29.56/15.48 YES 32.31/16.10 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 32.31/16.10 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 32.31/16.10 32.31/16.10 32.31/16.10 H-Termination with start terms of the given HASKELL could be proven: 32.31/16.10 32.31/16.10 (0) HASKELL 32.31/16.10 (1) BR [EQUIVALENT, 0 ms] 32.31/16.10 (2) HASKELL 32.31/16.10 (3) COR [EQUIVALENT, 0 ms] 32.31/16.10 (4) HASKELL 32.31/16.10 (5) Narrow [SOUND, 0 ms] 32.31/16.10 (6) QDP 32.31/16.10 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 32.31/16.10 (8) AND 32.31/16.10 (9) QDP 32.31/16.10 (10) MRRProof [EQUIVALENT, 97 ms] 32.31/16.10 (11) QDP 32.31/16.10 (12) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (13) QDP 32.31/16.10 (14) QReductionProof [EQUIVALENT, 0 ms] 32.31/16.10 (15) QDP 32.31/16.10 (16) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (17) QDP 32.31/16.10 (18) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (19) QDP 32.31/16.10 (20) UsableRulesProof [EQUIVALENT, 0 ms] 32.31/16.10 (21) QDP 32.31/16.10 (22) QReductionProof [EQUIVALENT, 0 ms] 32.31/16.10 (23) QDP 32.31/16.10 (24) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (25) QDP 32.31/16.10 (26) UsableRulesProof [EQUIVALENT, 0 ms] 32.31/16.10 (27) QDP 32.31/16.10 (28) QReductionProof [EQUIVALENT, 0 ms] 32.31/16.10 (29) QDP 32.31/16.10 (30) QDPOrderProof [EQUIVALENT, 35 ms] 32.31/16.10 (31) QDP 32.31/16.10 (32) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (33) QDP 32.31/16.10 (34) DependencyGraphProof [EQUIVALENT, 0 ms] 32.31/16.10 (35) TRUE 32.31/16.10 (36) QDP 32.31/16.10 (37) QDPPairToRuleProof [EQUIVALENT, 0 ms] 32.31/16.10 (38) AND 32.31/16.10 (39) QDP 32.31/16.10 (40) TransformationProof [EQUIVALENT, 0 ms] 32.31/16.10 (41) QDP 32.31/16.10 (42) DependencyGraphProof [EQUIVALENT, 0 ms] 32.31/16.10 (43) QDP 32.31/16.10 (44) UsableRulesProof [EQUIVALENT, 0 ms] 32.31/16.10 (45) QDP 32.31/16.10 (46) QReductionProof [EQUIVALENT, 0 ms] 32.31/16.10 (47) QDP 32.31/16.10 (48) InductionCalculusProof [EQUIVALENT, 0 ms] 32.31/16.10 (49) QDP 32.31/16.10 (50) NonInfProof [EQUIVALENT, 116 ms] 32.31/16.10 (51) QDP 32.31/16.10 (52) DependencyGraphProof [EQUIVALENT, 0 ms] 32.31/16.10 (53) TRUE 32.31/16.10 (54) QDP 32.31/16.10 (55) QDPSizeChangeProof [EQUIVALENT, 0 ms] 32.31/16.10 (56) YES 32.31/16.10 (57) QDP 32.31/16.10 (58) QDPSizeChangeProof [EQUIVALENT, 0 ms] 32.31/16.10 (59) YES 32.31/16.10 (60) QDP 32.31/16.10 (61) QDPSizeChangeProof [EQUIVALENT, 0 ms] 32.31/16.10 (62) YES 32.31/16.10 32.31/16.10 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (0) 32.31/16.10 Obligation: 32.31/16.10 mainModule Main 32.31/16.10 module Main where { 32.31/16.10 import qualified Prelude; 32.31/16.10 data List a = Cons a (List a) | Nil ; 32.31/16.10 32.31/16.10 data MyBool = MyTrue | MyFalse ; 32.31/16.10 32.31/16.10 data MyInt = Pos Main.Nat | Neg Main.Nat ; 32.31/16.10 32.31/16.10 data Main.Nat = Succ Main.Nat | Zero ; 32.31/16.10 32.31/16.10 data Ordering = LT | EQ | GT ; 32.31/16.10 32.31/16.10 data Main.WHNF a = WHNF a ; 32.31/16.10 32.31/16.10 compareMyInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 compareMyInt = primCmpInt; 32.31/16.10 32.31/16.10 dsEm :: (a -> b) -> a -> b; 32.31/16.10 dsEm f x = Main.seq x (f x); 32.31/16.10 32.31/16.10 enforceWHNF :: Main.WHNF b -> a -> a; 32.31/16.10 enforceWHNF (Main.WHNF x) y = y; 32.31/16.10 32.31/16.10 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 32.31/16.10 enumFromToMyInt = numericEnumFromTo; 32.31/16.10 32.31/16.10 esEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 esEsOrdering LT LT = MyTrue; 32.31/16.10 esEsOrdering LT EQ = MyFalse; 32.31/16.10 esEsOrdering LT GT = MyFalse; 32.31/16.10 esEsOrdering EQ LT = MyFalse; 32.31/16.10 esEsOrdering EQ EQ = MyTrue; 32.31/16.10 esEsOrdering EQ GT = MyFalse; 32.31/16.10 esEsOrdering GT LT = MyFalse; 32.31/16.10 esEsOrdering GT EQ = MyFalse; 32.31/16.10 esEsOrdering GT GT = MyTrue; 32.31/16.10 32.31/16.10 flip :: (c -> b -> a) -> b -> c -> a; 32.31/16.10 flip f x y = f y x; 32.31/16.10 32.31/16.10 fromIntMyInt :: MyInt -> MyInt; 32.31/16.10 fromIntMyInt x = x; 32.31/16.10 32.31/16.10 fsEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 fsEsOrdering x y = not (esEsOrdering x y); 32.31/16.10 32.31/16.10 ltEsMyInt :: MyInt -> MyInt -> MyBool; 32.31/16.10 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 32.31/16.10 32.31/16.10 not :: MyBool -> MyBool; 32.31/16.10 not MyTrue = MyFalse; 32.31/16.10 not MyFalse = MyTrue; 32.31/16.10 32.31/16.10 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 32.31/16.10 32.31/16.10 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 32.31/16.10 32.31/16.10 otherwise :: MyBool; 32.31/16.10 otherwise = MyTrue; 32.31/16.10 32.31/16.10 primCmpInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 32.31/16.10 32.31/16.10 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 32.31/16.10 primCmpNat Main.Zero Main.Zero = EQ; 32.31/16.10 primCmpNat Main.Zero (Main.Succ y) = LT; 32.31/16.10 primCmpNat (Main.Succ x) Main.Zero = GT; 32.31/16.10 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 32.31/16.10 32.31/16.10 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 32.31/16.10 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 32.31/16.10 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 32.31/16.10 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 32.31/16.10 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 32.31/16.10 32.31/16.10 primPlusInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 32.31/16.10 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 32.31/16.10 32.31/16.10 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 32.31/16.10 primPlusNat Main.Zero Main.Zero = Main.Zero; 32.31/16.10 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 32.31/16.10 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 32.31/16.10 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 32.31/16.10 32.31/16.10 psMyInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 psMyInt = primPlusInt; 32.31/16.10 32.31/16.10 seq :: a -> b -> b; 32.31/16.10 seq x y = Main.enforceWHNF (Main.WHNF x) y; 32.31/16.10 32.31/16.10 takeWhile :: (a -> MyBool) -> List a -> List a; 32.31/16.10 takeWhile p Nil = takeWhile3 p Nil; 32.31/16.10 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 32.31/16.10 32.31/16.10 takeWhile0 p x xs MyTrue = Nil; 32.31/16.10 32.31/16.10 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 32.31/16.10 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 32.31/16.10 32.31/16.10 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 32.31/16.10 32.31/16.10 takeWhile3 p Nil = Nil; 32.31/16.10 takeWhile3 vx vy = takeWhile2 vx vy; 32.31/16.10 32.31/16.10 } 32.31/16.10 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (1) BR (EQUIVALENT) 32.31/16.10 Replaced joker patterns by fresh variables and removed binding patterns. 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (2) 32.31/16.10 Obligation: 32.31/16.10 mainModule Main 32.31/16.10 module Main where { 32.31/16.10 import qualified Prelude; 32.31/16.10 data List a = Cons a (List a) | Nil ; 32.31/16.10 32.31/16.10 data MyBool = MyTrue | MyFalse ; 32.31/16.10 32.31/16.10 data MyInt = Pos Main.Nat | Neg Main.Nat ; 32.31/16.10 32.31/16.10 data Main.Nat = Succ Main.Nat | Zero ; 32.31/16.10 32.31/16.10 data Ordering = LT | EQ | GT ; 32.31/16.10 32.31/16.10 data Main.WHNF a = WHNF a ; 32.31/16.10 32.31/16.10 compareMyInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 compareMyInt = primCmpInt; 32.31/16.10 32.31/16.10 dsEm :: (b -> a) -> b -> a; 32.31/16.10 dsEm f x = Main.seq x (f x); 32.31/16.10 32.31/16.10 enforceWHNF :: Main.WHNF b -> a -> a; 32.31/16.10 enforceWHNF (Main.WHNF x) y = y; 32.31/16.10 32.31/16.10 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 32.31/16.10 enumFromToMyInt = numericEnumFromTo; 32.31/16.10 32.31/16.10 esEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 esEsOrdering LT LT = MyTrue; 32.31/16.10 esEsOrdering LT EQ = MyFalse; 32.31/16.10 esEsOrdering LT GT = MyFalse; 32.31/16.10 esEsOrdering EQ LT = MyFalse; 32.31/16.10 esEsOrdering EQ EQ = MyTrue; 32.31/16.10 esEsOrdering EQ GT = MyFalse; 32.31/16.10 esEsOrdering GT LT = MyFalse; 32.31/16.10 esEsOrdering GT EQ = MyFalse; 32.31/16.10 esEsOrdering GT GT = MyTrue; 32.31/16.10 32.31/16.10 flip :: (c -> b -> a) -> b -> c -> a; 32.31/16.10 flip f x y = f y x; 32.31/16.10 32.31/16.10 fromIntMyInt :: MyInt -> MyInt; 32.31/16.10 fromIntMyInt x = x; 32.31/16.10 32.31/16.10 fsEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 fsEsOrdering x y = not (esEsOrdering x y); 32.31/16.10 32.31/16.10 ltEsMyInt :: MyInt -> MyInt -> MyBool; 32.31/16.10 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 32.31/16.10 32.31/16.10 not :: MyBool -> MyBool; 32.31/16.10 not MyTrue = MyFalse; 32.31/16.10 not MyFalse = MyTrue; 32.31/16.10 32.31/16.10 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 32.31/16.10 32.31/16.10 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 32.31/16.10 32.31/16.10 otherwise :: MyBool; 32.31/16.10 otherwise = MyTrue; 32.31/16.10 32.31/16.10 primCmpInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 32.31/16.10 32.31/16.10 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 32.31/16.10 primCmpNat Main.Zero Main.Zero = EQ; 32.31/16.10 primCmpNat Main.Zero (Main.Succ y) = LT; 32.31/16.10 primCmpNat (Main.Succ x) Main.Zero = GT; 32.31/16.10 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 32.31/16.10 32.31/16.10 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 32.31/16.10 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 32.31/16.10 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 32.31/16.10 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 32.31/16.10 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 32.31/16.10 32.31/16.10 primPlusInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 32.31/16.10 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 32.31/16.10 32.31/16.10 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 32.31/16.10 primPlusNat Main.Zero Main.Zero = Main.Zero; 32.31/16.10 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 32.31/16.10 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 32.31/16.10 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 32.31/16.10 32.31/16.10 psMyInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 psMyInt = primPlusInt; 32.31/16.10 32.31/16.10 seq :: a -> b -> b; 32.31/16.10 seq x y = Main.enforceWHNF (Main.WHNF x) y; 32.31/16.10 32.31/16.10 takeWhile :: (a -> MyBool) -> List a -> List a; 32.31/16.10 takeWhile p Nil = takeWhile3 p Nil; 32.31/16.10 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 32.31/16.10 32.31/16.10 takeWhile0 p x xs MyTrue = Nil; 32.31/16.10 32.31/16.10 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 32.31/16.10 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 32.31/16.10 32.31/16.10 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 32.31/16.10 32.31/16.10 takeWhile3 p Nil = Nil; 32.31/16.10 takeWhile3 vx vy = takeWhile2 vx vy; 32.31/16.10 32.31/16.10 } 32.31/16.10 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (3) COR (EQUIVALENT) 32.31/16.10 Cond Reductions: 32.31/16.10 The following Function with conditions 32.31/16.10 "undefined |Falseundefined; 32.31/16.10 " 32.31/16.10 is transformed to 32.31/16.10 "undefined = undefined1; 32.31/16.10 " 32.31/16.10 "undefined0 True = undefined; 32.31/16.10 " 32.31/16.10 "undefined1 = undefined0 False; 32.31/16.10 " 32.31/16.10 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (4) 32.31/16.10 Obligation: 32.31/16.10 mainModule Main 32.31/16.10 module Main where { 32.31/16.10 import qualified Prelude; 32.31/16.10 data List a = Cons a (List a) | Nil ; 32.31/16.10 32.31/16.10 data MyBool = MyTrue | MyFalse ; 32.31/16.10 32.31/16.10 data MyInt = Pos Main.Nat | Neg Main.Nat ; 32.31/16.10 32.31/16.10 data Main.Nat = Succ Main.Nat | Zero ; 32.31/16.10 32.31/16.10 data Ordering = LT | EQ | GT ; 32.31/16.10 32.31/16.10 data Main.WHNF a = WHNF a ; 32.31/16.10 32.31/16.10 compareMyInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 compareMyInt = primCmpInt; 32.31/16.10 32.31/16.10 dsEm :: (b -> a) -> b -> a; 32.31/16.10 dsEm f x = Main.seq x (f x); 32.31/16.10 32.31/16.10 enforceWHNF :: Main.WHNF b -> a -> a; 32.31/16.10 enforceWHNF (Main.WHNF x) y = y; 32.31/16.10 32.31/16.10 enumFromToMyInt :: MyInt -> MyInt -> List MyInt; 32.31/16.10 enumFromToMyInt = numericEnumFromTo; 32.31/16.10 32.31/16.10 esEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 esEsOrdering LT LT = MyTrue; 32.31/16.10 esEsOrdering LT EQ = MyFalse; 32.31/16.10 esEsOrdering LT GT = MyFalse; 32.31/16.10 esEsOrdering EQ LT = MyFalse; 32.31/16.10 esEsOrdering EQ EQ = MyTrue; 32.31/16.10 esEsOrdering EQ GT = MyFalse; 32.31/16.10 esEsOrdering GT LT = MyFalse; 32.31/16.10 esEsOrdering GT EQ = MyFalse; 32.31/16.10 esEsOrdering GT GT = MyTrue; 32.31/16.10 32.31/16.10 flip :: (a -> b -> c) -> b -> a -> c; 32.31/16.10 flip f x y = f y x; 32.31/16.10 32.31/16.10 fromIntMyInt :: MyInt -> MyInt; 32.31/16.10 fromIntMyInt x = x; 32.31/16.10 32.31/16.10 fsEsOrdering :: Ordering -> Ordering -> MyBool; 32.31/16.10 fsEsOrdering x y = not (esEsOrdering x y); 32.31/16.10 32.31/16.10 ltEsMyInt :: MyInt -> MyInt -> MyBool; 32.31/16.10 ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT; 32.31/16.10 32.31/16.10 not :: MyBool -> MyBool; 32.31/16.10 not MyTrue = MyFalse; 32.31/16.10 not MyFalse = MyTrue; 32.31/16.10 32.31/16.10 numericEnumFrom n = Cons n (dsEm numericEnumFrom (psMyInt n (fromIntMyInt (Main.Pos (Main.Succ Main.Zero))))); 32.31/16.10 32.31/16.10 numericEnumFromTo n m = takeWhile (flip ltEsMyInt m) (numericEnumFrom n); 32.31/16.10 32.31/16.10 otherwise :: MyBool; 32.31/16.10 otherwise = MyTrue; 32.31/16.10 32.31/16.10 primCmpInt :: MyInt -> MyInt -> Ordering; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 32.31/16.10 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 32.31/16.10 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 32.31/16.10 32.31/16.10 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 32.31/16.10 primCmpNat Main.Zero Main.Zero = EQ; 32.31/16.10 primCmpNat Main.Zero (Main.Succ y) = LT; 32.31/16.10 primCmpNat (Main.Succ x) Main.Zero = GT; 32.31/16.10 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 32.31/16.10 32.31/16.10 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 32.31/16.10 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 32.31/16.10 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 32.31/16.10 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 32.31/16.10 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 32.31/16.10 32.31/16.10 primPlusInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x; 32.31/16.10 primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y); 32.31/16.10 primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y); 32.31/16.10 32.31/16.10 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 32.31/16.10 primPlusNat Main.Zero Main.Zero = Main.Zero; 32.31/16.10 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 32.31/16.10 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 32.31/16.10 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 32.31/16.10 32.31/16.10 psMyInt :: MyInt -> MyInt -> MyInt; 32.31/16.10 psMyInt = primPlusInt; 32.31/16.10 32.31/16.10 seq :: a -> b -> b; 32.31/16.10 seq x y = Main.enforceWHNF (Main.WHNF x) y; 32.31/16.10 32.31/16.10 takeWhile :: (a -> MyBool) -> List a -> List a; 32.31/16.10 takeWhile p Nil = takeWhile3 p Nil; 32.31/16.10 takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs); 32.31/16.10 32.31/16.10 takeWhile0 p x xs MyTrue = Nil; 32.31/16.10 32.31/16.10 takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs); 32.31/16.10 takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise; 32.31/16.10 32.31/16.10 takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x); 32.31/16.10 32.31/16.10 takeWhile3 p Nil = Nil; 32.31/16.10 takeWhile3 vx vy = takeWhile2 vx vy; 32.31/16.10 32.31/16.10 } 32.31/16.10 32.31/16.10 ---------------------------------------- 32.31/16.10 32.31/16.10 (5) Narrow (SOUND) 32.31/16.10 Haskell To QDPs 32.31/16.10 32.31/16.10 digraph dp_graph { 32.31/16.10 node [outthreshold=100, inthreshold=100];1[label="enumFromToMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 32.31/16.10 3[label="enumFromToMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 32.31/16.10 4[label="enumFromToMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 32.31/16.10 5[label="numericEnumFromTo vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 32.31/16.10 6[label="takeWhile (flip ltEsMyInt vz4) (numericEnumFrom vz3)",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3]; 32.31/16.10 7[label="takeWhile (flip ltEsMyInt vz4) (Cons vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 32.31/16.10 8[label="takeWhile2 (flip ltEsMyInt vz4) (Cons vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 32.31/16.10 9[label="takeWhile1 (flip ltEsMyInt vz4) vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))) (flip ltEsMyInt vz4 vz3)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 32.31/16.10 10[label="takeWhile1 (flip ltEsMyInt vz4) vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))) (ltEsMyInt vz3 vz4)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 32.31/16.10 11[label="takeWhile1 (flip ltEsMyInt vz4) vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))) (fsEsOrdering (compareMyInt vz3 vz4) GT)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 32.31/16.10 12[label="takeWhile1 (flip ltEsMyInt vz4) vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (compareMyInt vz3 vz4) GT))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 32.31/16.10 13[label="takeWhile1 (flip ltEsMyInt vz4) vz3 (dsEm numericEnumFrom (psMyInt vz3 (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt vz3 vz4) GT))",fontsize=16,color="burlywood",shape="box"];810[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];13 -> 810[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 810 -> 14[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 811[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];13 -> 811[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 811 -> 15[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 14[label="takeWhile1 (flip ltEsMyInt vz4) (Pos vz30) (dsEm numericEnumFrom (psMyInt (Pos vz30) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos vz30) vz4) GT))",fontsize=16,color="burlywood",shape="box"];812[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 812[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 812 -> 16[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 813[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 813[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 813 -> 17[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 15[label="takeWhile1 (flip ltEsMyInt vz4) (Neg vz30) (dsEm numericEnumFrom (psMyInt (Neg vz30) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg vz30) vz4) GT))",fontsize=16,color="burlywood",shape="box"];814[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];15 -> 814[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 814 -> 18[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 815[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 815[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 815 -> 19[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 16[label="takeWhile1 (flip ltEsMyInt vz4) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz300)) vz4) GT))",fontsize=16,color="burlywood",shape="box"];816[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];16 -> 816[label="",style="solid", color="burlywood", weight=9]; 32.31/16.10 816 -> 20[label="",style="solid", color="burlywood", weight=3]; 32.31/16.10 817[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];16 -> 817[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 817 -> 21[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 17[label="takeWhile1 (flip ltEsMyInt vz4) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) vz4) GT))",fontsize=16,color="burlywood",shape="box"];818[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 818[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 818 -> 22[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 819[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];17 -> 819[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 819 -> 23[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 18[label="takeWhile1 (flip ltEsMyInt vz4) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz300)) vz4) GT))",fontsize=16,color="burlywood",shape="box"];820[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 820[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 820 -> 24[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 821[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];18 -> 821[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 821 -> 25[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 19[label="takeWhile1 (flip ltEsMyInt vz4) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) vz4) GT))",fontsize=16,color="burlywood",shape="box"];822[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];19 -> 822[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 822 -> 26[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 823[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];19 -> 823[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 823 -> 27[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 20[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz300)) (Pos vz40)) GT))",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3]; 32.31/16.11 21[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos (Succ vz300)) (Neg vz40)) GT))",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3]; 32.31/16.11 22[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos vz40)) GT))",fontsize=16,color="burlywood",shape="box"];824[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 824[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 824 -> 30[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 825[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 825[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 825 -> 31[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 23[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Neg vz40)) GT))",fontsize=16,color="burlywood",shape="box"];826[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 826[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 826 -> 32[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 827[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 827[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 827 -> 33[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 24[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz300)) (Pos vz40)) GT))",fontsize=16,color="black",shape="box"];24 -> 34[label="",style="solid", color="black", weight=3]; 32.31/16.11 25[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg (Succ vz300)) (Neg vz40)) GT))",fontsize=16,color="black",shape="box"];25 -> 35[label="",style="solid", color="black", weight=3]; 32.31/16.11 26[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos vz40)) GT))",fontsize=16,color="burlywood",shape="box"];828[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];26 -> 828[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 828 -> 36[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 829[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 829[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 829 -> 37[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 27[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Neg vz40)) GT))",fontsize=16,color="burlywood",shape="box"];830[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];27 -> 830[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 830 -> 38[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 831[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 831[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 831 -> 39[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 28[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz300) vz40) GT))",fontsize=16,color="burlywood",shape="box"];832[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];28 -> 832[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 832 -> 40[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 833[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 833[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 833 -> 41[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 29[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3]; 32.31/16.11 30[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vz400))) GT))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3]; 32.31/16.11 31[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"];31 -> 44[label="",style="solid", color="black", weight=3]; 32.31/16.11 32[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Pos Zero) (Neg (Succ vz400))) GT))",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 32.31/16.11 33[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"];33 -> 46[label="",style="solid", color="black", weight=3]; 32.31/16.11 34[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];34 -> 47[label="",style="solid", color="black", weight=3]; 32.31/16.11 35[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz40 (Succ vz300)) GT))",fontsize=16,color="burlywood",shape="box"];834[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];35 -> 834[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 834 -> 48[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 835[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 835[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 835 -> 49[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 36[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vz400))) GT))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3]; 32.31/16.11 37[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"];37 -> 51[label="",style="solid", color="black", weight=3]; 32.31/16.11 38[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpInt (Neg Zero) (Neg (Succ vz400))) GT))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 32.31/16.11 39[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"];39 -> 53[label="",style="solid", color="black", weight=3]; 32.31/16.11 40[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz300) (Succ vz400)) GT))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3]; 32.31/16.11 41[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz300) Zero) GT))",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3]; 32.31/16.11 42[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3]; 32.31/16.11 43[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz400)) GT))",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3]; 32.31/16.11 44[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"];44 -> 58[label="",style="solid", color="black", weight=3]; 32.31/16.11 45[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3]; 32.31/16.11 46[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"];46 -> 60[label="",style="solid", color="black", weight=3]; 32.31/16.11 47[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3]; 32.31/16.11 48[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz400) (Succ vz300)) GT))",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3]; 32.31/16.11 49[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz300)) GT))",fontsize=16,color="black",shape="box"];49 -> 63[label="",style="solid", color="black", weight=3]; 32.31/16.11 50[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];50 -> 64[label="",style="solid", color="black", weight=3]; 32.31/16.11 51[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"];51 -> 65[label="",style="solid", color="black", weight=3]; 32.31/16.11 52[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz400) Zero) GT))",fontsize=16,color="black",shape="box"];52 -> 66[label="",style="solid", color="black", weight=3]; 32.31/16.11 53[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"];53 -> 67[label="",style="solid", color="black", weight=3]; 32.31/16.11 54 -> 446[label="",style="dashed", color="red", weight=0]; 32.31/16.11 54[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz300 vz400) GT))",fontsize=16,color="magenta"];54 -> 447[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 54 -> 448[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 54 -> 449[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 54 -> 450[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 55[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];55 -> 70[label="",style="solid", color="black", weight=3]; 32.31/16.11 56[label="takeWhile1 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];56 -> 71[label="",style="solid", color="black", weight=3]; 32.31/16.11 57[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];57 -> 72[label="",style="solid", color="black", weight=3]; 32.31/16.11 58[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"];58 -> 73[label="",style="solid", color="black", weight=3]; 32.31/16.11 59[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];59 -> 74[label="",style="solid", color="black", weight=3]; 32.31/16.11 60[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"];60 -> 75[label="",style="solid", color="black", weight=3]; 32.31/16.11 61[label="takeWhile1 (flip ltEsMyInt (Pos vz40)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];61 -> 76[label="",style="solid", color="black", weight=3]; 32.31/16.11 62 -> 722[label="",style="dashed", color="red", weight=0]; 32.31/16.11 62[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz400 vz300) GT))",fontsize=16,color="magenta"];62 -> 723[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 62 -> 724[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 62 -> 725[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 62 -> 726[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 62 -> 727[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 63[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3]; 32.31/16.11 64[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];64 -> 80[label="",style="solid", color="black", weight=3]; 32.31/16.11 65[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"];65 -> 81[label="",style="solid", color="black", weight=3]; 32.31/16.11 66[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];66 -> 82[label="",style="solid", color="black", weight=3]; 32.31/16.11 67[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"];67 -> 83[label="",style="solid", color="black", weight=3]; 32.31/16.11 447[label="vz300",fontsize=16,color="green",shape="box"];448[label="vz300",fontsize=16,color="green",shape="box"];449[label="vz400",fontsize=16,color="green",shape="box"];450[label="vz400",fontsize=16,color="green",shape="box"];446[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz24 vz25) GT))",fontsize=16,color="burlywood",shape="triangle"];836[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];446 -> 836[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 836 -> 475[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 837[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];446 -> 837[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 837 -> 476[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 70[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];70 -> 88[label="",style="solid", color="black", weight=3]; 32.31/16.11 71[label="takeWhile0 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];71 -> 89[label="",style="solid", color="black", weight=3]; 32.31/16.11 72[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];72 -> 90[label="",style="solid", color="black", weight=3]; 32.31/16.11 73[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];73 -> 91[label="",style="solid", color="black", weight=3]; 32.31/16.11 74[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];74 -> 92[label="",style="solid", color="black", weight=3]; 32.31/16.11 75[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];75 -> 93[label="",style="solid", color="black", weight=3]; 32.31/16.11 76[label="Cons (Neg (Succ vz300)) (takeWhile (flip ltEsMyInt (Pos vz40)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];76 -> 94[label="",style="dashed", color="green", weight=3]; 32.31/16.11 723[label="vz300",fontsize=16,color="green",shape="box"];724[label="vz400",fontsize=16,color="green",shape="box"];725 -> 179[label="",style="dashed", color="red", weight=0]; 32.31/16.11 725[label="psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];726[label="vz400",fontsize=16,color="green",shape="box"];727[label="vz300",fontsize=16,color="green",shape="box"];722[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat vz68 vz69) GT))",fontsize=16,color="burlywood",shape="triangle"];838[label="vz68/Succ vz680",fontsize=10,color="white",style="solid",shape="box"];722 -> 838[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 838 -> 778[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 839[label="vz68/Zero",fontsize=10,color="white",style="solid",shape="box"];722 -> 839[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 839 -> 779[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 79[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="box"];79 -> 99[label="",style="solid", color="black", weight=3]; 32.31/16.11 80[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];80 -> 100[label="",style="solid", color="black", weight=3]; 32.31/16.11 81[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (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]; 32.31/16.11 82[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];82 -> 102[label="",style="solid", color="black", weight=3]; 32.31/16.11 83[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];83 -> 103[label="",style="solid", color="black", weight=3]; 32.31/16.11 475[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz240) vz25) GT))",fontsize=16,color="burlywood",shape="box"];840[label="vz25/Succ vz250",fontsize=10,color="white",style="solid",shape="box"];475 -> 840[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 840 -> 482[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 841[label="vz25/Zero",fontsize=10,color="white",style="solid",shape="box"];475 -> 841[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 841 -> 483[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 476[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero vz25) GT))",fontsize=16,color="burlywood",shape="box"];842[label="vz25/Succ vz250",fontsize=10,color="white",style="solid",shape="box"];476 -> 842[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 842 -> 484[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 843[label="vz25/Zero",fontsize=10,color="white",style="solid",shape="box"];476 -> 843[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 843 -> 485[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 88[label="takeWhile1 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];88 -> 108[label="",style="solid", color="black", weight=3]; 32.31/16.11 89[label="takeWhile0 (flip ltEsMyInt (Neg vz40)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];89 -> 109[label="",style="solid", color="black", weight=3]; 32.31/16.11 90[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];90 -> 110[label="",style="solid", color="black", weight=3]; 32.31/16.11 91[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];91 -> 111[label="",style="dashed", color="green", weight=3]; 32.31/16.11 92[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];92 -> 112[label="",style="solid", color="black", weight=3]; 32.31/16.11 93[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];93 -> 113[label="",style="dashed", color="green", weight=3]; 32.31/16.11 94[label="takeWhile (flip ltEsMyInt (Pos vz40)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];94 -> 114[label="",style="solid", color="black", weight=3]; 32.31/16.11 179[label="psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];179 -> 205[label="",style="solid", color="black", weight=3]; 32.31/16.11 778[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat (Succ vz680) vz69) GT))",fontsize=16,color="burlywood",shape="box"];844[label="vz69/Succ vz690",fontsize=10,color="white",style="solid",shape="box"];778 -> 844[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 844 -> 781[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 845[label="vz69/Zero",fontsize=10,color="white",style="solid",shape="box"];778 -> 845[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 845 -> 782[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 779[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat Zero vz69) GT))",fontsize=16,color="burlywood",shape="box"];846[label="vz69/Succ vz690",fontsize=10,color="white",style="solid",shape="box"];779 -> 846[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 846 -> 783[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 847[label="vz69/Zero",fontsize=10,color="white",style="solid",shape="box"];779 -> 847[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 847 -> 784[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 99[label="takeWhile1 (flip ltEsMyInt (Neg Zero)) (Neg (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];99 -> 119[label="",style="solid", color="black", weight=3]; 32.31/16.11 100[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];100 -> 120[label="",style="dashed", color="green", weight=3]; 32.31/16.11 101[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];101 -> 121[label="",style="dashed", color="green", weight=3]; 32.31/16.11 102[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];102 -> 122[label="",style="solid", color="black", weight=3]; 32.31/16.11 103[label="Cons (Neg Zero) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];103 -> 123[label="",style="dashed", color="green", weight=3]; 32.31/16.11 482[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz240) (Succ vz250)) GT))",fontsize=16,color="black",shape="box"];482 -> 489[label="",style="solid", color="black", weight=3]; 32.31/16.11 483[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat (Succ vz240) Zero) GT))",fontsize=16,color="black",shape="box"];483 -> 490[label="",style="solid", color="black", weight=3]; 32.31/16.11 484[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero (Succ vz250)) GT))",fontsize=16,color="black",shape="box"];484 -> 491[label="",style="solid", color="black", weight=3]; 32.31/16.11 485[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat Zero Zero) GT))",fontsize=16,color="black",shape="box"];485 -> 492[label="",style="solid", color="black", weight=3]; 32.31/16.11 108[label="takeWhile0 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];108 -> 129[label="",style="solid", color="black", weight=3]; 32.31/16.11 109[label="Nil",fontsize=16,color="green",shape="box"];110[label="Cons (Pos Zero) (takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];110 -> 130[label="",style="dashed", color="green", weight=3]; 32.31/16.11 111[label="takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];111 -> 131[label="",style="solid", color="black", weight=3]; 32.31/16.11 112[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz400))) (Pos Zero) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];112 -> 132[label="",style="solid", color="black", weight=3]; 32.31/16.11 113[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];113 -> 133[label="",style="solid", color="black", weight=3]; 32.31/16.11 114[label="takeWhile (flip ltEsMyInt (Pos vz40)) (seq (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];114 -> 134[label="",style="solid", color="black", weight=3]; 32.31/16.11 205[label="primPlusInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];205 -> 244[label="",style="solid", color="black", weight=3]; 32.31/16.11 781[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat (Succ vz680) (Succ vz690)) GT))",fontsize=16,color="black",shape="box"];781 -> 787[label="",style="solid", color="black", weight=3]; 32.31/16.11 782[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat (Succ vz680) Zero) GT))",fontsize=16,color="black",shape="box"];782 -> 788[label="",style="solid", color="black", weight=3]; 32.31/16.11 783[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat Zero (Succ vz690)) GT))",fontsize=16,color="black",shape="box"];783 -> 789[label="",style="solid", color="black", weight=3]; 32.31/16.11 784[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat Zero Zero) GT))",fontsize=16,color="black",shape="box"];784 -> 790[label="",style="solid", color="black", weight=3]; 32.31/16.11 119[label="Cons (Neg (Succ vz300)) (takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];119 -> 140[label="",style="dashed", color="green", weight=3]; 32.31/16.11 120[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];120 -> 141[label="",style="solid", color="black", weight=3]; 32.31/16.11 121[label="takeWhile (flip ltEsMyInt (Pos Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];121 -> 142[label="",style="solid", color="black", weight=3]; 32.31/16.11 122[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];122 -> 143[label="",style="solid", color="black", weight=3]; 32.31/16.11 123[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];123 -> 144[label="",style="solid", color="black", weight=3]; 32.31/16.11 489 -> 446[label="",style="dashed", color="red", weight=0]; 32.31/16.11 489[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering (primCmpNat vz240 vz250) GT))",fontsize=16,color="magenta"];489 -> 504[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 489 -> 505[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 490[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];490 -> 506[label="",style="solid", color="black", weight=3]; 32.31/16.11 491[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];491 -> 507[label="",style="solid", color="black", weight=3]; 32.31/16.11 492[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];492 -> 508[label="",style="solid", color="black", weight=3]; 32.31/16.11 129[label="takeWhile0 (flip ltEsMyInt (Pos Zero)) (Pos (Succ vz300)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];129 -> 152[label="",style="solid", color="black", weight=3]; 32.31/16.11 130[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (dsEm numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];130 -> 153[label="",style="solid", color="black", weight=3]; 32.31/16.11 131[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"];131 -> 154[label="",style="solid", color="black", weight=3]; 32.31/16.11 132[label="Nil",fontsize=16,color="green",shape="box"];133[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"];133 -> 155[label="",style="solid", color="black", weight=3]; 32.31/16.11 134[label="takeWhile (flip ltEsMyInt (Pos vz40)) (enforceWHNF (WHNF (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];134 -> 156[label="",style="solid", color="black", weight=3]; 32.31/16.11 244[label="primPlusInt (Neg (Succ vz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];244 -> 272[label="",style="solid", color="black", weight=3]; 32.31/16.11 787 -> 722[label="",style="dashed", color="red", weight=0]; 32.31/16.11 787[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering (primCmpNat vz680 vz690) GT))",fontsize=16,color="magenta"];787 -> 793[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 787 -> 794[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 788[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];788 -> 795[label="",style="solid", color="black", weight=3]; 32.31/16.11 789[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];789 -> 796[label="",style="solid", color="black", weight=3]; 32.31/16.11 790[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];790 -> 797[label="",style="solid", color="black", weight=3]; 32.31/16.11 140[label="takeWhile (flip ltEsMyInt (Neg Zero)) (dsEm numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];140 -> 164[label="",style="solid", color="black", weight=3]; 32.31/16.11 141[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (seq (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];141 -> 165[label="",style="solid", color="black", weight=3]; 32.31/16.11 142[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"];142 -> 166[label="",style="solid", color="black", weight=3]; 32.31/16.11 143[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz400))) (Neg Zero) (dsEm numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];143 -> 167[label="",style="solid", color="black", weight=3]; 32.31/16.11 144[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"];144 -> 168[label="",style="solid", color="black", weight=3]; 32.31/16.11 504[label="vz240",fontsize=16,color="green",shape="box"];505[label="vz250",fontsize=16,color="green",shape="box"];506[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not MyTrue)",fontsize=16,color="black",shape="box"];506 -> 520[label="",style="solid", color="black", weight=3]; 32.31/16.11 507[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="black",shape="triangle"];507 -> 521[label="",style="solid", color="black", weight=3]; 32.31/16.11 508 -> 507[label="",style="dashed", color="red", weight=0]; 32.31/16.11 508[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (not MyFalse)",fontsize=16,color="magenta"];152[label="Nil",fontsize=16,color="green",shape="box"];153[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (seq (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];153 -> 176[label="",style="solid", color="black", weight=3]; 32.31/16.11 154[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"];154 -> 177[label="",style="solid", color="black", weight=3]; 32.31/16.11 155 -> 218[label="",style="dashed", color="red", weight=0]; 32.31/16.11 155[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"];155 -> 219[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 155 -> 220[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 156 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 156[label="takeWhile (flip ltEsMyInt (Pos vz40)) (numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];156 -> 179[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 156 -> 180[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 272[label="primMinusNat (Succ Zero) (Succ vz300)",fontsize=16,color="black",shape="box"];272 -> 296[label="",style="solid", color="black", weight=3]; 32.31/16.11 793[label="vz680",fontsize=16,color="green",shape="box"];794[label="vz690",fontsize=16,color="green",shape="box"];795[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not MyTrue)",fontsize=16,color="black",shape="box"];795 -> 798[label="",style="solid", color="black", weight=3]; 32.31/16.11 796[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not MyFalse)",fontsize=16,color="black",shape="triangle"];796 -> 799[label="",style="solid", color="black", weight=3]; 32.31/16.11 797 -> 796[label="",style="dashed", color="red", weight=0]; 32.31/16.11 797[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) (not MyFalse)",fontsize=16,color="magenta"];164[label="takeWhile (flip ltEsMyInt (Neg Zero)) (seq (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];164 -> 188[label="",style="solid", color="black", weight=3]; 32.31/16.11 165[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (enforceWHNF (WHNF (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];165 -> 189[label="",style="solid", color="black", weight=3]; 32.31/16.11 166[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"];166 -> 190[label="",style="solid", color="black", weight=3]; 32.31/16.11 167[label="Nil",fontsize=16,color="green",shape="box"];168 -> 218[label="",style="dashed", color="red", weight=0]; 32.31/16.11 168[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"];168 -> 221[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 168 -> 222[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 520[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) MyFalse",fontsize=16,color="black",shape="box"];520 -> 529[label="",style="solid", color="black", weight=3]; 32.31/16.11 521[label="takeWhile1 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];521 -> 530[label="",style="solid", color="black", weight=3]; 32.31/16.11 176[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (enforceWHNF (WHNF (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];176 -> 200[label="",style="solid", color="black", weight=3]; 32.31/16.11 177 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 177[label="takeWhile (flip ltEsMyInt (Pos Zero)) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];177 -> 201[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 177 -> 202[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 219[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];219 -> 225[label="",style="solid", color="black", weight=3]; 32.31/16.11 220 -> 219[label="",style="dashed", color="red", weight=0]; 32.31/16.11 220[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];218[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF vz11) (numericEnumFrom vz10))",fontsize=16,color="black",shape="triangle"];218 -> 226[label="",style="solid", color="black", weight=3]; 32.31/16.11 180[label="Pos vz40",fontsize=16,color="green",shape="box"];296[label="primMinusNat Zero vz300",fontsize=16,color="burlywood",shape="box"];848[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];296 -> 848[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 848 -> 321[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 849[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];296 -> 849[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 849 -> 322[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 798[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) MyFalse",fontsize=16,color="black",shape="box"];798 -> 800[label="",style="solid", color="black", weight=3]; 32.31/16.11 799[label="takeWhile1 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) MyTrue",fontsize=16,color="black",shape="box"];799 -> 801[label="",style="solid", color="black", weight=3]; 32.31/16.11 188 -> 218[label="",style="dashed", color="red", weight=0]; 32.31/16.11 188[label="takeWhile (flip ltEsMyInt (Neg Zero)) (enforceWHNF (WHNF (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];188 -> 223[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 188 -> 224[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 189 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 189[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];189 -> 227[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 189 -> 228[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 190 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 190[label="takeWhile (flip ltEsMyInt (Pos Zero)) (numericEnumFrom (psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];190 -> 229[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 190 -> 230[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 221[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="triangle"];221 -> 231[label="",style="solid", color="black", weight=3]; 32.31/16.11 222 -> 221[label="",style="dashed", color="red", weight=0]; 32.31/16.11 222[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];529[label="takeWhile0 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) otherwise",fontsize=16,color="black",shape="box"];529 -> 535[label="",style="solid", color="black", weight=3]; 32.31/16.11 530[label="Cons (Pos (Succ vz23)) (takeWhile (flip ltEsMyInt (Pos (Succ vz22))) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="green",shape="box"];530 -> 536[label="",style="dashed", color="green", weight=3]; 32.31/16.11 200 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 200[label="takeWhile (flip ltEsMyInt (Pos (Succ vz400))) (numericEnumFrom (psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];200 -> 242[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 200 -> 243[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 201 -> 219[label="",style="dashed", color="red", weight=0]; 32.31/16.11 201[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];202[label="Pos Zero",fontsize=16,color="green",shape="box"];225[label="primPlusInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];225 -> 259[label="",style="solid", color="black", weight=3]; 32.31/16.11 226 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 226[label="takeWhile (flip ltEsMyInt (Neg Zero)) (numericEnumFrom vz10)",fontsize=16,color="magenta"];226 -> 260[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 226 -> 261[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 321[label="primMinusNat Zero (Succ vz3000)",fontsize=16,color="black",shape="box"];321 -> 343[label="",style="solid", color="black", weight=3]; 32.31/16.11 322[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];322 -> 344[label="",style="solid", color="black", weight=3]; 32.31/16.11 800[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) otherwise",fontsize=16,color="black",shape="box"];800 -> 802[label="",style="solid", color="black", weight=3]; 32.31/16.11 801[label="Cons (Neg (Succ vz66)) (takeWhile (flip ltEsMyInt (Neg (Succ vz65))) (dsEm numericEnumFrom vz67))",fontsize=16,color="green",shape="box"];801 -> 803[label="",style="dashed", color="green", weight=3]; 32.31/16.11 223 -> 179[label="",style="dashed", color="red", weight=0]; 32.31/16.11 223[label="psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];224 -> 179[label="",style="dashed", color="red", weight=0]; 32.31/16.11 224[label="psMyInt (Neg (Succ vz300)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];227 -> 221[label="",style="dashed", color="red", weight=0]; 32.31/16.11 227[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];228[label="Pos (Succ vz400)",fontsize=16,color="green",shape="box"];229 -> 221[label="",style="dashed", color="red", weight=0]; 32.31/16.11 229[label="psMyInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];230[label="Pos Zero",fontsize=16,color="green",shape="box"];231[label="primPlusInt (Neg Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];231 -> 262[label="",style="solid", color="black", weight=3]; 32.31/16.11 535[label="takeWhile0 (flip ltEsMyInt (Pos (Succ vz22))) (Pos (Succ vz23)) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) MyTrue",fontsize=16,color="black",shape="box"];535 -> 541[label="",style="solid", color="black", weight=3]; 32.31/16.11 536[label="takeWhile (flip ltEsMyInt (Pos (Succ vz22))) (dsEm numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];536 -> 542[label="",style="solid", color="black", weight=3]; 32.31/16.11 242 -> 219[label="",style="dashed", color="red", weight=0]; 32.31/16.11 242[label="psMyInt (Pos Zero) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];243[label="Pos (Succ vz400)",fontsize=16,color="green",shape="box"];259[label="primPlusInt (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];259 -> 284[label="",style="solid", color="black", weight=3]; 32.31/16.11 260[label="vz10",fontsize=16,color="green",shape="box"];261[label="Neg Zero",fontsize=16,color="green",shape="box"];343[label="Neg (Succ vz3000)",fontsize=16,color="green",shape="box"];344[label="Pos Zero",fontsize=16,color="green",shape="box"];802[label="takeWhile0 (flip ltEsMyInt (Neg (Succ vz65))) (Neg (Succ vz66)) (dsEm numericEnumFrom vz67) MyTrue",fontsize=16,color="black",shape="box"];802 -> 804[label="",style="solid", color="black", weight=3]; 32.31/16.11 803[label="takeWhile (flip ltEsMyInt (Neg (Succ vz65))) (dsEm numericEnumFrom vz67)",fontsize=16,color="black",shape="box"];803 -> 805[label="",style="solid", color="black", weight=3]; 32.31/16.11 262[label="primPlusInt (Neg Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];262 -> 285[label="",style="solid", color="black", weight=3]; 32.31/16.11 541[label="Nil",fontsize=16,color="green",shape="box"];542[label="takeWhile (flip ltEsMyInt (Pos (Succ vz22))) (seq (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero)))) (numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];542 -> 546[label="",style="solid", color="black", weight=3]; 32.31/16.11 284[label="Pos (primPlusNat Zero (Succ Zero))",fontsize=16,color="green",shape="box"];284 -> 305[label="",style="dashed", color="green", weight=3]; 32.31/16.11 804[label="Nil",fontsize=16,color="green",shape="box"];805[label="takeWhile (flip ltEsMyInt (Neg (Succ vz65))) (seq vz67 (numericEnumFrom vz67))",fontsize=16,color="black",shape="box"];805 -> 806[label="",style="solid", color="black", weight=3]; 32.31/16.11 285[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];285 -> 306[label="",style="solid", color="black", weight=3]; 32.31/16.11 546[label="takeWhile (flip ltEsMyInt (Pos (Succ vz22))) (enforceWHNF (WHNF (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))) (numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];546 -> 582[label="",style="solid", color="black", weight=3]; 32.31/16.11 305[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];305 -> 332[label="",style="solid", color="black", weight=3]; 32.31/16.11 806[label="takeWhile (flip ltEsMyInt (Neg (Succ vz65))) (enforceWHNF (WHNF vz67) (numericEnumFrom vz67))",fontsize=16,color="black",shape="box"];806 -> 807[label="",style="solid", color="black", weight=3]; 32.31/16.11 306[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];582 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 582[label="takeWhile (flip ltEsMyInt (Pos (Succ vz22))) (numericEnumFrom (psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];582 -> 618[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 582 -> 619[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 332[label="Succ Zero",fontsize=16,color="green",shape="box"];807 -> 6[label="",style="dashed", color="red", weight=0]; 32.31/16.11 807[label="takeWhile (flip ltEsMyInt (Neg (Succ vz65))) (numericEnumFrom vz67)",fontsize=16,color="magenta"];807 -> 808[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 807 -> 809[label="",style="dashed", color="magenta", weight=3]; 32.31/16.11 618[label="psMyInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];618 -> 649[label="",style="solid", color="black", weight=3]; 32.31/16.11 619[label="Pos (Succ vz22)",fontsize=16,color="green",shape="box"];808[label="vz67",fontsize=16,color="green",shape="box"];809[label="Neg (Succ vz65)",fontsize=16,color="green",shape="box"];649[label="primPlusInt (Pos (Succ vz23)) (fromIntMyInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];649 -> 662[label="",style="solid", color="black", weight=3]; 32.31/16.11 662[label="primPlusInt (Pos (Succ vz23)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];662 -> 700[label="",style="solid", color="black", weight=3]; 32.31/16.11 700[label="Pos (primPlusNat (Succ vz23) (Succ Zero))",fontsize=16,color="green",shape="box"];700 -> 712[label="",style="dashed", color="green", weight=3]; 32.31/16.11 712[label="primPlusNat (Succ vz23) (Succ Zero)",fontsize=16,color="black",shape="box"];712 -> 721[label="",style="solid", color="black", weight=3]; 32.31/16.11 721[label="Succ (Succ (primPlusNat vz23 Zero))",fontsize=16,color="green",shape="box"];721 -> 780[label="",style="dashed", color="green", weight=3]; 32.31/16.11 780[label="primPlusNat vz23 Zero",fontsize=16,color="burlywood",shape="box"];850[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];780 -> 850[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 850 -> 785[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 851[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];780 -> 851[label="",style="solid", color="burlywood", weight=9]; 32.31/16.11 851 -> 786[label="",style="solid", color="burlywood", weight=3]; 32.31/16.11 785[label="primPlusNat (Succ vz230) Zero",fontsize=16,color="black",shape="box"];785 -> 791[label="",style="solid", color="black", weight=3]; 32.31/16.11 786[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];786 -> 792[label="",style="solid", color="black", weight=3]; 32.31/16.11 791[label="Succ vz230",fontsize=16,color="green",shape="box"];792[label="Zero",fontsize=16,color="green",shape="box"];} 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (6) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile12(vz65, vz66, vz67) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) 32.31/16.11 new_takeWhile(Main.Pos(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Zero), new_psMyInt) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> new_takeWhile1(vz400, vz300, vz300, vz400) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Neg(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Succ(vz400)), new_psMyInt1) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Succ(vz240), Main.Succ(vz250)) -> new_takeWhile1(vz22, vz23, vz240, vz250) 32.31/16.11 new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Succ(vz690)) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Pos(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Zero), new_psMyInt1) 32.31/16.11 new_takeWhile11(vz65, vz66, vz67, Main.Succ(vz680), Main.Succ(vz690)) -> new_takeWhile11(vz65, vz66, vz67, vz680, vz690) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Zero)) -> new_takeWhile(Main.Pos(Main.Succ(vz400)), new_psMyInt) 32.31/16.11 new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Zero) -> new_takeWhile12(vz65, vz66, vz67) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Succ(vz400)), Main.Neg(Main.Succ(vz300))) -> new_takeWhile11(vz400, vz300, new_psMyInt0(vz300), vz400, vz300) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Pos(vz40), Main.Neg(Main.Succ(vz300))) -> new_takeWhile(Main.Pos(vz40), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (7) DependencyGraphProof (EQUIVALENT) 32.31/16.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (8) 32.31/16.11 Complex Obligation (AND) 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (9) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (10) MRRProof (EQUIVALENT) 32.31/16.11 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. 32.31/16.11 32.31/16.11 32.31/16.11 Strictly oriented rules of the TRS R: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 32.31/16.11 Used ordering: Polynomial interpretation [POLO]: 32.31/16.11 32.31/16.11 POL(Main.Neg(x_1)) = x_1 32.31/16.11 POL(Main.Pos(x_1)) = x_1 32.31/16.11 POL(Main.Succ(x_1)) = 2*x_1 32.31/16.11 POL(Main.Zero) = 0 32.31/16.11 POL(new_primPlusNat(x_1)) = 2 + 2*x_1 32.31/16.11 POL(new_psMyInt) = 0 32.31/16.11 POL(new_psMyInt0(x_1)) = x_1 32.31/16.11 POL(new_psMyInt1) = 0 32.31/16.11 POL(new_takeWhile(x_1, x_2)) = 2 + x_1 + x_2 32.31/16.11 POL(new_takeWhile0(x_1, x_2)) = 2 + x_1 + x_2 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (11) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(new_psMyInt, new_psMyInt) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (12) TransformationProof (EQUIVALENT) 32.31/16.11 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]: 32.31/16.11 32.31/16.11 (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)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (13) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (14) QReductionProof (EQUIVALENT) 32.31/16.11 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (15) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(new_psMyInt1, new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (16) TransformationProof (EQUIVALENT) 32.31/16.11 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]: 32.31/16.11 32.31/16.11 (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)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (17) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (18) TransformationProof (EQUIVALENT) 32.31/16.11 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]: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (19) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (20) UsableRulesProof (EQUIVALENT) 32.31/16.11 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. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (21) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (22) QReductionProof (EQUIVALENT) 32.31/16.11 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 32.31/16.11 32.31/16.11 new_psMyInt 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (23) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), new_psMyInt1) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (24) TransformationProof (EQUIVALENT) 32.31/16.11 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]: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (25) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (26) UsableRulesProof (EQUIVALENT) 32.31/16.11 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. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (27) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (28) QReductionProof (EQUIVALENT) 32.31/16.11 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (29) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (30) QDPOrderProof (EQUIVALENT) 32.31/16.11 We use the reduction pair processor [LPAR04,JAR06]. 32.31/16.11 32.31/16.11 32.31/16.11 The following pairs can be oriented strictly and are deleted. 32.31/16.11 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Succ(vz300))) -> new_takeWhile0(new_psMyInt0(vz300), new_psMyInt0(vz300)) 32.31/16.11 The remaining pairs can at least be oriented weakly. 32.31/16.11 Used ordering: Polynomial interpretation [POLO]: 32.31/16.11 32.31/16.11 POL(Main.Neg(x_1)) = x_1 32.31/16.11 POL(Main.Pos(x_1)) = 0 32.31/16.11 POL(Main.Succ(x_1)) = 1 + x_1 32.31/16.11 POL(Main.Zero) = 0 32.31/16.11 POL(new_psMyInt0(x_1)) = x_1 32.31/16.11 POL(new_takeWhile(x_1, x_2)) = x_2 32.31/16.11 POL(new_takeWhile0(x_1, x_2)) = x_2 32.31/16.11 32.31/16.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (31) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (32) TransformationProof (EQUIVALENT) 32.31/16.11 By instantiating [LPAR04] the rule new_takeWhile0(vz11, vz10) -> new_takeWhile(Main.Neg(Main.Zero), vz10) we obtained the following new rules [LPAR04]: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (33) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Pos(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 new_takeWhile(Main.Neg(Main.Zero), Main.Neg(Main.Zero)) -> new_takeWhile0(Main.Pos(Main.Succ(Main.Zero)), Main.Pos(Main.Succ(Main.Zero))) 32.31/16.11 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))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (34) DependencyGraphProof (EQUIVALENT) 32.31/16.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (35) 32.31/16.11 TRUE 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (36) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Succ(vz240), Main.Succ(vz250)) -> new_takeWhile1(vz22, vz23, vz240, vz250) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> new_takeWhile1(vz400, vz300, vz300, vz400) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (37) QDPPairToRuleProof (EQUIVALENT) 32.31/16.11 The dependency pair new_takeWhile1(vz22, vz23, Main.Succ(vz240), Main.Succ(vz250)) -> new_takeWhile1(vz22, vz23, vz240, vz250) was transformed to the following new rules: 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 the following new pairs maintain the fan-in: 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 32.31/16.11 the following new pairs maintain the fan-out: 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (38) 32.31/16.11 Complex Obligation (AND) 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (39) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> new_takeWhile1(vz400, vz300, vz300, vz400) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (40) TransformationProof (EQUIVALENT) 32.31/16.11 By instantiating [LPAR04] the rule new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> new_takeWhile1(vz400, vz300, vz300, vz400) we obtained the following new rules [LPAR04]: 32.31/16.11 32.31/16.11 (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)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (41) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 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) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (42) DependencyGraphProof (EQUIVALENT) 32.31/16.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (43) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.11 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (44) UsableRulesProof (EQUIVALENT) 32.31/16.11 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. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (45) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (46) QReductionProof (EQUIVALENT) 32.31/16.11 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 32.31/16.11 32.31/16.11 new_psMyInt1 32.31/16.11 new_psMyInt0(Main.Zero) 32.31/16.11 new_psMyInt 32.31/16.11 new_psMyInt0(Main.Succ(x0)) 32.31/16.11 32.31/16.11 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (47) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (48) InductionCalculusProof (EQUIVALENT) 32.31/16.11 Note that final constraints are written in bold face. 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We solved constraint (6) using rules (I), (II). 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (1) (new_takeWhile10(x82, x83)=new_takeWhile10(x84, x85) ==> new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 To summarize, we get the following constraints P__>=_ for the following pairs. 32.31/16.11 32.31/16.11 *new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 32.31/16.11 *(H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 *(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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 *(H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 32.31/16.11 *(new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 *(new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (49) 32.31/16.11 Obligation: 32.31/16.11 Q DP problem: 32.31/16.11 The TRS P consists of the following rules: 32.31/16.11 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 The TRS R consists of the following rules: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.11 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.11 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 The set Q consists of the following terms: 32.31/16.11 32.31/16.11 new_primPlusNat(Main.Zero) 32.31/16.11 new_primPlusNat(Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.11 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 We have to consider all minimal (P,Q,R)-chains. 32.31/16.11 ---------------------------------------- 32.31/16.11 32.31/16.11 (50) NonInfProof (EQUIVALENT) 32.31/16.11 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 32.31/16.11 32.31/16.11 Note that final constraints are written in bold face. 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We solved constraint (6) using rules (I), (II). 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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: 32.31/16.11 32.31/16.11 (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))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (1) (new_takeWhile10(x82, x83)=new_takeWhile10(x84, x85) ==> new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 For Pair new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) the following chains were created: 32.31/16.11 *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: 32.31/16.11 32.31/16.11 (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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 32.31/16.11 32.31/16.11 (2) (new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 To summarize, we get the following constraints P__>=_ for the following pairs. 32.31/16.11 32.31/16.11 *new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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))))) 32.31/16.11 32.31/16.11 32.31/16.11 *(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)))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.11 32.31/16.11 *(H(x23, x24, cons_new_takeWhile1(Main.Zero, Main.Succ(x25)))_>=_new_takeWhile1(x23, x24, Main.Zero, Main.Succ(x25))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 *(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)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.11 32.31/16.11 *(H(x66, x67, cons_new_takeWhile1(Main.Zero, Main.Zero))_>=_new_takeWhile1(x66, x67, Main.Zero, Main.Zero)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.11 32.31/16.11 *(new_takeWhile1(x82, x83, Main.Zero, Main.Zero)_>=_new_takeWhile10(x82, x83)) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 *new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.11 32.31/16.11 *(new_takeWhile10(x86, x87)_>=_new_takeWhile(Main.Pos(Main.Succ(x86)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(x87)))))) 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 32.31/16.11 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. 32.31/16.11 32.31/16.11 Using the following integer polynomial ordering the resulting constraints can be solved 32.31/16.11 32.31/16.11 Polynomial interpretation [NONINF]: 32.31/16.11 32.31/16.11 POL(H(x_1, x_2, x_3)) = -1 + x_1 - x_2 - x_3 32.31/16.11 POL(Main.Pos(x_1)) = 1 + x_1 32.31/16.11 POL(Main.Succ(x_1)) = 1 + x_1 32.31/16.11 POL(Main.Zero) = 0 32.31/16.11 POL(anew_new_takeWhile1(x_1, x_2)) = 0 32.31/16.11 POL(c) = -1 32.31/16.11 POL(cons_new_takeWhile1(x_1, x_2)) = 0 32.31/16.11 POL(new_new_takeWhile1(x_1, x_2)) = 0 32.31/16.11 POL(new_primPlusNat(x_1)) = x_1 32.31/16.11 POL(new_takeWhile(x_1, x_2)) = x_1 - x_2 32.31/16.11 POL(new_takeWhile1(x_1, x_2, x_3, x_4)) = -1 + x_1 - x_2 + x_3 32.31/16.11 POL(new_takeWhile10(x_1, x_2)) = -1 + x_1 - x_2 32.31/16.11 32.31/16.11 32.31/16.11 The following pairs are in P_>: 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 The following pairs are in P_bound: 32.31/16.11 new_takeWhile(Main.Pos(Main.Succ(vz400)), Main.Pos(Main.Succ(vz300))) -> H(vz400, vz300, anew_new_takeWhile1(vz300, vz400)) 32.31/16.11 The following rules are usable: 32.31/16.11 new_new_takeWhile1(vz240, vz250) -> anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) 32.31/16.12 Main.Zero -> new_primPlusNat(Main.Zero) 32.31/16.12 Main.Succ(vz230) -> new_primPlusNat(Main.Succ(vz230)) 32.31/16.12 new_new_takeWhile1(vz240, vz250) -> new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) 32.31/16.12 cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.12 cons_new_takeWhile1(Main.Zero, Main.Zero) -> new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (51) 32.31/16.12 Obligation: 32.31/16.12 Q DP problem: 32.31/16.12 The TRS P consists of the following rules: 32.31/16.12 32.31/16.12 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Succ(vz250))) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) 32.31/16.12 new_takeWhile1(vz22, vz23, Main.Zero, Main.Succ(vz250)) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.12 H(vz22, vz23, cons_new_takeWhile1(Main.Zero, Main.Zero)) -> new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) 32.31/16.12 new_takeWhile1(vz22, vz23, Main.Zero, Main.Zero) -> new_takeWhile10(vz22, vz23) 32.31/16.12 new_takeWhile10(vz22, vz23) -> new_takeWhile(Main.Pos(Main.Succ(vz22)), Main.Pos(Main.Succ(Main.Succ(new_primPlusNat(vz23))))) 32.31/16.12 32.31/16.12 The TRS R consists of the following rules: 32.31/16.12 32.31/16.12 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.12 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.12 anew_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.12 new_new_takeWhile1(Main.Succ(vz240), Main.Succ(vz250)) -> new_new_takeWhile1(vz240, vz250) 32.31/16.12 new_new_takeWhile1(Main.Zero, Main.Succ(vz250)) -> cons_new_takeWhile1(Main.Zero, Main.Succ(vz250)) 32.31/16.12 new_new_takeWhile1(Main.Zero, Main.Zero) -> cons_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.12 32.31/16.12 The set Q consists of the following terms: 32.31/16.12 32.31/16.12 new_primPlusNat(Main.Zero) 32.31/16.12 new_primPlusNat(Main.Succ(x0)) 32.31/16.12 new_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.12 anew_new_takeWhile1(Main.Succ(x0), Main.Succ(x1)) 32.31/16.12 new_new_takeWhile1(Main.Zero, Main.Succ(x0)) 32.31/16.12 new_new_takeWhile1(Main.Zero, Main.Zero) 32.31/16.12 32.31/16.12 We have to consider all minimal (P,Q,R)-chains. 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (52) DependencyGraphProof (EQUIVALENT) 32.31/16.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes. 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (53) 32.31/16.12 TRUE 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (54) 32.31/16.12 Obligation: 32.31/16.12 Q DP problem: 32.31/16.12 The TRS P consists of the following rules: 32.31/16.12 32.31/16.12 new_takeWhile1(vz22, vz23, Main.Succ(vz240), Main.Succ(vz250)) -> new_takeWhile1(vz22, vz23, vz240, vz250) 32.31/16.12 32.31/16.12 R is empty. 32.31/16.12 Q is empty. 32.31/16.12 We have to consider all minimal (P,Q,R)-chains. 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (55) QDPSizeChangeProof (EQUIVALENT) 32.31/16.12 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. 32.31/16.12 32.31/16.12 From the DPs we obtained the following set of size-change graphs: 32.31/16.12 *new_takeWhile1(vz22, vz23, Main.Succ(vz240), Main.Succ(vz250)) -> new_takeWhile1(vz22, vz23, vz240, vz250) 32.31/16.12 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 32.31/16.12 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (56) 32.31/16.12 YES 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (57) 32.31/16.12 Obligation: 32.31/16.12 Q DP problem: 32.31/16.12 The TRS P consists of the following rules: 32.31/16.12 32.31/16.12 new_takeWhile(Main.Pos(vz40), Main.Neg(Main.Succ(vz300))) -> new_takeWhile(Main.Pos(vz40), new_psMyInt0(vz300)) 32.31/16.12 32.31/16.12 The TRS R consists of the following rules: 32.31/16.12 32.31/16.12 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.12 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.12 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.12 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.12 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.12 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.12 32.31/16.12 The set Q consists of the following terms: 32.31/16.12 32.31/16.12 new_psMyInt1 32.31/16.12 new_psMyInt0(Main.Zero) 32.31/16.12 new_psMyInt 32.31/16.12 new_psMyInt0(Main.Succ(x0)) 32.31/16.12 new_primPlusNat(Main.Zero) 32.31/16.12 new_primPlusNat(Main.Succ(x0)) 32.31/16.12 32.31/16.12 We have to consider all minimal (P,Q,R)-chains. 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (58) QDPSizeChangeProof (EQUIVALENT) 32.31/16.12 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 32.31/16.12 32.31/16.12 Order:Polynomial interpretation [POLO]: 32.31/16.12 32.31/16.12 POL(Main.Neg(x_1)) = x_1 32.31/16.12 POL(Main.Pos(x_1)) = 1 32.31/16.12 POL(Main.Succ(x_1)) = 1 + x_1 32.31/16.12 POL(Main.Zero) = 1 32.31/16.12 POL(new_psMyInt0(x_1)) = x_1 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 From the DPs we obtained the following set of size-change graphs: 32.31/16.12 *new_takeWhile(Main.Pos(vz40), Main.Neg(Main.Succ(vz300))) -> new_takeWhile(Main.Pos(vz40), new_psMyInt0(vz300)) (allowed arguments on rhs = {1, 2}) 32.31/16.12 The graph contains the following edges 1 >= 1, 2 > 2 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 We oriented the following set of usable rules [AAECC05,FROCOS05]. 32.31/16.12 32.31/16.12 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.12 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (59) 32.31/16.12 YES 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (60) 32.31/16.12 Obligation: 32.31/16.12 Q DP problem: 32.31/16.12 The TRS P consists of the following rules: 32.31/16.12 32.31/16.12 new_takeWhile(Main.Neg(Main.Succ(vz400)), Main.Neg(Main.Succ(vz300))) -> new_takeWhile11(vz400, vz300, new_psMyInt0(vz300), vz400, vz300) 32.31/16.12 new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Succ(vz690)) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) 32.31/16.12 new_takeWhile11(vz65, vz66, vz67, Main.Succ(vz680), Main.Succ(vz690)) -> new_takeWhile11(vz65, vz66, vz67, vz680, vz690) 32.31/16.12 new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Zero) -> new_takeWhile12(vz65, vz66, vz67) 32.31/16.12 new_takeWhile12(vz65, vz66, vz67) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) 32.31/16.12 32.31/16.12 The TRS R consists of the following rules: 32.31/16.12 32.31/16.12 new_psMyInt1 -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.12 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.12 new_primPlusNat(Main.Zero) -> Main.Zero 32.31/16.12 new_primPlusNat(Main.Succ(vz230)) -> Main.Succ(vz230) 32.31/16.12 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.12 new_psMyInt -> Main.Pos(Main.Succ(Main.Zero)) 32.31/16.12 32.31/16.12 The set Q consists of the following terms: 32.31/16.12 32.31/16.12 new_psMyInt1 32.31/16.12 new_psMyInt0(Main.Zero) 32.31/16.12 new_psMyInt 32.31/16.12 new_psMyInt0(Main.Succ(x0)) 32.31/16.12 new_primPlusNat(Main.Zero) 32.31/16.12 new_primPlusNat(Main.Succ(x0)) 32.31/16.12 32.31/16.12 We have to consider all minimal (P,Q,R)-chains. 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (61) QDPSizeChangeProof (EQUIVALENT) 32.31/16.12 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 32.31/16.12 32.31/16.12 Order:Polynomial interpretation [POLO]: 32.31/16.12 32.31/16.12 POL(Main.Neg(x_1)) = x_1 32.31/16.12 POL(Main.Pos(x_1)) = 0 32.31/16.12 POL(Main.Succ(x_1)) = 1 + x_1 32.31/16.12 POL(Main.Zero) = 1 32.31/16.12 POL(new_psMyInt0(x_1)) = x_1 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 From the DPs we obtained the following set of size-change graphs: 32.31/16.12 *new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Succ(vz690)) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) (allowed arguments on rhs = {1, 2}) 32.31/16.12 The graph contains the following edges 3 >= 2 32.31/16.12 32.31/16.12 32.31/16.12 *new_takeWhile12(vz65, vz66, vz67) -> new_takeWhile(Main.Neg(Main.Succ(vz65)), vz67) (allowed arguments on rhs = {1, 2}) 32.31/16.12 The graph contains the following edges 3 >= 2 32.31/16.12 32.31/16.12 32.31/16.12 *new_takeWhile(Main.Neg(Main.Succ(vz400)), Main.Neg(Main.Succ(vz300))) -> new_takeWhile11(vz400, vz300, new_psMyInt0(vz300), vz400, vz300) (allowed arguments on rhs = {1, 2, 3, 4, 5}) 32.31/16.12 The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 1 > 4, 2 > 5 32.31/16.12 32.31/16.12 32.31/16.12 *new_takeWhile11(vz65, vz66, vz67, Main.Succ(vz680), Main.Succ(vz690)) -> new_takeWhile11(vz65, vz66, vz67, vz680, vz690) (allowed arguments on rhs = {1, 2, 3, 4, 5}) 32.31/16.12 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5 32.31/16.12 32.31/16.12 32.31/16.12 *new_takeWhile11(vz65, vz66, vz67, Main.Zero, Main.Zero) -> new_takeWhile12(vz65, vz66, vz67) (allowed arguments on rhs = {1, 2, 3}) 32.31/16.12 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 32.31/16.12 32.31/16.12 32.31/16.12 32.31/16.12 We oriented the following set of usable rules [AAECC05,FROCOS05]. 32.31/16.12 32.31/16.12 new_psMyInt0(Main.Zero) -> Main.Pos(Main.Zero) 32.31/16.12 new_psMyInt0(Main.Succ(vz3000)) -> Main.Neg(Main.Succ(vz3000)) 32.31/16.12 32.31/16.12 ---------------------------------------- 32.31/16.12 32.31/16.12 (62) 32.31/16.12 YES 32.31/16.17 EOF