/16.20 76[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];76 -> 94[label="",style="solid", color="black", weight=3]; 31.93/16.20 77[label="Cons (Neg (Succ vz3000)) (takeWhile (flip ltEsMyInt (Pos vz310)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];77 -> 95[label="",style="dashed", color="green", weight=3]; 31.93/16.20 724[label="vz3100",fontsize=16,color="green",shape="box"];725[label="vz3000",fontsize=16,color="green",shape="box"];726 -> 180[label="",style="dashed", color="red", weight=0]; 31.93/16.20 726[label="psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];727[label="vz3100",fontsize=16,color="green",shape="box"];728[label="vz3000",fontsize=16,color="green",shape="box"];723[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat vz67 vz68) GT))",fontsize=16,color="burlywood",shape="triangle"];840[label="vz67/Succ vz670",fontsize=10,color="white",style="solid",shape="box"];723 -> 840[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 840 -> 779[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 841[label="vz67/Zero",fontsize=10,color="white",style="solid",shape="box"];723 -> 841[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 841 -> 780[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 80[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];80 -> 100[label="",style="solid", color="black", weight=3]; 31.93/16.20 81[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];81 -> 101[label="",style="solid", color="black", weight=3]; 31.93/16.20 82[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];82 -> 102[label="",style="solid", color="black", weight=3]; 31.93/16.20 83[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];83 -> 103[label="",style="solid", color="black", weight=3]; 31.93/16.20 84[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];84 -> 104[label="",style="solid", color="black", weight=3]; 31.93/16.20 476[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz230) vz24) GT))",fontsize=16,color="burlywood",shape="box"];842[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];476 -> 842[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 842 -> 483[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 843[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];476 -> 843[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 843 -> 484[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 477[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero vz24) GT))",fontsize=16,color="burlywood",shape="box"];844[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];477 -> 844[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 844 -> 485[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 845[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];477 -> 845[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 845 -> 486[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 89[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];89 -> 109[label="",style="solid", color="black", weight=3]; 31.93/16.20 90[label="takeWhile0 (flip ltEsMyInt (Neg vz310)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];90 -> 110[label="",style="solid", color="black", weight=3]; 31.93/16.20 91[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];91 -> 111[label="",style="solid", color="black", weight=3]; 31.93/16.20 92[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];92 -> 112[label="",style="dashed", color="green", weight=3]; 31.93/16.20 93[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];93 -> 113[label="",style="solid", color="black", weight=3]; 31.93/16.20 94[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];94 -> 114[label="",style="dashed", color="green", weight=3]; 31.93/16.20 95[label="takeWhile (flip ltEsMyInt (Pos vz310)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];95 -> 115[label="",style="solid", color="black", weight=3]; 31.93/16.20 180[label="psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];180 -> 206[label="",style="solid", color="black", weight=3]; 31.93/16.20 779[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat (Succ vz670) vz68) GT))",fontsize=16,color="burlywood",shape="box"];846[label="vz68/Succ vz680",fontsize=10,color="white",style="solid",shape="box"];779 -> 846[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 846 -> 782[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 847[label="vz68/Zero",fontsize=10,color="white",style="solid",shape="box"];779 -> 847[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 847 -> 783[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 780[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat Zero vz68) GT))",fontsize=16,color="burlywood",shape="box"];848[label="vz68/Succ vz680",fontsize=10,color="white",style="solid",shape="box"];780 -> 848[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 848 -> 784[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 849[label="vz68/Zero",fontsize=10,color="white",style="solid",shape="box"];780 -> 849[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 849 -> 785[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 100[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];100 -> 120[label="",style="solid", color="black", weight=3]; 31.93/16.20 101[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];101 -> 121[label="",style="dashed", color="green", weight=3]; 31.93/16.20 102[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];102 -> 122[label="",style="dashed", color="green", weight=3]; 31.93/16.20 103[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];103 -> 123[label="",style="solid", color="black", weight=3]; 31.93/16.20 104[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];104 -> 124[label="",style="dashed", color="green", weight=3]; 31.93/16.20 483[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz230) (Succ vz240)) GT))",fontsize=16,color="black",shape="box"];483 -> 490[label="",style="solid", color="black", weight=3]; 31.93/16.20 484[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz230) Zero) GT))",fontsize=16,color="black",shape="box"];484 -> 491[label="",style="solid", color="black", weight=3]; 31.93/16.20 485[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz240)) GT))",fontsize=16,color="black",shape="box"];485 -> 492[label="",style="solid", color="black", weight=3]; 31.93/16.20 486[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero Zero) GT))",fontsize=16,color="black",shape="box"];486 -> 493[label="",style="solid", color="black", weight=3]; 31.93/16.20 109[label="takeWhile0 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];109 -> 130[label="",style="solid", color="black", weight=3]; 31.93/16.20 110[label="Nil",fontsize=16,color="green",shape="box"];111[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];111 -> 131[label="",style="dashed", color="green", weight=3]; 31.93/16.20 112[label="takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];112 -> 132[label="",style="solid", color="black", weight=3]; 31.93/16.20 113[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz3100))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];113 -> 133[label="",style="solid", color="black", weight=3]; 31.93/16.20 114[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];114 -> 134[label="",style="solid", color="black", weight=3]; 31.93/16.20 115[label="takeWhile (flip ltEsMyInt (Pos vz310)) (seq (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];115 -> 135[label="",style="solid", color="black", weight=3]; 31.93/16.20 206[label="primPlusInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];206 -> 245[label="",style="solid", color="black", weight=3]; 31.93/16.20 782[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat (Succ vz670) (Succ vz680)) GT))",fontsize=16,color="black",shape="box"];782 -> 788[label="",style="solid", color="black", weight=3]; 31.93/16.20 783[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat (Succ vz670) Zero) GT))",fontsize=16,color="black",shape="box"];783 -> 789[label="",style="solid", color="black", weight=3]; 31.93/16.20 784[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat Zero (Succ vz680)) GT))",fontsize=16,color="black",shape="box"];784 -> 790[label="",style="solid", color="black", weight=3]; 31.93/16.20 785[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat Zero Zero) GT))",fontsize=16,color="black",shape="box"];785 -> 791[label="",style="solid", color="black", weight=3]; 31.93/16.20 120[label="Cons (Neg (Succ vz3000)) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];120 -> 141[label="",style="dashed", color="green", weight=3]; 31.93/16.20 121[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];121 -> 142[label="",style="solid", color="black", weight=3]; 31.93/16.20 122[label="takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];122 -> 143[label="",style="solid", color="black", weight=3]; 31.93/16.20 123[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];123 -> 144[label="",style="solid", color="black", weight=3]; 31.93/16.20 124[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];124 -> 145[label="",style="solid", color="black", weight=3]; 31.93/16.20 490 -> 447[label="",style="dashed", color="red", weight=0]; 31.93/16.20 490[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz230 vz240) GT))",fontsize=16,color="magenta"];490 -> 505[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 490 -> 506[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 491[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];491 -> 507[label="",style="solid", color="black", weight=3]; 31.93/16.20 492[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];492 -> 508[label="",style="solid", color="black", weight=3]; 31.93/16.20 493[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];493 -> 509[label="",style="solid", color="black", weight=3]; 31.93/16.20 130[label="takeWhile0 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz3000)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];130 -> 153[label="",style="solid", color="black", weight=3]; 31.93/16.20 131[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];131 -> 154[label="",style="solid", color="black", weight=3]; 31.93/16.20 132[label="takeWhile (flip ltEsMyInt (Pos Zero)) (seq (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];132 -> 155[label="",style="solid", color="black", weight=3]; 31.93/16.20 133[label="Nil",fontsize=16,color="green",shape="box"];134[label="takeWhile (flip ltEsMyInt (Neg Zero)) (seq (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];134 -> 156[label="",style="solid", color="black", weight=3]; 31.93/16.20 135[label="takeWhile (flip ltEsMyInt (Pos vz310)) (enforceWHNF (WHNF (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];135 -> 157[label="",style="solid", color="black", weight=3]; 31.93/16.20 245[label="primPlusInt (Neg (Succ vz3000)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];245 -> 273[label="",style="solid", color="black", weight=3]; 31.93/16.20 788 -> 723[label="",style="dashed", color="red", weight=0]; 31.93/16.20 788[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering (primCmpNat vz670 vz680) GT))",fontsize=16,color="magenta"];788 -> 794[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 788 -> 795[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 789[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];789 -> 796[label="",style="solid", color="black", weight=3]; 31.93/16.20 790[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];790 -> 797[label="",style="solid", color="black", weight=3]; 31.93/16.20 791[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];791 -> 798[label="",style="solid", color="black", weight=3]; 31.93/16.20 141[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];141 -> 165[label="",style="solid", color="black", weight=3]; 31.93/16.20 142[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (seq (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];142 -> 166[label="",style="solid", color="black", weight=3]; 31.93/16.20 143[label="takeWhile (flip ltEsMyInt (Pos Zero)) (seq (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];143 -> 167[label="",style="solid", color="black", weight=3]; 31.93/16.20 144[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz3100))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];144 -> 168[label="",style="solid", color="black", weight=3]; 31.93/16.20 145[label="takeWhile (flip ltEsMyInt (Neg Zero)) (seq (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];145 -> 169[label="",style="solid", color="black", weight=3]; 31.93/16.20 505[label="vz240",fontsize=16,color="green",shape="box"];506[label="vz230",fontsize=16,color="green",shape="box"];507[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];507 -> 521[label="",style="solid", color="black", weight=3]; 31.93/16.20 508[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="triangle"];508 -> 522[label="",style="solid", color="black", weight=3]; 31.93/16.20 509 -> 508[label="",style="dashed", color="red", weight=0]; 31.93/16.20 509[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="magenta"];153[label="Nil",fontsize=16,color="green",shape="box"];154[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (seq (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];154 -> 177[label="",style="solid", color="black", weight=3]; 31.93/16.20 155[label="takeWhile (flip ltEsMyInt (Pos Zero)) (enforceWHNF (WHNF (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];155 -> 178[label="",style="solid", color="black", weight=3]; 31.93/16.20 156 -> 219[label="",style="dashed", color="red", weight=0]; 31.93/16.20 156[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];156 -> 220[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 156 -> 221[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 157 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 157[label="takeWhile (flip ltEsMyInt (Pos vz310)) (numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];157 -> 180[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 157 -> 181[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 273[label="primMinusNat (Succ Zero) (Succ vz3000)",fontsize=16,color="black",shape="box"];273 -> 297[label="",style="solid", color="black", weight=3]; 31.93/16.20 794[label="vz680",fontsize=16,color="green",shape="box"];795[label="vz670",fontsize=16,color="green",shape="box"];796[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not MyTrue)",fontsize=16,color="black",shape="box"];796 -> 799[label="",style="solid", color="black", weight=3]; 31.93/16.20 797[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not MyFalse)",fontsize=16,color="black",shape="triangle"];797 -> 800[label="",style="solid", color="black", weight=3]; 31.93/16.20 798 -> 797[label="",style="dashed", color="red", weight=0]; 31.93/16.20 798[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) (not MyFalse)",fontsize=16,color="magenta"];165[label="takeWhile (flip ltEsMyInt (Neg Zero)) (seq (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];165 -> 189[label="",style="solid", color="black", weight=3]; 31.93/16.20 166[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (enforceWHNF (WHNF (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];166 -> 190[label="",style="solid", color="black", weight=3]; 31.93/16.20 167[label="takeWhile (flip ltEsMyInt (Pos Zero)) (enforceWHNF (WHNF (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];167 -> 191[label="",style="solid", color="black", weight=3]; 31.93/16.20 168[label="Nil",fontsize=16,color="green",shape="box"];169 -> 219[label="",style="dashed", color="red", weight=0]; 31.93/16.20 169[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];169 -> 222[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 169 -> 223[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 521[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];521 -> 530[label="",style="solid", color="black", weight=3]; 31.93/16.20 522[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];522 -> 531[label="",style="solid", color="black", weight=3]; 31.93/16.20 177[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (enforceWHNF (WHNF (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];177 -> 201[label="",style="solid", color="black", weight=3]; 31.93/16.20 178 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 178[label="takeWhile (flip ltEsMyInt (Pos Zero)) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];178 -> 202[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 178 -> 203[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 220[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];220 -> 226[label="",style="solid", color="black", weight=3]; 31.93/16.20 221 -> 220[label="",style="dashed", color="red", weight=0]; 31.93/16.20 221[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];219[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF vz10) (numericEnumFrom vz9))",fontsize=16,color="black",shape="triangle"];219 -> 227[label="",style="solid", color="black", weight=3]; 31.93/16.20 181[label="Pos vz310",fontsize=16,color="green",shape="box"];297[label="primMinusNat Zero vz3000",fontsize=16,color="burlywood",shape="box"];850[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];297 -> 850[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 850 -> 322[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 851[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];297 -> 851[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 851 -> 323[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 799[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) MyFalse",fontsize=16,color="black",shape="box"];799 -> 801[label="",style="solid", color="black", weight=3]; 31.93/16.20 800[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) MyTrue",fontsize=16,color="black",shape="box"];800 -> 802[label="",style="solid", color="black", weight=3]; 31.93/16.20 189 -> 219[label="",style="dashed", color="red", weight=0]; 31.93/16.20 189[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];189 -> 224[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 189 -> 225[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 190 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 190[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];190 -> 228[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 190 -> 229[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 191 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 191[label="takeWhile (flip ltEsMyInt (Pos Zero)) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];191 -> 230[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 191 -> 231[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 222[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];222 -> 232[label="",style="solid", color="black", weight=3]; 31.93/16.20 223 -> 222[label="",style="dashed", color="red", weight=0]; 31.93/16.20 223[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];530[label="takeWhile0 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];530 -> 536[label="",style="solid", color="black", weight=3]; 31.93/16.20 531[label="Cons (Pos (Succ vz22)) (takeWhile (flip ltEsMyInt (Pos (Succ vz21))) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];531 -> 537[label="",style="dashed", color="green", weight=3]; 31.93/16.20 201 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 201[label="takeWhile (flip ltEsMyInt (Pos (Succ vz3100))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];201 -> 243[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 201 -> 244[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 202 -> 220[label="",style="dashed", color="red", weight=0]; 31.93/16.20 202[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];203[label="Pos Zero",fontsize=16,color="green",shape="box"];226[label="primPlusInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];226 -> 260[label="",style="solid", color="black", weight=3]; 31.93/16.20 227 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 227[label="takeWhile (flip ltEsMyInt (Neg Zero)) (numericEnumFrom vz9)",fontsize=16,color="magenta"];227 -> 261[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 227 -> 262[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 322[label="primMinusNat Zero (Succ vz30000)",fontsize=16,color="black",shape="box"];322 -> 344[label="",style="solid", color="black", weight=3]; 31.93/16.20 323[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];323 -> 345[label="",style="solid", color="black", weight=3]; 31.93/16.20 801[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) otherwise",fontsize=16,color="black",shape="box"];801 -> 803[label="",style="solid", color="black", weight=3]; 31.93/16.20 802[label="Cons (Neg (Succ vz65)) (takeWhile (flip ltEsMyInt (Neg (Succ vz64))) (dsEm numericEnumFrom vz66))",fontsize=16,color="green",shape="box"];802 -> 804[label="",style="dashed", color="green", weight=3]; 31.93/16.20 224 -> 180[label="",style="dashed", color="red", weight=0]; 31.93/16.20 224[label="psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];225 -> 180[label="",style="dashed", color="red", weight=0]; 31.93/16.20 225[label="psMyInt (Neg (Succ vz3000)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];228 -> 222[label="",style="dashed", color="red", weight=0]; 31.93/16.20 228[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];229[label="Pos (Succ vz3100)",fontsize=16,color="green",shape="box"];230 -> 222[label="",style="dashed", color="red", weight=0]; 31.93/16.20 230[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];231[label="Pos Zero",fontsize=16,color="green",shape="box"];232[label="primPlusInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];232 -> 263[label="",style="solid", color="black", weight=3]; 31.93/16.20 536[label="takeWhile0 (flip ltEsMyInt (Pos (Succ vz21))) (Pos (Succ vz22)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];536 -> 542[label="",style="solid", color="black", weight=3]; 31.93/16.20 537[label="takeWhile (flip ltEsMyInt (Pos (Succ vz21))) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];537 -> 543[label="",style="solid", color="black", weight=3]; 31.93/16.20 243 -> 220[label="",style="dashed", color="red", weight=0]; 31.93/16.20 243[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];244[label="Pos (Succ vz3100)",fontsize=16,color="green",shape="box"];260[label="primPlusInt (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];260 -> 285[label="",style="solid", color="black", weight=3]; 31.93/16.20 261[label="vz9",fontsize=16,color="green",shape="box"];262[label="Neg Zero",fontsize=16,color="green",shape="box"];344[label="Neg (Succ vz30000)",fontsize=16,color="green",shape="box"];345[label="Pos Zero",fontsize=16,color="green",shape="box"];803[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz64))) (Neg (Succ vz65)) (dsEm numericEnumFrom vz66) MyTrue",fontsize=16,color="black",shape="box"];803 -> 805[label="",style="solid", color="black", weight=3]; 31.93/16.20 804[label="takeWhile (flip ltEsMyInt (Neg (Succ vz64))) (dsEm numericEnumFrom vz66)",fontsize=16,color="black",shape="box"];804 -> 806[label="",style="solid", color="black", weight=3]; 31.93/16.20 263[label="primPlusInt (Neg Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];263 -> 286[label="",style="solid", color="black", weight=3]; 31.93/16.20 542[label="Nil",fontsize=16,color="green",shape="box"];543[label="takeWhile (flip ltEsMyInt (Pos (Succ vz21))) (seq (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];543 -> 547[label="",style="solid", color="black", weight=3]; 31.93/16.20 285[label="Pos (primPlusNat Zero (Succ Zero))",fontsize=16,color="green",shape="box"];285 -> 306[label="",style="dashed", color="green", weight=3]; 31.93/16.20 805[label="Nil",fontsize=16,color="green",shape="box"];806[label="takeWhile (flip ltEsMyInt (Neg (Succ vz64))) (seq vz66 (numericEnumFrom vz66))",fontsize=16,color="black",shape="box"];806 -> 807[label="",style="solid", color="black", weight=3]; 31.93/16.20 286[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];286 -> 307[label="",style="solid", color="black", weight=3]; 31.93/16.20 547[label="takeWhile (flip ltEsMyInt (Pos (Succ vz21))) (enforceWHNF (WHNF (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];547 -> 583[label="",style="solid", color="black", weight=3]; 31.93/16.20 306[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];306 -> 333[label="",style="solid", color="black", weight=3]; 31.93/16.20 807[label="takeWhile (flip ltEsMyInt (Neg (Succ vz64))) (enforceWHNF (WHNF vz66) (numericEnumFrom vz66))",fontsize=16,color="black",shape="box"];807 -> 808[label="",style="solid", color="black", weight=3]; 31.93/16.20 307[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];583 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 583[label="takeWhile (flip ltEsMyInt (Pos (Succ vz21))) (numericEnumFrom (psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];583 -> 619[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 583 -> 620[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 333[label="Succ Zero",fontsize=16,color="green",shape="box"];808 -> 7[label="",style="dashed", color="red", weight=0]; 31.93/16.20 808[label="takeWhile (flip ltEsMyInt (Neg (Succ vz64))) (numericEnumFrom vz66)",fontsize=16,color="magenta"];808 -> 809[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 808 -> 810[label="",style="dashed", color="magenta", weight=3]; 31.93/16.20 619[label="psMyInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];619 -> 650[label="",style="solid", color="black", weight=3]; 31.93/16.20 620[label="Pos (Succ vz21)",fontsize=16,color="green",shape="box"];809[label="vz66",fontsize=16,color="green",shape="box"];810[label="Neg (Succ vz64)",fontsize=16,color="green",shape="box"];650[label="primPlusInt (Pos (Succ vz22)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];650 -> 663[label="",style="solid", color="black", weight=3]; 31.93/16.20 663[label="primPlusInt (Pos (Succ vz22)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];663 -> 701[label="",style="solid", color="black", weight=3]; 31.93/16.20 701[label="Pos (primPlusNat (Succ vz22) (Succ Zero))",fontsize=16,color="green",shape="box"];701 -> 713[label="",style="dashed", color="green", weight=3]; 31.93/16.20 713[label="primPlusNat (Succ vz22) (Succ Zero)",fontsize=16,color="black",shape="box"];713 -> 722[label="",style="solid", color="black", weight=3]; 31.93/16.20 722[label="Succ (Succ (primPlusNat vz22 Zero))",fontsize=16,color="green",shape="box"];722 -> 781[label="",style="dashed", color="green", weight=3]; 31.93/16.20 781[label="primPlusNat vz22 Zero",fontsize=16,color="burlywood",shape="box"];852[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];781 -> 852[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 852 -> 786[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 853[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];781 -> 853[label="",style="solid", color="burlywood", weight=9]; 31.93/16.20 853 -> 787[label="",style="solid", color="burlywood", weight=3]; 31.93/16.20 786[label="primPlusNat (Succ vz220) Zero",fontsize=16,color="black",shape="box"];786 -> 792[label="",style="solid", color="black", weight=3]; 31.93/16.20 787[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];787 -> 793[label="",style="solid", color="black", weight=3]; 31.93/16.20 792[label="Succ vz220",fontsize=16,color="green",shape="box"];793[label="Zero",fontsize=16,color="green",shape="box"];} 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (6) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile12(vz64, vz65, vz66) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) 31.93/16.20 new_takeWhile(Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Zero), new_psMyInt) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> new_takeWhile1(vz3100, vz3000, vz3000, vz3100) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Neg(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Succ(vz3100)), new_psMyInt1) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_takeWhile1(vz21, vz22, vz230, vz240) 31.93/16.20 new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Succ(vz680)) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Pos(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Zero), new_psMyInt1) 31.93/16.20 new_takeWhile11(vz64, vz65, vz66, Main.Succ(vz670), Main.Succ(vz680)) -> new_takeWhile11(vz64, vz65, vz66, vz670, vz680) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Succ(vz3100)), new_psMyInt) 31.93/16.20 new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Zero) -> new_takeWhile12(vz64, vz65, vz66) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Succ(vz3100)), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile11(vz3100, vz3000, new_psMyInt0(vz3000), vz3100, vz3000) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Pos(vz310), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile(Main.Pos(vz310), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (7) DependencyGraphProof (EQUIVALENT) 31.93/16.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (8) 31.93/16.20 Complex Obligation (AND) 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (9) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (10) MRRProof (EQUIVALENT) 31.93/16.20 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 31.93/16.20 31.93/16.20 31.93/16.20 Strictly oriented rules of the TRS R: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 31.93/16.20 Used ordering: Polynomial interpretation [POLO]: 31.93/16.20 31.93/16.20 POL(Main.Neg(x_1)) = x_1 31.93/16.20 POL(Main.Pos(x_1)) = x_1 31.93/16.20 POL(Main.Succ(x_1)) = 2*x_1 31.93/16.20 POL(Main.Zero) = 0 31.93/16.20 POL(new_primPlusNat(x_1)) = 2 + 2*x_1 31.93/16.20 POL(new_psMyInt) = 0 31.93/16.20 POL(new_psMyInt0(x_1)) = x_1 31.93/16.20 POL(new_psMyInt1) = 0 31.93/16.20 POL(new_takeWhile(x_1, x_2)) = 2 + x_1 + x_2 31.93/16.20 POL(new_takeWhile0(x_1, x_2)) = 2 + x_1 + x_2 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (11) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (12) TransformationProof (EQUIVALENT) 31.93/16.20 By rewriting [LPAR04] the rule new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) at position [0] we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt),new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt)) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (13) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (14) QReductionProof (EQUIVALENT) 31.93/16.20 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (15) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (16) TransformationProof (EQUIVALENT) 31.93/16.20 By rewriting [LPAR04] the rule new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) at position [0] we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1),new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1)) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (17) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (18) TransformationProof (EQUIVALENT) 31.93/16.20 By rewriting [LPAR04] the rule new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) at position [1] we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))),new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (19) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (20) UsableRulesProof (EQUIVALENT) 31.93/16.20 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. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (21) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (22) QReductionProof (EQUIVALENT) 31.93/16.20 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 31.93/16.20 31.93/16.20 new_psMyInt 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (23) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (24) TransformationProof (EQUIVALENT) 31.93/16.20 By rewriting [LPAR04] the rule new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) at position [1] we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))),new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (25) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (26) UsableRulesProof (EQUIVALENT) 31.93/16.20 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. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (27) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (28) QReductionProof (EQUIVALENT) 31.93/16.20 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (29) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (30) QDPOrderProof (EQUIVALENT) 31.93/16.20 We use the reduction pair processor [LPAR04,JAR06]. 31.93/16.20 31.93/16.20 31.93/16.20 The following pairs can be oriented strictly and are deleted. 31.93/16.20 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile0(new_psMyInt0(vz3000), new_psMyInt0(vz3000)) 31.93/16.20 The remaining pairs can at least be oriented weakly. 31.93/16.20 Used ordering: Polynomial interpretation [POLO]: 31.93/16.20 31.93/16.20 POL(Main.Neg(x_1)) = x_1 31.93/16.20 POL(Main.Pos(x_1)) = 0 31.93/16.20 POL(Main.Succ(x_1)) = 1 + x_1 31.93/16.20 POL(Main.Zero) = 0 31.93/16.20 POL(new_psMyInt0(x_1)) = x_1 31.93/16.20 POL(new_takeWhile(x_1, x_2)) = x_2 31.93/16.20 POL(new_takeWhile0(x_1, x_2)) = x_2 31.93/16.20 31.93/16.20 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (31) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (32) TransformationProof (EQUIVALENT) 31.93/16.20 By instantiating [LPAR04] the rule new_takeWhile0(vz10, vz9) -> new_takeWhile(Main.Neg(Main.Zero), vz9) we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) -> new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Succ(Main.Zero))),new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) -> new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (33) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) -> new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Succ(Main.Zero))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (34) DependencyGraphProof (EQUIVALENT) 31.93/16.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (35) 31.93/16.20 TRUE 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (36) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_takeWhile1(vz21, vz22, vz230, vz240) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> new_takeWhile1(vz3100, vz3000, vz3000, vz3100) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (37) QDPPairToRuleProof (EQUIVALENT) 31.93/16.20 The dependency pair new_takeWhile1(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_takeWhile1(vz21, vz22, vz230, vz240) was transformed to the following new rules: 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 the following new pairs maintain the fan-in: 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 31.93/16.20 the following new pairs maintain the fan-out: 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (38) 31.93/16.20 Complex Obligation (AND) 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (39) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> new_takeWhile1(vz3100, vz3000, vz3000, vz3100) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (40) TransformationProof (EQUIVALENT) 31.93/16.20 By instantiating [LPAR04] the rule new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> new_takeWhile1(vz3100, vz3000, vz3000, vz3100) we obtained the following new rules [LPAR04]: 31.93/16.20 31.93/16.20 (new_takeWhile(Main.Pos(Main.Succ(z0)), Main.Pos(Main.Succ(Main.Succ(y_0)))) -> new_takeWhile1(z0, Main.Succ(y_0), Main.Succ(y_0), z0),new_takeWhile(Main.Pos(Main.Succ(z0)), Main.Pos(Main.Succ(Main.Succ(y_0)))) -> new_takeWhile1(z0, Main.Succ(y_0), Main.Succ(y_0), z0)) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (41) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(z0)), Main.Pos(Main.Succ(Main.Succ(y_0)))) -> new_takeWhile1(z0, Main.Succ(y_0), Main.Succ(y_0), z0) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (42) DependencyGraphProof (EQUIVALENT) 31.93/16.20 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (43) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.20 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (44) UsableRulesProof (EQUIVALENT) 31.93/16.20 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. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (45) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (46) QReductionProof (EQUIVALENT) 31.93/16.20 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 31.93/16.20 31.93/16.20 new_psMyInt1 31.93/16.20 new_psMyInt0(Main.Zero) 31.93/16.20 new_psMyInt 31.93/16.20 new_psMyInt0(Main.Succ(x0)) 31.93/16.20 31.93/16.20 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (47) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (48) InductionCalculusProof (EQUIVALENT) 31.93/16.20 Note that final constraints are written in bold face. 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) the following chains were created: 31.93/16.20 *We consider the chain new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3))) -> H(x2, x3, anew_new_takeWhile1(x3, x2)), H(x4, x5, cons_new_takeWhile1(Main.Zero, Main.Succ(x6))) -> new_takeWhile1(x4, x5, Main.Zero, Main.Succ(x6)) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (H(x2, x3, anew_new_takeWhile1(x3, x2))=H(x4, x5, cons_new_takeWhile1(Main.Zero, Main.Succ(x6))) ==> new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3)))_>=_H(x2, x3, anew_new_takeWhile1(x3, x2))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (anew_new_takeWhile1(x3, x2)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3)))_>=_H(x2, x3, anew_new_takeWhile1(x3, x2))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_takeWhile1(x3, x2)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) which results in the following new constraint: 31.93/16.20 31.93/16.20 (3) (new_new_takeWhile1(x101, x100)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x100))), Main.Pos(Main.Succ(Main.Succ(x101))))_>=_H(Main.Succ(x100), Main.Succ(x101), anew_new_takeWhile1(Main.Succ(x101), Main.Succ(x100)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_takeWhile1(x101, x100)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) which results in the following new constraints: 31.93/16.20 31.93/16.20 (4) (new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) & (\/x104:new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x104)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102)))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.20 31.93/16.20 (5) (cons_new_takeWhile1(Main.Zero, Main.Succ(x105))=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.20 31.93/16.20 (6) (cons_new_takeWhile1(Main.Zero, Main.Zero)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x104:new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x104)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102)))) with sigma = [x104 / x6] which results in the following new constraint: 31.93/16.20 31.93/16.20 (7) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (8) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We solved constraint (6) using rules (I), (II). 31.93/16.20 *We consider the chain new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10))) -> H(x9, x10, anew_new_takeWhile1(x10, x9)), H(x11, x12, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(x11, x12, Main.Zero, Main.Zero) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (H(x9, x10, anew_new_takeWhile1(x10, x9))=H(x11, x12, cons_new_takeWhile1(Main.Zero, Main.Zero)) ==> new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10)))_>=_H(x9, x10, anew_new_takeWhile1(x10, x9))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (anew_new_takeWhile1(x10, x9)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10)))_>=_H(x9, x10, anew_new_takeWhile1(x10, x9))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_takeWhile1(x10, x9)=cons_new_takeWhile1(Main.Zero, Main.Zero) which results in the following new constraint: 31.93/16.20 31.93/16.20 (3) (new_new_takeWhile1(x107, x106)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x106))), Main.Pos(Main.Succ(Main.Succ(x107))))_>=_H(Main.Succ(x106), Main.Succ(x107), anew_new_takeWhile1(Main.Succ(x107), Main.Succ(x106)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_takeWhile1(x107, x106)=cons_new_takeWhile1(Main.Zero, Main.Zero) which results in the following new constraints: 31.93/16.20 31.93/16.20 (4) (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) & (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108)))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.20 31.93/16.20 (5) (cons_new_takeWhile1(Main.Zero, Main.Succ(x110))=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x110)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x110)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x110))))) 31.93/16.20 31.93/16.20 (6) (cons_new_takeWhile1(Main.Zero, Main.Zero)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108)))) with sigma = [ ] which results in the following new constraint: 31.93/16.20 31.93/16.20 (7) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 31.93/16.20 31.93/16.20 (8) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) the following chains were created: 31.93/16.20 *We consider the chain H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25))) -> new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25)), new_takeWhile1(x26, x27, Main.Zero, Main.Succ(x28)) -> new_takeWhile(Main.Pos(Main.Succ(x26)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x27))))) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))=new_takeWhile1(x26, x27, Main.Zero, Main.Succ(x28)) ==> H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) the following chains were created: 31.93/16.20 *We consider the chain new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40)) -> new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39))))), new_takeWhile(Main.Pos(Main.Succ(x41)), Main.Pos(Main.Succ(x42))) -> H(x41, x42, anew_new_takeWhile1(x42, x41)) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))=new_takeWhile(Main.Pos(Main.Succ(x41)), Main.Pos(Main.Succ(x42))) ==> new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) the following chains were created: 31.93/16.20 *We consider the chain H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(x66, x67, Main.Zero, Main.Zero), new_takeWhile1(x68, x69, Main.Zero, Main.Zero) -> new_takeWhile10(x68, x69) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (new_takeWhile1(x66, x67, Main.Zero, Main.Zero)=new_takeWhile1(x68, x69, Main.Zero, Main.Zero) ==> H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) the following chains were created: 31.93/16.20 *We consider the chain new_takeWhile1(x82, x83, Main.Zero, Main.Zero) -> new_takeWhile10(x82, x83), new_takeWhile10(x84, x85) -> new_takeWhile(Main.Pos(Main.Succ(x84)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x85))))) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (new_takeWhile10(x82, x83)=new_takeWhile10(x84, x85) ==> new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) the following chains were created: 31.93/16.20 *We consider the chain new_takeWhile10(x86, x87) -> new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87))))), new_takeWhile(Main.Pos(Main.Succ(x88)), Main.Pos(Main.Succ(x89))) -> H(x88, x89, anew_new_takeWhile1(x89, x88)) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))=new_takeWhile(Main.Pos(Main.Succ(x88)), Main.Pos(Main.Succ(x89))) ==> new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 To summarize, we get the following constraints P__>=_ for the following pairs. 31.93/16.20 31.93/16.20 *new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 31.93/16.20 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.20 31.93/16.20 31.93/16.20 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.20 31.93/16.20 31.93/16.20 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.20 31.93/16.20 31.93/16.20 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 *H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 31.93/16.20 *(H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 *new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 *(new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 *H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 *(H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 *new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 31.93/16.20 *(new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 *new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 *(new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 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. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (49) 31.93/16.20 Obligation: 31.93/16.20 Q DP problem: 31.93/16.20 The TRS P consists of the following rules: 31.93/16.20 31.93/16.20 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.20 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.20 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.20 31.93/16.20 The TRS R consists of the following rules: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.20 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.20 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 The set Q consists of the following terms: 31.93/16.20 31.93/16.20 new_primPlusNat(Main.Succ(x0)) 31.93/16.20 new_primPlusNat(Main.Zero) 31.93/16.20 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.20 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.20 31.93/16.20 We have to consider all minimal (P,Q,R)-chains. 31.93/16.20 ---------------------------------------- 31.93/16.20 31.93/16.20 (50) NonInfProof (EQUIVALENT) 31.93/16.20 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 31.93/16.20 31.93/16.20 Note that final constraints are written in bold face. 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 For Pair new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) the following chains were created: 31.93/16.20 *We consider the chain new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3))) -> H(x2, x3, anew_new_takeWhile1(x3, x2)), H(x4, x5, cons_new_takeWhile1(Main.Zero, Main.Succ(x6))) -> new_takeWhile1(x4, x5, Main.Zero, Main.Succ(x6)) which results in the following constraint: 31.93/16.20 31.93/16.20 (1) (H(x2, x3, anew_new_takeWhile1(x3, x2))=H(x4, x5, cons_new_takeWhile1(Main.Zero, Main.Succ(x6))) ==> new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3)))_>=_H(x2, x3, anew_new_takeWhile1(x3, x2))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.20 31.93/16.20 (2) (anew_new_takeWhile1(x3, x2)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(x2)), Main.Pos(Main.Succ(x3)))_>=_H(x2, x3, anew_new_takeWhile1(x3, x2))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_takeWhile1(x3, x2)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) which results in the following new constraint: 31.93/16.20 31.93/16.20 (3) (new_new_takeWhile1(x101, x100)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x100))), Main.Pos(Main.Succ(Main.Succ(x101))))_>=_H(Main.Succ(x100), Main.Succ(x101), anew_new_takeWhile1(Main.Succ(x101), Main.Succ(x100)))) 31.93/16.20 31.93/16.20 31.93/16.20 31.93/16.20 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_takeWhile1(x101, x100)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) which results in the following new constraints: 31.93/16.21 31.93/16.21 (4) (new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) & (\/x104:new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x104)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102)))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.21 31.93/16.21 (5) (cons_new_takeWhile1(Main.Zero, Main.Succ(x105))=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.21 31.93/16.21 (6) (cons_new_takeWhile1(Main.Zero, Main.Zero)=cons_new_takeWhile1(Main.Zero, Main.Succ(x6)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x104:new_new_takeWhile1(x103, x102)=cons_new_takeWhile1(Main.Zero, Main.Succ(x104)) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102)))) with sigma = [x104 / x6] which results in the following new constraint: 31.93/16.21 31.93/16.21 (7) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (8) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We solved constraint (6) using rules (I), (II). 31.93/16.21 *We consider the chain new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10))) -> H(x9, x10, anew_new_takeWhile1(x10, x9)), H(x11, x12, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(x11, x12, Main.Zero, Main.Zero) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (H(x9, x10, anew_new_takeWhile1(x10, x9))=H(x11, x12, cons_new_takeWhile1(Main.Zero, Main.Zero)) ==> new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10)))_>=_H(x9, x10, anew_new_takeWhile1(x10, x9))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (anew_new_takeWhile1(x10, x9)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(x9)), Main.Pos(Main.Succ(x10)))_>=_H(x9, x10, anew_new_takeWhile1(x10, x9))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_takeWhile1(x10, x9)=cons_new_takeWhile1(Main.Zero, Main.Zero) which results in the following new constraint: 31.93/16.21 31.93/16.21 (3) (new_new_takeWhile1(x107, x106)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x106))), Main.Pos(Main.Succ(Main.Succ(x107))))_>=_H(Main.Succ(x106), Main.Succ(x107), anew_new_takeWhile1(Main.Succ(x107), Main.Succ(x106)))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_takeWhile1(x107, x106)=cons_new_takeWhile1(Main.Zero, Main.Zero) which results in the following new constraints: 31.93/16.21 31.93/16.21 (4) (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) & (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108)))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.21 31.93/16.21 (5) (cons_new_takeWhile1(Main.Zero, Main.Succ(x110))=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x110)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x110)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x110))))) 31.93/16.21 31.93/16.21 (6) (cons_new_takeWhile1(Main.Zero, Main.Zero)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (new_new_takeWhile1(x109, x108)=cons_new_takeWhile1(Main.Zero, Main.Zero) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108)))) with sigma = [ ] which results in the following new constraint: 31.93/16.21 31.93/16.21 (7) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 31.93/16.21 31.93/16.21 (8) (new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 For Pair H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) the following chains were created: 31.93/16.21 *We consider the chain H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25))) -> new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25)), new_takeWhile1(x26, x27, Main.Zero, Main.Succ(x28)) -> new_takeWhile(Main.Pos(Main.Succ(x26)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x27))))) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))=new_takeWhile1(x26, x27, Main.Zero, Main.Succ(x28)) ==> H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 For Pair new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) the following chains were created: 31.93/16.21 *We consider the chain new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40)) -> new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39))))), new_takeWhile(Main.Pos(Main.Succ(x41)), Main.Pos(Main.Succ(x42))) -> H(x41, x42, anew_new_takeWhile1(x42, x41)) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))=new_takeWhile(Main.Pos(Main.Succ(x41)), Main.Pos(Main.Succ(x42))) ==> new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 For Pair H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) the following chains were created: 31.93/16.21 *We consider the chain H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(x66, x67, Main.Zero, Main.Zero), new_takeWhile1(x68, x69, Main.Zero, Main.Zero) -> new_takeWhile10(x68, x69) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (new_takeWhile1(x66, x67, Main.Zero, Main.Zero)=new_takeWhile1(x68, x69, Main.Zero, Main.Zero) ==> H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 For Pair new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) the following chains were created: 31.93/16.21 *We consider the chain new_takeWhile1(x82, x83, Main.Zero, Main.Zero) -> new_takeWhile10(x82, x83), new_takeWhile10(x84, x85) -> new_takeWhile(Main.Pos(Main.Succ(x84)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x85))))) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (new_takeWhile10(x82, x83)=new_takeWhile10(x84, x85) ==> new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 For Pair new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) the following chains were created: 31.93/16.21 *We consider the chain new_takeWhile10(x86, x87) -> new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87))))), new_takeWhile(Main.Pos(Main.Succ(x88)), Main.Pos(Main.Succ(x89))) -> H(x88, x89, anew_new_takeWhile1(x89, x88)) which results in the following constraint: 31.93/16.21 31.93/16.21 (1) (new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))=new_takeWhile(Main.Pos(Main.Succ(x88)), Main.Pos(Main.Succ(x89))) ==> new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 31.93/16.21 31.93/16.21 (2) (new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 To summarize, we get the following constraints P__>=_ for the following pairs. 31.93/16.21 31.93/16.21 *new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.21 31.93/16.21 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x102))), Main.Pos(Main.Succ(Main.Succ(x103))))_>=_H(Main.Succ(x102), Main.Succ(x103), anew_new_takeWhile1(Main.Succ(x103), Main.Succ(x102))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x102)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x103)))))_>=_H(Main.Succ(Main.Succ(x102)), Main.Succ(Main.Succ(x103)), anew_new_takeWhile1(Main.Succ(Main.Succ(x103)), Main.Succ(Main.Succ(x102))))) 31.93/16.21 31.93/16.21 31.93/16.21 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x105)))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Succ(x105)), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Succ(x105))))) 31.93/16.21 31.93/16.21 31.93/16.21 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(x108))), Main.Pos(Main.Succ(Main.Succ(x109))))_>=_H(Main.Succ(x108), Main.Succ(x109), anew_new_takeWhile1(Main.Succ(x109), Main.Succ(x108))) ==> new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Succ(x108)))), Main.Pos(Main.Succ(Main.Succ(Main.Succ(x109)))))_>=_H(Main.Succ(Main.Succ(x108)), Main.Succ(Main.Succ(x109)), anew_new_takeWhile1(Main.Succ(Main.Succ(x109)), Main.Succ(Main.Succ(x108))))) 31.93/16.21 31.93/16.21 31.93/16.21 *(new_takeWhile(Main.Pos(Main.Succ(Main.Succ(Main.Zero))), Main.Pos(Main.Succ(Main.Succ(Main.Zero))))_>=_H(Main.Succ(Main.Zero), Main.Succ(Main.Zero), anew_new_takeWhile1(Main.Succ(Main.Zero), Main.Succ(Main.Zero)))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 *H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.21 31.93/16.21 *(H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.21 31.93/16.21 *(new_takeWhile1(x38, x39, Main.Zero, Main.Succ(x40))_>=_new_takeWhile(Main.Pos(Main.Succ(x38)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x39)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 *H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.21 31.93/16.21 *(H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.21 31.93/16.21 *(new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.21 31.93/16.21 *(new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 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. 31.93/16.21 31.93/16.21 Using the following integer polynomial ordering the resulting constraints can be solved 31.93/16.21 31.93/16.21 Polynomial interpretation [NONINF]: 31.93/16.21 31.93/16.21 POL(H(x_1, x_2, x_3)) = -1 + x_1 - x_2 - x_3 31.93/16.21 POL(Main.Pos(x_1)) = 1 + x_1 31.93/16.21 POL(Main.Succ(x_1)) = 1 + x_1 31.93/16.21 POL(Main.Zero) = 0 31.93/16.21 POL(anew_new_takeWhile1(x_1, x_2)) = 0 31.93/16.21 POL(c) = -1 31.93/16.21 POL(cons_new_takeWhile1(x_1, x_2)) = 0 31.93/16.21 POL(new_new_takeWhile1(x_1, x_2)) = 0 31.93/16.21 POL(new_primPlusNat(x_1)) = x_1 31.93/16.21 POL(new_takeWhile(x_1, x_2)) = x_1 - x_2 31.93/16.21 POL(new_takeWhile1(x_1, x_2, x_3, x_4)) = -1 + x_1 - x_2 + x_3 31.93/16.21 POL(new_takeWhile10(x_1, x_2)) = -1 + x_1 - x_2 31.93/16.21 31.93/16.21 31.93/16.21 The following pairs are in P_>: 31.93/16.21 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.21 The following pairs are in P_bound: 31.93/16.21 new_takeWhile(Main.Pos(Main.Succ(vz3100)), Main.Pos(Main.Succ(vz3000))) -> H(vz3100, vz3000, anew_new_takeWhile1(vz3000, vz3100)) 31.93/16.21 The following rules are usable: 31.93/16.21 new_new_takeWhile1(vz230, vz240) -> anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) 31.93/16.21 Main.Zero -> new_primPlusNat(Main.Zero) 31.93/16.21 Main.Succ(vz220) -> new_primPlusNat(Main.Succ(vz220)) 31.93/16.21 new_new_takeWhile1(vz230, vz240) -> new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) 31.93/16.21 cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.21 cons_new_takeWhile1(Main.Zero, Main.Zero) -> new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (51) 31.93/16.21 Obligation: 31.93/16.21 Q DP problem: 31.93/16.21 The TRS P consists of the following rules: 31.93/16.21 31.93/16.21 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Succ(vz240))) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) 31.93/16.21 new_takeWhile1(vz21, vz22, Main.Zero, Main.Succ(vz240)) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.21 H(vz21, vz22, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) 31.93/16.21 new_takeWhile1(vz21, vz22, Main.Zero, Main.Zero) -> new_takeWhile10(vz21, vz22) 31.93/16.21 new_takeWhile10(vz21, vz22) -> new_takeWhile(Main.Pos(Main.Succ(vz21)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz22))))) 31.93/16.21 31.93/16.21 The TRS R consists of the following rules: 31.93/16.21 31.93/16.21 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.21 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.21 anew_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.21 new_new_takeWhile1(Main.Succ(vz230), Main.Succ(vz240)) -> new_new_takeWhile1(vz230, vz240) 31.93/16.21 new_new_takeWhile1(Main.Zero, Main.Succ(vz240)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz240)) 31.93/16.21 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.21 31.93/16.21 The set Q consists of the following terms: 31.93/16.21 31.93/16.21 new_primPlusNat(Main.Succ(x0)) 31.93/16.21 new_primPlusNat(Main.Zero) 31.93/16.21 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.21 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 31.93/16.21 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 31.93/16.21 new_new_takeWhile1(Main.Zero, Main.Zero) 31.93/16.21 31.93/16.21 We have to consider all minimal (P,Q,R)-chains. 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (52) DependencyGraphProof (EQUIVALENT) 31.93/16.21 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes. 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (53) 31.93/16.21 TRUE 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (54) 31.93/16.21 Obligation: 31.93/16.21 Q DP problem: 31.93/16.21 The TRS P consists of the following rules: 31.93/16.21 31.93/16.21 new_takeWhile1(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_takeWhile1(vz21, vz22, vz230, vz240) 31.93/16.21 31.93/16.21 R is empty. 31.93/16.21 Q is empty. 31.93/16.21 We have to consider all minimal (P,Q,R)-chains. 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (55) QDPSizeChangeProof (EQUIVALENT) 31.93/16.21 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. 31.93/16.21 31.93/16.21 From the DPs we obtained the following set of size-change graphs: 31.93/16.21 *new_takeWhile1(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_takeWhile1(vz21, vz22, vz230, vz240) 31.93/16.21 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 31.93/16.21 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (56) 31.93/16.21 YES 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (57) 31.93/16.21 Obligation: 31.93/16.21 Q DP problem: 31.93/16.21 The TRS P consists of the following rules: 31.93/16.21 31.93/16.21 new_takeWhile(Main.Pos(vz310), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile(Main.Pos(vz310), new_psMyInt0(vz3000)) 31.93/16.21 31.93/16.21 The TRS R consists of the following rules: 31.93/16.21 31.93/16.21 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.21 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.21 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.21 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.21 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.21 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.21 31.93/16.21 The set Q consists of the following terms: 31.93/16.21 31.93/16.21 new_psMyInt1 31.93/16.21 new_psMyInt0(Main.Zero) 31.93/16.21 new_primPlusNat(Main.Succ(x0)) 31.93/16.21 new_psMyInt 31.93/16.21 new_psMyInt0(Main.Succ(x0)) 31.93/16.21 new_primPlusNat(Main.Zero) 31.93/16.21 31.93/16.21 We have to consider all minimal (P,Q,R)-chains. 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (58) QDPSizeChangeProof (EQUIVALENT) 31.93/16.21 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 31.93/16.21 31.93/16.21 Order:Polynomial interpretation [POLO]: 31.93/16.21 31.93/16.21 POL(Main.Neg(x_1)) = x_1 31.93/16.21 POL(Main.Pos(x_1)) = 1 31.93/16.21 POL(Main.Succ(x_1)) = 1 + x_1 31.93/16.21 POL(Main.Zero) = 1 31.93/16.21 POL(new_psMyInt0(x_1)) = x_1 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 From the DPs we obtained the following set of size-change graphs: 31.93/16.21 *new_takeWhile(Main.Pos(vz310), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile(Main.Pos(vz310), new_psMyInt0(vz3000)) (allowed arguments on rhs = {1, 2}) 31.93/16.21 The graph contains the following edges 1 >= 1, 2 > 2 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We oriented the following set of usable rules [AAECC05,FROCOS05]. 31.93/16.21 31.93/16.21 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.21 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (59) 31.93/16.21 YES 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (60) 31.93/16.21 Obligation: 31.93/16.21 Q DP problem: 31.93/16.21 The TRS P consists of the following rules: 31.93/16.21 31.93/16.21 new_takeWhile(Main.Neg(Main.Succ(vz3100)), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile11(vz3100, vz3000, new_psMyInt0(vz3000), vz3100, vz3000) 31.93/16.21 new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Succ(vz680)) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) 31.93/16.21 new_takeWhile11(vz64, vz65, vz66, Main.Succ(vz670), Main.Succ(vz680)) -> new_takeWhile11(vz64, vz65, vz66, vz670, vz680) 31.93/16.21 new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Zero) -> new_takeWhile12(vz64, vz65, vz66) 31.93/16.21 new_takeWhile12(vz64, vz65, vz66) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) 31.93/16.21 31.93/16.21 The TRS R consists of the following rules: 31.93/16.21 31.93/16.21 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.21 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.21 new_primPlusNat(Main.Zero) -> Main.Zero 31.93/16.21 new_primPlusNat(Main.Succ(vz220)) -> Main.Succ(vz220) 31.93/16.21 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.21 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 31.93/16.21 31.93/16.21 The set Q consists of the following terms: 31.93/16.21 31.93/16.21 new_psMyInt1 31.93/16.21 new_psMyInt0(Main.Zero) 31.93/16.21 new_primPlusNat(Main.Succ(x0)) 31.93/16.21 new_psMyInt 31.93/16.21 new_psMyInt0(Main.Succ(x0)) 31.93/16.21 new_primPlusNat(Main.Zero) 31.93/16.21 31.93/16.21 We have to consider all minimal (P,Q,R)-chains. 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (61) QDPSizeChangeProof (EQUIVALENT) 31.93/16.21 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 31.93/16.21 31.93/16.21 Order:Polynomial interpretation [POLO]: 31.93/16.21 31.93/16.21 POL(Main.Neg(x_1)) = x_1 31.93/16.21 POL(Main.Pos(x_1)) = 0 31.93/16.21 POL(Main.Succ(x_1)) = 1 + x_1 31.93/16.21 POL(Main.Zero) = 1 31.93/16.21 POL(new_psMyInt0(x_1)) = x_1 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 From the DPs we obtained the following set of size-change graphs: 31.93/16.21 *new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Succ(vz680)) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) (allowed arguments on rhs = {1, 2}) 31.93/16.21 The graph contains the following edges 3 >= 2 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile12(vz64, vz65, vz66) -> new_takeWhile(Main.Neg(Main.Succ(vz64)), vz66) (allowed arguments on rhs = {1, 2}) 31.93/16.21 The graph contains the following edges 3 >= 2 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile(Main.Neg(Main.Succ(vz3100)), Main.Neg(Main.Succ(vz3000))) -> new_takeWhile11(vz3100, vz3000, new_psMyInt0(vz3000), vz3100, vz3000) (allowed arguments on rhs = {1, 2, 3, 4, 5}) 31.93/16.21 The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 1 > 4, 2 > 5 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile11(vz64, vz65, vz66, Main.Succ(vz670), Main.Succ(vz680)) -> new_takeWhile11(vz64, vz65, vz66, vz670, vz680) (allowed arguments on rhs = {1, 2, 3, 4, 5}) 31.93/16.21 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5 31.93/16.21 31.93/16.21 31.93/16.21 *new_takeWhile11(vz64, vz65, vz66, Main.Zero, Main.Zero) -> new_takeWhile12(vz64, vz65, vz66) (allowed arguments on rhs = {1, 2, 3}) 31.93/16.21 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 31.93/16.21 31.93/16.21 31.93/16.21 31.93/16.21 We oriented the following set of usable rules [AAECC05,FROCOS05]. 31.93/16.21 31.93/16.21 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 31.93/16.21 new_psMyInt0(Main.Succ(vz30000)) -> Main.Neg(Main.Succ(vz30000)) 31.93/16.21 31.93/16.21 ---------------------------------------- 31.93/16.21 31.93/16.21 (62) 31.93/16.21 YES 31.93/16.26 EOF