7.87/3.54 YES 9.92/4.06 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.92/4.06 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.92/4.06 9.92/4.06 9.92/4.06 H-Termination with start terms of the given HASKELL could be proven: 9.92/4.06 9.92/4.06 (0) HASKELL 9.92/4.06 (1) BR [EQUIVALENT, 0 ms] 9.92/4.06 (2) HASKELL 9.92/4.06 (3) COR [EQUIVALENT, 0 ms] 9.92/4.06 (4) HASKELL 9.92/4.06 (5) Narrow [EQUIVALENT, 24 ms] 9.92/4.06 (6) YES 9.92/4.06 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (0) 9.92/4.06 Obligation: 9.92/4.06 mainModule Main 9.92/4.06 module Main where { 9.92/4.06 import qualified Prelude; 9.92/4.06 data MyBool = MyTrue | MyFalse ; 9.92/4.06 9.92/4.06 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.92/4.06 9.92/4.06 data Main.Nat = Succ Main.Nat | Zero ; 9.92/4.06 9.92/4.06 data Ordering = LT | EQ | GT ; 9.92/4.06 9.92/4.06 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 esEsMyInt = primEqInt; 9.92/4.06 9.92/4.06 primEqInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt vv vw = MyFalse; 9.92/4.06 9.92/4.06 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.92/4.06 primEqNat Main.Zero Main.Zero = MyTrue; 9.92/4.06 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.92/4.06 9.92/4.06 toEnum0 MyTrue vx = GT; 9.92/4.06 9.92/4.06 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ (Main.Succ Main.Zero)))) vx; 9.92/4.06 9.92/4.06 toEnum2 MyTrue vy = EQ; 9.92/4.06 toEnum2 vz wu = toEnum1 wu; 9.92/4.06 9.92/4.06 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos (Main.Succ Main.Zero))) vy; 9.92/4.06 toEnum3 wv = toEnum1 wv; 9.92/4.06 9.92/4.06 toEnum4 MyTrue ww = LT; 9.92/4.06 toEnum4 wx wy = toEnum3 wy; 9.92/4.06 9.92/4.06 toEnum5 ww = toEnum4 (esEsMyInt ww (Main.Pos Main.Zero)) ww; 9.92/4.06 toEnum5 wz = toEnum3 wz; 9.92/4.06 9.92/4.06 toEnumOrdering :: MyInt -> Ordering; 9.92/4.06 toEnumOrdering ww = toEnum5 ww; 9.92/4.06 toEnumOrdering vy = toEnum3 vy; 9.92/4.06 toEnumOrdering vx = toEnum1 vx; 9.92/4.06 9.92/4.06 } 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (1) BR (EQUIVALENT) 9.92/4.06 Replaced joker patterns by fresh variables and removed binding patterns. 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (2) 9.92/4.06 Obligation: 9.92/4.06 mainModule Main 9.92/4.06 module Main where { 9.92/4.06 import qualified Prelude; 9.92/4.06 data MyBool = MyTrue | MyFalse ; 9.92/4.06 9.92/4.06 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.92/4.06 9.92/4.06 data Main.Nat = Succ Main.Nat | Zero ; 9.92/4.06 9.92/4.06 data Ordering = LT | EQ | GT ; 9.92/4.06 9.92/4.06 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 esEsMyInt = primEqInt; 9.92/4.06 9.92/4.06 primEqInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt vv vw = MyFalse; 9.92/4.06 9.92/4.06 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.92/4.06 primEqNat Main.Zero Main.Zero = MyTrue; 9.92/4.06 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.92/4.06 9.92/4.06 toEnum0 MyTrue vx = GT; 9.92/4.06 9.92/4.06 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ (Main.Succ Main.Zero)))) vx; 9.92/4.06 9.92/4.06 toEnum2 MyTrue vy = EQ; 9.92/4.06 toEnum2 vz wu = toEnum1 wu; 9.92/4.06 9.92/4.06 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos (Main.Succ Main.Zero))) vy; 9.92/4.06 toEnum3 wv = toEnum1 wv; 9.92/4.06 9.92/4.06 toEnum4 MyTrue ww = LT; 9.92/4.06 toEnum4 wx wy = toEnum3 wy; 9.92/4.06 9.92/4.06 toEnum5 ww = toEnum4 (esEsMyInt ww (Main.Pos Main.Zero)) ww; 9.92/4.06 toEnum5 wz = toEnum3 wz; 9.92/4.06 9.92/4.06 toEnumOrdering :: MyInt -> Ordering; 9.92/4.06 toEnumOrdering ww = toEnum5 ww; 9.92/4.06 toEnumOrdering vy = toEnum3 vy; 9.92/4.06 toEnumOrdering vx = toEnum1 vx; 9.92/4.06 9.92/4.06 } 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (3) COR (EQUIVALENT) 9.92/4.06 Cond Reductions: 9.92/4.06 The following Function with conditions 9.92/4.06 "undefined |Falseundefined; 9.92/4.06 " 9.92/4.06 is transformed to 9.92/4.06 "undefined = undefined1; 9.92/4.06 " 9.92/4.06 "undefined0 True = undefined; 9.92/4.06 " 9.92/4.06 "undefined1 = undefined0 False; 9.92/4.06 " 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (4) 9.92/4.06 Obligation: 9.92/4.06 mainModule Main 9.92/4.06 module Main where { 9.92/4.06 import qualified Prelude; 9.92/4.06 data MyBool = MyTrue | MyFalse ; 9.92/4.06 9.92/4.06 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.92/4.06 9.92/4.06 data Main.Nat = Succ Main.Nat | Zero ; 9.92/4.06 9.92/4.06 data Ordering = LT | EQ | GT ; 9.92/4.06 9.92/4.06 esEsMyInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 esEsMyInt = primEqInt; 9.92/4.06 9.92/4.06 primEqInt :: MyInt -> MyInt -> MyBool; 9.92/4.06 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.92/4.06 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.92/4.06 primEqInt vv vw = MyFalse; 9.92/4.06 9.92/4.06 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.92/4.06 primEqNat Main.Zero Main.Zero = MyTrue; 9.92/4.06 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.92/4.06 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.92/4.06 9.92/4.06 toEnum0 MyTrue vx = GT; 9.92/4.06 9.92/4.06 toEnum1 vx = toEnum0 (esEsMyInt vx (Main.Pos (Main.Succ (Main.Succ Main.Zero)))) vx; 9.92/4.06 9.92/4.06 toEnum2 MyTrue vy = EQ; 9.92/4.06 toEnum2 vz wu = toEnum1 wu; 9.92/4.06 9.92/4.06 toEnum3 vy = toEnum2 (esEsMyInt vy (Main.Pos (Main.Succ Main.Zero))) vy; 9.92/4.06 toEnum3 wv = toEnum1 wv; 9.92/4.06 9.92/4.06 toEnum4 MyTrue ww = LT; 9.92/4.06 toEnum4 wx wy = toEnum3 wy; 9.92/4.06 9.92/4.06 toEnum5 ww = toEnum4 (esEsMyInt ww (Main.Pos Main.Zero)) ww; 9.92/4.06 toEnum5 wz = toEnum3 wz; 9.92/4.06 9.92/4.06 toEnumOrdering :: MyInt -> Ordering; 9.92/4.06 toEnumOrdering ww = toEnum5 ww; 9.92/4.06 toEnumOrdering vy = toEnum3 vy; 9.92/4.06 toEnumOrdering vx = toEnum1 vx; 9.92/4.06 9.92/4.06 } 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (5) Narrow (EQUIVALENT) 9.92/4.06 Haskell To QDPs 9.92/4.06 9.92/4.06 digraph dp_graph { 9.92/4.06 node [outthreshold=100, inthreshold=100];1[label="toEnumOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.92/4.06 3[label="toEnumOrdering xw3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 9.92/4.06 4[label="toEnum5 xw3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.92/4.06 5[label="toEnum4 (esEsMyInt xw3 (Pos Zero)) xw3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.92/4.06 6[label="toEnum4 (primEqInt xw3 (Pos Zero)) xw3",fontsize=16,color="burlywood",shape="box"];48[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];6 -> 48[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 48 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 49[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 49 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 7[label="toEnum4 (primEqInt (Pos xw30) (Pos Zero)) (Pos xw30)",fontsize=16,color="burlywood",shape="box"];50[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];7 -> 50[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 50 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 51[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 51[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 51 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 8[label="toEnum4 (primEqInt (Neg xw30) (Pos Zero)) (Neg xw30)",fontsize=16,color="burlywood",shape="box"];52[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];8 -> 52[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 52 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 53[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 53[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 53 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 9[label="toEnum4 (primEqInt (Pos (Succ xw300)) (Pos Zero)) (Pos (Succ xw300))",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3]; 9.92/4.06 10[label="toEnum4 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3]; 9.92/4.06 11[label="toEnum4 (primEqInt (Neg (Succ xw300)) (Pos Zero)) (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 9.92/4.06 12[label="toEnum4 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 9.92/4.06 13[label="toEnum4 MyFalse (Pos (Succ xw300))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 9.92/4.06 14[label="toEnum4 MyTrue (Pos Zero)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 9.92/4.06 15[label="toEnum4 MyFalse (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 9.92/4.06 16[label="toEnum4 MyTrue (Neg Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 9.92/4.06 17[label="toEnum3 (Pos (Succ xw300))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 9.92/4.06 18[label="LT",fontsize=16,color="green",shape="box"];19[label="toEnum3 (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3]; 9.92/4.06 20[label="LT",fontsize=16,color="green",shape="box"];21[label="toEnum2 (esEsMyInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos (Succ xw300))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3]; 9.92/4.06 22[label="toEnum2 (esEsMyInt (Neg (Succ xw300)) (Pos (Succ Zero))) (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3]; 9.92/4.06 23[label="toEnum2 (primEqInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos (Succ xw300))",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3]; 9.92/4.06 24[label="toEnum2 (primEqInt (Neg (Succ xw300)) (Pos (Succ Zero))) (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3]; 9.92/4.06 25[label="toEnum2 (primEqNat xw300 Zero) (Pos (Succ xw300))",fontsize=16,color="burlywood",shape="box"];54[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 54[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 54 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 55[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 55[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 55 -> 28[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 26[label="toEnum2 MyFalse (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3]; 9.92/4.06 27[label="toEnum2 (primEqNat (Succ xw3000) Zero) (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3]; 9.92/4.06 28[label="toEnum2 (primEqNat Zero Zero) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3]; 9.92/4.06 29[label="toEnum1 (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3]; 9.92/4.06 30[label="toEnum2 MyFalse (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3]; 9.92/4.06 31[label="toEnum2 MyTrue (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3]; 9.92/4.06 32[label="toEnum0 (esEsMyInt (Neg (Succ xw300)) (Pos (Succ (Succ Zero)))) (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3]; 9.92/4.06 33[label="toEnum1 (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3]; 9.92/4.06 34[label="EQ",fontsize=16,color="green",shape="box"];35[label="toEnum0 (primEqInt (Neg (Succ xw300)) (Pos (Succ (Succ Zero)))) (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 9.92/4.06 36[label="toEnum0 (esEsMyInt (Pos (Succ (Succ xw3000))) (Pos (Succ (Succ Zero)))) (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3]; 9.92/4.06 37[label="toEnum0 MyFalse (Neg (Succ xw300))",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3]; 9.92/4.06 38[label="toEnum0 (primEqInt (Pos (Succ (Succ xw3000))) (Pos (Succ (Succ Zero)))) (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3]; 9.92/4.06 39[label="error []",fontsize=16,color="red",shape="box"];40[label="toEnum0 (primEqNat (Succ xw3000) (Succ Zero)) (Pos (Succ (Succ xw3000)))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3]; 9.92/4.06 41[label="toEnum0 (primEqNat xw3000 Zero) (Pos (Succ (Succ xw3000)))",fontsize=16,color="burlywood",shape="box"];56[label="xw3000/Succ xw30000",fontsize=10,color="white",style="solid",shape="box"];41 -> 56[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 56 -> 42[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 57[label="xw3000/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 57[label="",style="solid", color="burlywood", weight=9]; 9.92/4.06 57 -> 43[label="",style="solid", color="burlywood", weight=3]; 9.92/4.06 42[label="toEnum0 (primEqNat (Succ xw30000) Zero) (Pos (Succ (Succ (Succ xw30000))))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3]; 9.92/4.06 43[label="toEnum0 (primEqNat Zero Zero) (Pos (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3]; 9.92/4.06 44[label="toEnum0 MyFalse (Pos (Succ (Succ (Succ xw30000))))",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 9.92/4.06 45[label="toEnum0 MyTrue (Pos (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3]; 9.92/4.06 46[label="error []",fontsize=16,color="red",shape="box"];47[label="GT",fontsize=16,color="green",shape="box"];} 9.92/4.06 9.92/4.06 ---------------------------------------- 9.92/4.06 9.92/4.06 (6) 9.92/4.06 YES 9.92/4.11 EOF