15.52/6.89 YES 18.07/7.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 18.07/7.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 18.07/7.58 18.07/7.58 18.07/7.58 H-Termination with start terms of the given HASKELL could be proven: 18.07/7.58 18.07/7.58 (0) HASKELL 18.07/7.58 (1) BR [EQUIVALENT, 0 ms] 18.07/7.58 (2) HASKELL 18.07/7.58 (3) COR [EQUIVALENT, 0 ms] 18.07/7.58 (4) HASKELL 18.07/7.58 (5) Narrow [SOUND, 0 ms] 18.07/7.58 (6) AND 18.07/7.58 (7) QDP 18.07/7.58 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 18.07/7.58 (9) YES 18.07/7.58 (10) QDP 18.07/7.58 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 18.07/7.58 (12) YES 18.07/7.58 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (0) 18.07/7.58 Obligation: 18.07/7.58 mainModule Main 18.07/7.58 module Main where { 18.07/7.58 import qualified Prelude; 18.07/7.58 data Main.Char = Char MyInt ; 18.07/7.58 18.07/7.58 data List a = Cons a (List a) | Nil ; 18.07/7.58 18.07/7.58 data MyBool = MyTrue | MyFalse ; 18.07/7.58 18.07/7.58 data MyInt = Pos Main.Nat | Neg Main.Nat ; 18.07/7.58 18.07/7.58 data Main.Nat = Succ Main.Nat | Zero ; 18.07/7.58 18.07/7.58 data Ordering = LT | EQ | GT ; 18.07/7.58 18.07/7.58 compareMyInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 compareMyInt = primCmpInt; 18.07/7.58 18.07/7.58 esEsOrdering :: Ordering -> Ordering -> MyBool; 18.07/7.58 esEsOrdering LT LT = MyTrue; 18.07/7.58 esEsOrdering LT EQ = MyFalse; 18.07/7.58 esEsOrdering LT GT = MyFalse; 18.07/7.58 esEsOrdering EQ LT = MyFalse; 18.07/7.58 esEsOrdering EQ EQ = MyTrue; 18.07/7.58 esEsOrdering EQ GT = MyFalse; 18.07/7.58 esEsOrdering GT LT = MyFalse; 18.07/7.58 esEsOrdering GT EQ = MyFalse; 18.07/7.58 esEsOrdering GT GT = MyTrue; 18.07/7.58 18.07/7.58 gtMyInt :: MyInt -> MyInt -> MyBool; 18.07/7.58 gtMyInt x y = esEsOrdering (compareMyInt x y) GT; 18.07/7.58 18.07/7.58 primCmpInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 18.07/7.58 18.07/7.58 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 18.07/7.58 primCmpNat Main.Zero Main.Zero = EQ; 18.07/7.58 primCmpNat Main.Zero (Main.Succ y) = LT; 18.07/7.58 primCmpNat (Main.Succ x) Main.Zero = GT; 18.07/7.58 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 18.07/7.58 18.07/7.58 psPs :: List a -> List a -> List a; 18.07/7.58 psPs Nil ys = ys; 18.07/7.58 psPs (Cons x xs) ys = Cons x (psPs xs ys); 18.07/7.58 18.07/7.58 pt :: (c -> b) -> (a -> c) -> a -> b; 18.07/7.58 pt f g x = f (g x); 18.07/7.58 18.07/7.58 showChar :: Main.Char -> List Main.Char -> List Main.Char; 18.07/7.58 showChar = Cons; 18.07/7.58 18.07/7.58 showParen :: MyBool -> (List Main.Char -> List Main.Char) -> List Main.Char -> List Main.Char; 18.07/7.58 showParen b p = showParen0 p b; 18.07/7.58 18.07/7.58 showParen0 p MyTrue = pt (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))) (pt p (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ 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(Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))))))))))))))))))))))))) Nil)))))); 18.07/7.58 18.07/7.58 } 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (1) BR (EQUIVALENT) 18.07/7.58 Replaced joker patterns by fresh variables and removed binding patterns. 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (2) 18.07/7.58 Obligation: 18.07/7.58 mainModule Main 18.07/7.58 module Main where { 18.07/7.58 import qualified Prelude; 18.07/7.58 data Main.Char = Char MyInt ; 18.07/7.58 18.07/7.58 data List a = Cons a (List a) | Nil ; 18.07/7.58 18.07/7.58 data MyBool = MyTrue | MyFalse ; 18.07/7.58 18.07/7.58 data MyInt = Pos Main.Nat | Neg Main.Nat ; 18.07/7.58 18.07/7.58 data Main.Nat = Succ Main.Nat | Zero ; 18.07/7.58 18.07/7.58 data Ordering = LT | EQ | GT ; 18.07/7.58 18.07/7.58 compareMyInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 compareMyInt = primCmpInt; 18.07/7.58 18.07/7.58 esEsOrdering :: Ordering -> Ordering -> MyBool; 18.07/7.58 esEsOrdering LT LT = MyTrue; 18.07/7.58 esEsOrdering LT EQ = MyFalse; 18.07/7.58 esEsOrdering LT GT = MyFalse; 18.07/7.58 esEsOrdering EQ LT = MyFalse; 18.07/7.58 esEsOrdering EQ EQ = MyTrue; 18.07/7.58 esEsOrdering EQ GT = MyFalse; 18.07/7.58 esEsOrdering GT LT = MyFalse; 18.07/7.58 esEsOrdering GT EQ = MyFalse; 18.07/7.58 esEsOrdering GT GT = MyTrue; 18.07/7.58 18.07/7.58 gtMyInt :: MyInt -> MyInt -> MyBool; 18.07/7.58 gtMyInt x y = esEsOrdering (compareMyInt x y) GT; 18.07/7.58 18.07/7.58 primCmpInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 18.07/7.58 18.07/7.58 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 18.07/7.58 primCmpNat Main.Zero Main.Zero = EQ; 18.07/7.58 primCmpNat Main.Zero (Main.Succ y) = LT; 18.07/7.58 primCmpNat (Main.Succ x) Main.Zero = GT; 18.07/7.58 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 18.07/7.58 18.07/7.58 psPs :: List a -> List a -> List a; 18.07/7.58 psPs Nil ys = ys; 18.07/7.58 psPs (Cons x xs) ys = Cons x (psPs xs ys); 18.07/7.58 18.07/7.58 pt :: (a -> c) -> (b -> a) -> b -> c; 18.07/7.58 pt f g x = f (g x); 18.07/7.58 18.07/7.58 showChar :: Main.Char -> List Main.Char -> List Main.Char; 18.07/7.58 showChar = Cons; 18.07/7.58 18.07/7.58 showParen :: MyBool -> (List Main.Char -> List Main.Char) -> List Main.Char -> List Main.Char; 18.07/7.58 showParen b p = showParen0 p b; 18.07/7.58 18.07/7.58 showParen0 p MyTrue = pt (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))) (pt p (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))); 18.07/7.58 showParen0 p MyFalse = p; 18.07/7.58 18.07/7.58 showString :: List Main.Char -> List Main.Char -> List Main.Char; 18.07/7.58 showString = psPs; 18.07/7.58 18.07/7.58 showsPrecMyBool :: MyInt -> MyBool -> List Main.Char -> List Main.Char; 18.07/7.58 showsPrecMyBool d MyFalse = showParen (gtMyInt d (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))) (showString (Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ 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(Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) (Cons (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero)))))))))))))))))))))))))))))))))) Nil)))))); 18.07/7.58 18.07/7.58 } 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (3) COR (EQUIVALENT) 18.07/7.58 Cond Reductions: 18.07/7.58 The following Function with conditions 18.07/7.58 "undefined |Falseundefined; 18.07/7.58 " 18.07/7.58 is transformed to 18.07/7.58 "undefined = undefined1; 18.07/7.58 " 18.07/7.58 "undefined0 True = undefined; 18.07/7.58 " 18.07/7.58 "undefined1 = undefined0 False; 18.07/7.58 " 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (4) 18.07/7.58 Obligation: 18.07/7.58 mainModule Main 18.07/7.58 module Main where { 18.07/7.58 import qualified Prelude; 18.07/7.58 data Main.Char = Char MyInt ; 18.07/7.58 18.07/7.58 data List a = Cons a (List a) | Nil ; 18.07/7.58 18.07/7.58 data MyBool = MyTrue | MyFalse ; 18.07/7.58 18.07/7.58 data MyInt = Pos Main.Nat | Neg Main.Nat ; 18.07/7.58 18.07/7.58 data Main.Nat = Succ Main.Nat | Zero ; 18.07/7.58 18.07/7.58 data Ordering = LT | EQ | GT ; 18.07/7.58 18.07/7.58 compareMyInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 compareMyInt = primCmpInt; 18.07/7.58 18.07/7.58 esEsOrdering :: Ordering -> Ordering -> MyBool; 18.07/7.58 esEsOrdering LT LT = MyTrue; 18.07/7.58 esEsOrdering LT EQ = MyFalse; 18.07/7.58 esEsOrdering LT GT = MyFalse; 18.07/7.58 esEsOrdering EQ LT = MyFalse; 18.07/7.58 esEsOrdering EQ EQ = MyTrue; 18.07/7.58 esEsOrdering EQ GT = MyFalse; 18.07/7.58 esEsOrdering GT LT = MyFalse; 18.07/7.58 esEsOrdering GT EQ = MyFalse; 18.07/7.58 esEsOrdering GT GT = MyTrue; 18.07/7.58 18.07/7.58 gtMyInt :: MyInt -> MyInt -> MyBool; 18.07/7.58 gtMyInt x y = esEsOrdering (compareMyInt x y) GT; 18.07/7.58 18.07/7.58 primCmpInt :: MyInt -> MyInt -> Ordering; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 18.07/7.58 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 18.07/7.58 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 18.07/7.58 18.07/7.58 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 18.07/7.58 primCmpNat Main.Zero Main.Zero = EQ; 18.07/7.58 primCmpNat Main.Zero (Main.Succ y) = LT; 18.07/7.58 primCmpNat (Main.Succ x) Main.Zero = GT; 18.07/7.58 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 18.07/7.58 18.07/7.58 psPs :: List a -> List a -> List a; 18.07/7.58 psPs Nil ys = ys; 18.07/7.58 psPs (Cons x xs) ys = Cons x (psPs xs ys); 18.07/7.58 18.07/7.58 pt :: (a -> c) -> (b -> a) -> b -> c; 18.07/7.58 pt f g x = f (g x); 18.07/7.58 18.07/7.58 showChar :: Main.Char -> List Main.Char -> List Main.Char; 18.07/7.58 showChar = Cons; 18.07/7.58 18.07/7.58 showParen :: MyBool -> (List Main.Char -> List Main.Char) -> List Main.Char -> List Main.Char; 18.07/7.58 showParen b p = showParen0 p b; 18.07/7.58 18.07/7.58 showParen0 p MyTrue = pt (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ Main.Zero))))))))))))))))))))))))))))))))))))))))))) (pt p (showChar (Main.Char (Main.Pos (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ (Main.Succ 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vx58))) GT) vx59",fontsize=16,color="burlywood",shape="box"];174[label="vx570/Succ vx5700",fontsize=10,color="white",style="solid",shape="box"];108 -> 174[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 174 -> 110[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 175[label="vx570/Zero",fontsize=10,color="white",style="solid",shape="box"];108 -> 175[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 175 -> 111[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 109[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpInt (Neg vx570) (Pos (Succ vx58))) GT) vx59",fontsize=16,color="burlywood",shape="box"];176[label="vx570/Succ vx5700",fontsize=10,color="white",style="solid",shape="box"];109 -> 176[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 176 -> 112[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 177[label="vx570/Zero",fontsize=10,color="white",style="solid",shape="box"];109 -> 177[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 177 -> 113[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 110[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpInt (Pos (Succ vx5700)) (Pos (Succ vx58))) GT) vx59",fontsize=16,color="black",shape="box"];110 -> 114[label="",style="solid", color="black", weight=3]; 18.07/7.58 111[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpInt (Pos Zero) (Pos (Succ vx58))) GT) vx59",fontsize=16,color="black",shape="box"];111 -> 115[label="",style="solid", color="black", weight=3]; 18.07/7.58 112[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpInt (Neg (Succ vx5700)) (Pos (Succ vx58))) GT) vx59",fontsize=16,color="black",shape="box"];112 -> 116[label="",style="solid", color="black", weight=3]; 18.07/7.58 113[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpInt (Neg Zero) (Pos (Succ vx58))) GT) vx59",fontsize=16,color="black",shape="box"];113 -> 117[label="",style="solid", color="black", weight=3]; 18.07/7.58 114[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat (Succ vx5700) (Succ vx58)) GT) vx59",fontsize=16,color="black",shape="box"];114 -> 118[label="",style="solid", color="black", weight=3]; 18.07/7.58 115[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat Zero (Succ vx58)) GT) vx59",fontsize=16,color="black",shape="box"];115 -> 119[label="",style="solid", color="black", weight=3]; 18.07/7.58 116[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering LT GT) vx59",fontsize=16,color="black",shape="triangle"];116 -> 120[label="",style="solid", color="black", weight=3]; 18.07/7.58 117 -> 116[label="",style="dashed", color="red", weight=0]; 18.07/7.58 117[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering LT GT) vx59",fontsize=16,color="magenta"];118[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat vx5700 vx58) GT) vx59",fontsize=16,color="burlywood",shape="triangle"];178[label="vx5700/Succ vx57000",fontsize=10,color="white",style="solid",shape="box"];118 -> 178[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 178 -> 121[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 179[label="vx5700/Zero",fontsize=10,color="white",style="solid",shape="box"];118 -> 179[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 179 -> 122[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 119 -> 116[label="",style="dashed", color="red", weight=0]; 18.07/7.58 119[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering LT GT) vx59",fontsize=16,color="magenta"];120[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) MyFalse vx59",fontsize=16,color="black",shape="triangle"];120 -> 123[label="",style="solid", color="black", weight=3]; 18.07/7.58 121[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat (Succ vx57000) vx58) GT) vx59",fontsize=16,color="burlywood",shape="box"];180[label="vx58/Succ vx580",fontsize=10,color="white",style="solid",shape="box"];121 -> 180[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 180 -> 124[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 181[label="vx58/Zero",fontsize=10,color="white",style="solid",shape="box"];121 -> 181[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 181 -> 125[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 122[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat Zero vx58) GT) vx59",fontsize=16,color="burlywood",shape="box"];182[label="vx58/Succ vx580",fontsize=10,color="white",style="solid",shape="box"];122 -> 182[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 182 -> 126[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 183[label="vx58/Zero",fontsize=10,color="white",style="solid",shape="box"];122 -> 183[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 183 -> 127[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 123[label="showString (Cons (Char (Pos (Succ vx55))) vx56) vx59",fontsize=16,color="black",shape="triangle"];123 -> 128[label="",style="solid", color="black", weight=3]; 18.07/7.58 124[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat (Succ vx57000) (Succ vx580)) GT) vx59",fontsize=16,color="black",shape="box"];124 -> 129[label="",style="solid", color="black", weight=3]; 18.07/7.58 125[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat (Succ vx57000) Zero) GT) vx59",fontsize=16,color="black",shape="box"];125 -> 130[label="",style="solid", color="black", weight=3]; 18.07/7.58 126[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat Zero (Succ vx580)) GT) vx59",fontsize=16,color="black",shape="box"];126 -> 131[label="",style="solid", color="black", weight=3]; 18.07/7.58 127[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat Zero Zero) GT) vx59",fontsize=16,color="black",shape="box"];127 -> 132[label="",style="solid", color="black", weight=3]; 18.07/7.58 128[label="psPs (Cons (Char (Pos (Succ vx55))) vx56) vx59",fontsize=16,color="black",shape="box"];128 -> 133[label="",style="solid", color="black", weight=3]; 18.07/7.58 129 -> 118[label="",style="dashed", color="red", weight=0]; 18.07/7.58 129[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering (primCmpNat vx57000 vx580) GT) vx59",fontsize=16,color="magenta"];129 -> 134[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 129 -> 135[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 130[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering GT GT) vx59",fontsize=16,color="black",shape="box"];130 -> 136[label="",style="solid", color="black", weight=3]; 18.07/7.58 131 -> 116[label="",style="dashed", color="red", weight=0]; 18.07/7.58 131[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering LT GT) vx59",fontsize=16,color="magenta"];132[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) (esEsOrdering EQ GT) vx59",fontsize=16,color="black",shape="box"];132 -> 137[label="",style="solid", color="black", weight=3]; 18.07/7.58 133[label="Cons (Char (Pos (Succ vx55))) (psPs vx56 vx59)",fontsize=16,color="green",shape="box"];133 -> 138[label="",style="dashed", color="green", weight=3]; 18.07/7.58 134[label="vx57000",fontsize=16,color="green",shape="box"];135[label="vx580",fontsize=16,color="green",shape="box"];136[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) MyTrue vx59",fontsize=16,color="black",shape="box"];136 -> 139[label="",style="solid", color="black", weight=3]; 18.07/7.58 137 -> 120[label="",style="dashed", color="red", weight=0]; 18.07/7.58 137[label="showParen0 (showString (Cons (Char (Pos (Succ vx55))) vx56)) MyFalse vx59",fontsize=16,color="magenta"];138[label="psPs vx56 vx59",fontsize=16,color="burlywood",shape="triangle"];184[label="vx56/Cons vx560 vx561",fontsize=10,color="white",style="solid",shape="box"];138 -> 184[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 184 -> 140[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 185[label="vx56/Nil",fontsize=10,color="white",style="solid",shape="box"];138 -> 185[label="",style="solid", color="burlywood", weight=9]; 18.07/7.58 185 -> 141[label="",style="solid", color="burlywood", weight=3]; 18.07/7.58 139 -> 149[label="",style="dashed", color="red", weight=0]; 18.07/7.58 139[label="pt (showChar (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))) (pt (showString (Cons (Char (Pos (Succ vx55))) vx56)) (showChar (Char (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))) vx59",fontsize=16,color="magenta"];139 -> 150[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 139 -> 151[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 139 -> 152[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 139 -> 153[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 139 -> 154[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 140[label="psPs (Cons vx560 vx561) vx59",fontsize=16,color="black",shape="box"];140 -> 147[label="",style="solid", color="black", weight=3]; 18.07/7.58 141[label="psPs Nil vx59",fontsize=16,color="black",shape="box"];141 -> 148[label="",style="solid", color="black", weight=3]; 18.07/7.58 150[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];151[label="vx59",fontsize=16,color="green",shape="box"];152[label="vx56",fontsize=16,color="green",shape="box"];153[label="vx55",fontsize=16,color="green",shape="box"];154[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];149[label="pt (showChar (Char (Pos (Succ vx66)))) (pt (showString (Cons (Char (Pos (Succ vx67))) vx68)) (showChar (Char (Pos (Succ vx69))))) vx70",fontsize=16,color="black",shape="triangle"];149 -> 160[label="",style="solid", color="black", weight=3]; 18.07/7.58 147[label="Cons vx560 (psPs vx561 vx59)",fontsize=16,color="green",shape="box"];147 -> 161[label="",style="dashed", color="green", weight=3]; 18.07/7.58 148[label="vx59",fontsize=16,color="green",shape="box"];160[label="showChar (Char (Pos (Succ vx66))) (pt (showString (Cons (Char (Pos (Succ vx67))) vx68)) (showChar (Char (Pos (Succ vx69)))) vx70)",fontsize=16,color="black",shape="box"];160 -> 162[label="",style="solid", color="black", weight=3]; 18.07/7.58 161 -> 138[label="",style="dashed", color="red", weight=0]; 18.07/7.58 161[label="psPs vx561 vx59",fontsize=16,color="magenta"];161 -> 163[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 162[label="Cons (Char (Pos (Succ vx66))) (pt (showString (Cons (Char (Pos (Succ vx67))) vx68)) (showChar (Char (Pos (Succ vx69)))) vx70)",fontsize=16,color="green",shape="box"];162 -> 164[label="",style="dashed", color="green", weight=3]; 18.07/7.58 163[label="vx561",fontsize=16,color="green",shape="box"];164[label="pt (showString (Cons (Char (Pos (Succ vx67))) vx68)) (showChar (Char (Pos (Succ vx69)))) vx70",fontsize=16,color="black",shape="box"];164 -> 165[label="",style="solid", color="black", weight=3]; 18.07/7.58 165 -> 123[label="",style="dashed", color="red", weight=0]; 18.07/7.58 165[label="showString (Cons (Char (Pos (Succ vx67))) vx68) (showChar (Char (Pos (Succ vx69))) vx70)",fontsize=16,color="magenta"];165 -> 166[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 165 -> 167[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 165 -> 168[label="",style="dashed", color="magenta", weight=3]; 18.07/7.58 166[label="vx67",fontsize=16,color="green",shape="box"];167[label="showChar (Char (Pos (Succ vx69))) vx70",fontsize=16,color="black",shape="box"];167 -> 169[label="",style="solid", color="black", weight=3]; 18.07/7.58 168[label="vx68",fontsize=16,color="green",shape="box"];169[label="Cons (Char (Pos (Succ vx69))) vx70",fontsize=16,color="green",shape="box"];} 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (6) 18.07/7.58 Complex Obligation (AND) 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (7) 18.07/7.58 Obligation: 18.07/7.58 Q DP problem: 18.07/7.58 The TRS P consists of the following rules: 18.07/7.58 18.07/7.58 new_showParen0(vx55, vx56, Main.Succ(vx57000), Main.Succ(vx580), vx59) -> new_showParen0(vx55, vx56, vx57000, vx580, vx59) 18.07/7.58 18.07/7.58 R is empty. 18.07/7.58 Q is empty. 18.07/7.58 We have to consider all minimal (P,Q,R)-chains. 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (8) QDPSizeChangeProof (EQUIVALENT) 18.07/7.58 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. 18.07/7.58 18.07/7.58 From the DPs we obtained the following set of size-change graphs: 18.07/7.58 *new_showParen0(vx55, vx56, Main.Succ(vx57000), Main.Succ(vx580), vx59) -> new_showParen0(vx55, vx56, vx57000, vx580, vx59) 18.07/7.58 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5 18.07/7.58 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (9) 18.07/7.58 YES 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (10) 18.07/7.58 Obligation: 18.07/7.58 Q DP problem: 18.07/7.58 The TRS P consists of the following rules: 18.07/7.58 18.07/7.58 new_psPs(Cons(vx560, vx561), vx59) -> new_psPs(vx561, vx59) 18.07/7.58 18.07/7.58 R is empty. 18.07/7.58 Q is empty. 18.07/7.58 We have to consider all minimal (P,Q,R)-chains. 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (11) QDPSizeChangeProof (EQUIVALENT) 18.07/7.58 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. 18.07/7.58 18.07/7.58 From the DPs we obtained the following set of size-change graphs: 18.07/7.58 *new_psPs(Cons(vx560, vx561), vx59) -> new_psPs(vx561, vx59) 18.07/7.58 The graph contains the following edges 1 > 1, 2 >= 2 18.07/7.58 18.07/7.58 18.07/7.58 ---------------------------------------- 18.07/7.58 18.07/7.58 (12) 18.07/7.58 YES 18.47/8.27 EOF