7.53/3.46 YES 9.19/3.96 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.19/3.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.19/3.96 9.19/3.96 9.19/3.96 H-Termination with start terms of the given HASKELL could be proven: 9.19/3.96 9.19/3.96 (0) HASKELL 9.19/3.96 (1) BR [EQUIVALENT, 0 ms] 9.19/3.96 (2) HASKELL 9.19/3.96 (3) COR [EQUIVALENT, 0 ms] 9.19/3.96 (4) HASKELL 9.19/3.96 (5) Narrow [EQUIVALENT, 31 ms] 9.19/3.96 (6) YES 9.19/3.96 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (0) 9.19/3.96 Obligation: 9.19/3.96 mainModule Main 9.19/3.96 module Main where { 9.19/3.96 import qualified Prelude; 9.19/3.96 data MyBool = MyTrue | MyFalse ; 9.19/3.96 9.19/3.96 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.19/3.96 9.19/3.96 data Main.Nat = Succ Main.Nat | Zero ; 9.19/3.96 9.19/3.96 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 esEsMyInt = primEqInt; 9.19/3.96 9.19/3.96 primEqInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt vv vw = MyFalse; 9.19/3.96 9.19/3.96 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.19/3.96 primEqNat Main.Zero Main.Zero = MyTrue; 9.19/3.96 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.19/3.96 9.19/3.96 toEnum0 MyTrue vx = MyTrue; 9.19/3.96 9.19/3.96 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ Main.Zero))) vx; 9.19/3.96 9.19/3.96 toEnum2 MyTrue vy = MyFalse; 9.19/3.96 toEnum2 vz wu = toEnum1 wu; 9.19/3.96 9.19/3.96 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos Main.Zero)) vy; 9.19/3.96 toEnum3 wv = toEnum1 wv; 9.19/3.96 9.19/3.96 toEnumMyBool :: MyInt -> MyBool; 9.19/3.96 toEnumMyBool vy = toEnum3 vy; 9.19/3.96 toEnumMyBool vx = toEnum1 vx; 9.19/3.96 9.19/3.96 } 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (1) BR (EQUIVALENT) 9.19/3.96 Replaced joker patterns by fresh variables and removed binding patterns. 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (2) 9.19/3.96 Obligation: 9.19/3.96 mainModule Main 9.19/3.96 module Main where { 9.19/3.96 import qualified Prelude; 9.19/3.96 data MyBool = MyTrue | MyFalse ; 9.19/3.96 9.19/3.96 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.19/3.96 9.19/3.96 data Main.Nat = Succ Main.Nat | Zero ; 9.19/3.96 9.19/3.96 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 esEsMyInt = primEqInt; 9.19/3.96 9.19/3.96 primEqInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt vv vw = MyFalse; 9.19/3.96 9.19/3.96 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.19/3.96 primEqNat Main.Zero Main.Zero = MyTrue; 9.19/3.96 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.19/3.96 9.19/3.96 toEnum0 MyTrue vx = MyTrue; 9.19/3.96 9.19/3.96 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ Main.Zero))) vx; 9.19/3.96 9.19/3.96 toEnum2 MyTrue vy = MyFalse; 9.19/3.96 toEnum2 vz wu = toEnum1 wu; 9.19/3.96 9.19/3.96 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos Main.Zero)) vy; 9.19/3.96 toEnum3 wv = toEnum1 wv; 9.19/3.96 9.19/3.96 toEnumMyBool :: MyInt -> MyBool; 9.19/3.96 toEnumMyBool vy = toEnum3 vy; 9.19/3.96 toEnumMyBool vx = toEnum1 vx; 9.19/3.96 9.19/3.96 } 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (3) COR (EQUIVALENT) 9.19/3.96 Cond Reductions: 9.19/3.96 The following Function with conditions 9.19/3.96 "undefined |Falseundefined; 9.19/3.96 " 9.19/3.96 is transformed to 9.19/3.96 "undefined = undefined1; 9.19/3.96 " 9.19/3.96 "undefined0 True = undefined; 9.19/3.96 " 9.19/3.96 "undefined1 = undefined0 False; 9.19/3.96 " 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (4) 9.19/3.96 Obligation: 9.19/3.96 mainModule Main 9.19/3.96 module Main where { 9.19/3.96 import qualified Prelude; 9.19/3.96 data MyBool = MyTrue | MyFalse ; 9.19/3.96 9.19/3.96 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.19/3.96 9.19/3.96 data Main.Nat = Succ Main.Nat | Zero ; 9.19/3.96 9.19/3.96 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 esEsMyInt = primEqInt; 9.19/3.96 9.19/3.96 primEqInt :: MyInt -> MyInt -> MyBool; 9.19/3.96 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.19/3.96 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.19/3.96 primEqInt vv vw = MyFalse; 9.19/3.96 9.19/3.96 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.19/3.96 primEqNat Main.Zero Main.Zero = MyTrue; 9.19/3.96 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.19/3.96 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.19/3.96 9.19/3.96 toEnum0 MyTrue vx = MyTrue; 9.19/3.96 9.19/3.96 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ Main.Zero))) vx; 9.19/3.96 9.19/3.96 toEnum2 MyTrue vy = MyFalse; 9.19/3.96 toEnum2 vz wu = toEnum1 wu; 9.19/3.96 9.19/3.96 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos Main.Zero)) vy; 9.19/3.96 toEnum3 wv = toEnum1 wv; 9.19/3.96 9.19/3.96 toEnumMyBool :: MyInt -> MyBool; 9.19/3.96 toEnumMyBool vy = toEnum3 vy; 9.19/3.96 toEnumMyBool vx = toEnum1 vx; 9.19/3.96 9.19/3.96 } 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (5) Narrow (EQUIVALENT) 9.19/3.96 Haskell To QDPs 9.19/3.96 9.19/3.96 digraph dp_graph { 9.19/3.96 node [outthreshold=100, inthreshold=100];1[label="toEnumMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.19/3.96 3[label="toEnumMyBool wy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.19/3.96 4[label="toEnum3 wy3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.19/3.96 5[label="toEnum2 (esEsMyInt wy3 (Pos Zero)) wy3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.19/3.96 6[label="toEnum2 (primEqInt wy3 (Pos Zero)) wy3",fontsize=16,color="burlywood",shape="box"];34[label="wy3/Pos wy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 34[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 34 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 35[label="wy3/Neg wy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 35[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 35 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 7[label="toEnum2 (primEqInt (Pos wy30) (Pos Zero)) (Pos wy30)",fontsize=16,color="burlywood",shape="box"];36[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 36 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 37[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 37 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 8[label="toEnum2 (primEqInt (Neg wy30) (Pos Zero)) (Neg wy30)",fontsize=16,color="burlywood",shape="box"];38[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];8 -> 38[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 38 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 39[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 39[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 39 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 9[label="toEnum2 (primEqInt (Pos (Succ wy300)) (Pos Zero)) (Pos (Succ wy300))",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3]; 9.19/3.96 10[label="toEnum2 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3]; 9.19/3.96 11[label="toEnum2 (primEqInt (Neg (Succ wy300)) (Pos Zero)) (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 9.19/3.96 12[label="toEnum2 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 9.19/3.96 13[label="toEnum2 MyFalse (Pos (Succ wy300))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 9.19/3.96 14[label="toEnum2 MyTrue (Pos Zero)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 9.19/3.96 15[label="toEnum2 MyFalse (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 9.19/3.96 16[label="toEnum2 MyTrue (Neg Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 9.19/3.96 17[label="toEnum1 (Pos (Succ wy300))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 9.19/3.96 18[label="MyFalse",fontsize=16,color="green",shape="box"];19[label="toEnum1 (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3]; 9.19/3.96 20[label="MyFalse",fontsize=16,color="green",shape="box"];21[label="toEnum0 (esEsMyInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos (Succ wy300))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3]; 9.19/3.96 22[label="toEnum0 (esEsMyInt (Neg (Succ wy300)) (Pos (Succ Zero))) (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3]; 9.19/3.96 23[label="toEnum0 (primEqInt (Pos (Succ wy300)) (Pos (Succ Zero))) (Pos (Succ wy300))",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3]; 9.19/3.96 24[label="toEnum0 (primEqInt (Neg (Succ wy300)) (Pos (Succ Zero))) (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3]; 9.19/3.96 25[label="toEnum0 (primEqNat wy300 Zero) (Pos (Succ wy300))",fontsize=16,color="burlywood",shape="box"];40[label="wy300/Succ wy3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 40[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 40 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 41[label="wy300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 41[label="",style="solid", color="burlywood", weight=9]; 9.19/3.96 41 -> 28[label="",style="solid", color="burlywood", weight=3]; 9.19/3.96 26[label="toEnum0 MyFalse (Neg (Succ wy300))",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3]; 9.19/3.96 27[label="toEnum0 (primEqNat (Succ wy3000) Zero) (Pos (Succ (Succ wy3000)))",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3]; 9.19/3.96 28[label="toEnum0 (primEqNat Zero Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3]; 9.19/3.96 29[label="error []",fontsize=16,color="red",shape="box"];30[label="toEnum0 MyFalse (Pos (Succ (Succ wy3000)))",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3]; 9.19/3.96 31[label="toEnum0 MyTrue (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3]; 9.19/3.96 32[label="error []",fontsize=16,color="red",shape="box"];33[label="MyTrue",fontsize=16,color="green",shape="box"];} 9.19/3.96 9.19/3.96 ---------------------------------------- 9.19/3.96 9.19/3.96 (6) 9.19/3.96 YES 9.54/4.00 EOF