7.72/3.56 YES 9.67/4.11 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.67/4.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.67/4.11 9.67/4.11 9.67/4.11 H-Termination with start terms of the given HASKELL could be proven: 9.67/4.11 9.67/4.11 (0) HASKELL 9.67/4.11 (1) BR [EQUIVALENT, 0 ms] 9.67/4.11 (2) HASKELL 9.67/4.11 (3) COR [EQUIVALENT, 0 ms] 9.67/4.11 (4) HASKELL 9.67/4.11 (5) Narrow [EQUIVALENT, 27 ms] 9.67/4.11 (6) YES 9.67/4.11 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (0) 9.67/4.11 Obligation: 9.67/4.11 mainModule Main 9.67/4.11 module Main where { 9.67/4.11 import qualified Prelude; 9.67/4.11 data MyBool = MyTrue | MyFalse ; 9.67/4.11 9.67/4.11 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.67/4.11 9.67/4.11 data Main.Nat = Succ Main.Nat | Zero ; 9.67/4.11 9.67/4.11 data Tup0 = Tup0 ; 9.67/4.11 9.67/4.11 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 esEsMyInt = primEqInt; 9.67/4.11 9.67/4.11 flip :: (b -> c -> a) -> c -> b -> a; 9.67/4.11 flip f x y = f y x; 9.67/4.11 9.67/4.11 fromEnumTup0 :: Tup0 -> MyInt; 9.67/4.11 fromEnumTup0 Tup0 = Main.Pos Main.Zero; 9.67/4.11 9.67/4.11 msMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 msMyInt = primMinusInt; 9.67/4.11 9.67/4.11 predTup0 :: Tup0 -> Tup0; 9.67/4.11 predTup0 = pt toEnumTup0 (pt (subtractMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0); 9.67/4.11 9.67/4.11 primEqInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt vv vw = MyFalse; 9.67/4.11 9.67/4.11 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.67/4.11 primEqNat Main.Zero Main.Zero = MyTrue; 9.67/4.11 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.67/4.11 9.67/4.11 primMinusInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 9.67/4.11 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 9.67/4.11 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 9.67/4.11 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 9.67/4.11 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 9.67/4.11 primPlusNat Main.Zero Main.Zero = Main.Zero; 9.67/4.11 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 9.67/4.11 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 9.67/4.11 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 9.67/4.11 9.67/4.11 pt :: (a -> c) -> (b -> a) -> b -> c; 9.67/4.11 pt f g x = f (g x); 9.67/4.11 9.67/4.11 subtractMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 subtractMyInt = flip msMyInt; 9.67/4.11 9.67/4.11 toEnum0 MyTrue vx = Tup0; 9.67/4.11 9.67/4.11 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx; 9.67/4.11 9.67/4.11 toEnumTup0 :: MyInt -> Tup0; 9.67/4.11 toEnumTup0 vx = toEnum1 vx; 9.67/4.11 9.67/4.11 } 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (1) BR (EQUIVALENT) 9.67/4.11 Replaced joker patterns by fresh variables and removed binding patterns. 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (2) 9.67/4.11 Obligation: 9.67/4.11 mainModule Main 9.67/4.11 module Main where { 9.67/4.11 import qualified Prelude; 9.67/4.11 data MyBool = MyTrue | MyFalse ; 9.67/4.11 9.67/4.11 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.67/4.11 9.67/4.11 data Main.Nat = Succ Main.Nat | Zero ; 9.67/4.11 9.67/4.11 data Tup0 = Tup0 ; 9.67/4.11 9.67/4.11 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 esEsMyInt = primEqInt; 9.67/4.11 9.67/4.11 flip :: (c -> a -> b) -> a -> c -> b; 9.67/4.11 flip f x y = f y x; 9.67/4.11 9.67/4.11 fromEnumTup0 :: Tup0 -> MyInt; 9.67/4.11 fromEnumTup0 Tup0 = Main.Pos Main.Zero; 9.67/4.11 9.67/4.11 msMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 msMyInt = primMinusInt; 9.67/4.11 9.67/4.11 predTup0 :: Tup0 -> Tup0; 9.67/4.11 predTup0 = pt toEnumTup0 (pt (subtractMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0); 9.67/4.11 9.67/4.11 primEqInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt vv vw = MyFalse; 9.67/4.11 9.67/4.11 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.67/4.11 primEqNat Main.Zero Main.Zero = MyTrue; 9.67/4.11 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.67/4.11 9.67/4.11 primMinusInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 9.67/4.11 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 9.67/4.11 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 9.67/4.11 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 9.67/4.11 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 9.67/4.11 primPlusNat Main.Zero Main.Zero = Main.Zero; 9.67/4.11 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 9.67/4.11 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 9.67/4.11 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 9.67/4.11 9.67/4.11 pt :: (b -> c) -> (a -> b) -> a -> c; 9.67/4.11 pt f g x = f (g x); 9.67/4.11 9.67/4.11 subtractMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 subtractMyInt = flip msMyInt; 9.67/4.11 9.67/4.11 toEnum0 MyTrue vx = Tup0; 9.67/4.11 9.67/4.11 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx; 9.67/4.11 9.67/4.11 toEnumTup0 :: MyInt -> Tup0; 9.67/4.11 toEnumTup0 vx = toEnum1 vx; 9.67/4.11 9.67/4.11 } 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (3) COR (EQUIVALENT) 9.67/4.11 Cond Reductions: 9.67/4.11 The following Function with conditions 9.67/4.11 "undefined |Falseundefined; 9.67/4.11 " 9.67/4.11 is transformed to 9.67/4.11 "undefined = undefined1; 9.67/4.11 " 9.67/4.11 "undefined0 True = undefined; 9.67/4.11 " 9.67/4.11 "undefined1 = undefined0 False; 9.67/4.11 " 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (4) 9.67/4.11 Obligation: 9.67/4.11 mainModule Main 9.67/4.11 module Main where { 9.67/4.11 import qualified Prelude; 9.67/4.11 data MyBool = MyTrue | MyFalse ; 9.67/4.11 9.67/4.11 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.67/4.11 9.67/4.11 data Main.Nat = Succ Main.Nat | Zero ; 9.67/4.11 9.67/4.11 data Tup0 = Tup0 ; 9.67/4.11 9.67/4.11 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 esEsMyInt = primEqInt; 9.67/4.11 9.67/4.11 flip :: (b -> c -> a) -> c -> b -> a; 9.67/4.11 flip f x y = f y x; 9.67/4.11 9.67/4.11 fromEnumTup0 :: Tup0 -> MyInt; 9.67/4.11 fromEnumTup0 Tup0 = Main.Pos Main.Zero; 9.67/4.11 9.67/4.11 msMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 msMyInt = primMinusInt; 9.67/4.11 9.67/4.11 predTup0 :: Tup0 -> Tup0; 9.67/4.11 predTup0 = pt toEnumTup0 (pt (subtractMyInt (Main.Pos (Main.Succ Main.Zero))) fromEnumTup0); 9.67/4.11 9.67/4.11 primEqInt :: MyInt -> MyInt -> MyBool; 9.67/4.11 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.67/4.11 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.67/4.11 primEqInt vv vw = MyFalse; 9.67/4.11 9.67/4.11 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.67/4.11 primEqNat Main.Zero Main.Zero = MyTrue; 9.67/4.11 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.67/4.11 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.67/4.11 9.67/4.11 primMinusInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); 9.67/4.11 primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; 9.67/4.11 primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primMinusNat :: Main.Nat -> Main.Nat -> MyInt; 9.67/4.11 primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; 9.67/4.11 primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); 9.67/4.11 primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); 9.67/4.11 primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; 9.67/4.11 9.67/4.11 primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; 9.67/4.11 primPlusNat Main.Zero Main.Zero = Main.Zero; 9.67/4.11 primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; 9.67/4.11 primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; 9.67/4.11 primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); 9.67/4.11 9.67/4.11 pt :: (a -> b) -> (c -> a) -> c -> b; 9.67/4.11 pt f g x = f (g x); 9.67/4.11 9.67/4.11 subtractMyInt :: MyInt -> MyInt -> MyInt; 9.67/4.11 subtractMyInt = flip msMyInt; 9.67/4.11 9.67/4.11 toEnum0 MyTrue vx = Tup0; 9.67/4.11 9.67/4.11 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos Main.Zero)) vx; 9.67/4.11 9.67/4.11 toEnumTup0 :: MyInt -> Tup0; 9.67/4.11 toEnumTup0 vx = toEnum1 vx; 9.67/4.11 9.67/4.11 } 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (5) Narrow (EQUIVALENT) 9.67/4.11 Haskell To QDPs 9.67/4.11 9.67/4.11 digraph dp_graph { 9.67/4.11 node [outthreshold=100, inthreshold=100];1[label="predTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.67/4.11 3[label="predTup0 wu3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.67/4.11 4[label="pt toEnumTup0 (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0) wu3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.67/4.11 5[label="toEnumTup0 (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.67/4.11 6[label="toEnum1 (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 9.67/4.11 7[label="toEnum0 (esEsMyInt (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3) (Pos Zero)) (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 9.67/4.11 8[label="toEnum0 (primEqInt (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3) (Pos Zero)) (pt (subtractMyInt (Pos (Succ Zero))) fromEnumTup0 wu3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9.67/4.11 9[label="toEnum0 (primEqInt (subtractMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3)) (Pos Zero)) (subtractMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 9.67/4.11 10[label="toEnum0 (primEqInt (flip msMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3)) (Pos Zero)) (flip msMyInt (Pos (Succ Zero)) (fromEnumTup0 wu3))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 9.67/4.11 11[label="toEnum0 (primEqInt (msMyInt (fromEnumTup0 wu3) (Pos (Succ Zero))) (Pos Zero)) (msMyInt (fromEnumTup0 wu3) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 9.67/4.11 12[label="toEnum0 (primEqInt (primMinusInt (fromEnumTup0 wu3) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnumTup0 wu3) (Pos (Succ Zero)))",fontsize=16,color="burlywood",shape="box"];19[label="wu3/Tup0",fontsize=10,color="white",style="solid",shape="box"];12 -> 19[label="",style="solid", color="burlywood", weight=9]; 9.67/4.11 19 -> 13[label="",style="solid", color="burlywood", weight=3]; 9.67/4.11 13[label="toEnum0 (primEqInt (primMinusInt (fromEnumTup0 Tup0) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (fromEnumTup0 Tup0) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 9.67/4.11 14[label="toEnum0 (primEqInt (primMinusInt (Pos Zero) (Pos (Succ Zero))) (Pos Zero)) (primMinusInt (Pos Zero) (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 9.67/4.11 15[label="toEnum0 (primEqInt (primMinusNat Zero (Succ Zero)) (Pos Zero)) (primMinusNat Zero (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 9.67/4.11 16[label="toEnum0 (primEqInt (Neg (Succ Zero)) (Pos Zero)) (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3]; 9.67/4.11 17[label="toEnum0 MyFalse (Neg (Succ Zero))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 9.67/4.11 18[label="error []",fontsize=16,color="red",shape="box"];} 9.67/4.11 9.67/4.11 ---------------------------------------- 9.67/4.11 9.67/4.11 (6) 9.67/4.11 YES 9.98/5.88 EOF