8.36/3.63 YES 9.93/4.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.93/4.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.93/4.16 9.93/4.16 9.93/4.16 H-Termination with start terms of the given HASKELL could be proven: 9.93/4.16 9.93/4.16 (0) HASKELL 9.93/4.16 (1) BR [EQUIVALENT, 0 ms] 9.93/4.16 (2) HASKELL 9.93/4.16 (3) COR [EQUIVALENT, 0 ms] 9.93/4.16 (4) HASKELL 9.93/4.16 (5) Narrow [SOUND, 0 ms] 9.93/4.16 (6) AND 9.93/4.16 (7) QDP 9.93/4.16 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.93/4.16 (9) YES 9.93/4.16 (10) QDP 9.93/4.16 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.93/4.16 (12) YES 9.93/4.16 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (0) 9.93/4.16 Obligation: 9.93/4.16 mainModule Main 9.93/4.16 module Main where { 9.93/4.16 import qualified Prelude; 9.93/4.16 data Main.Char = Char MyInt ; 9.93/4.16 9.93/4.16 data List a = Cons a (List a) | Nil ; 9.93/4.16 9.93/4.16 data MyBool = MyTrue | MyFalse ; 9.93/4.16 9.93/4.16 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.93/4.16 9.93/4.16 data Main.Nat = Succ Main.Nat | Zero ; 9.93/4.16 9.93/4.16 all :: (a -> MyBool) -> List a -> MyBool; 9.93/4.16 all p = pt and (map p); 9.93/4.16 9.93/4.16 and :: List MyBool -> MyBool; 9.93/4.16 and = foldr asAs MyTrue; 9.93/4.16 9.93/4.16 asAs :: MyBool -> MyBool -> MyBool; 9.93/4.16 asAs MyFalse x = MyFalse; 9.93/4.16 asAs MyTrue x = x; 9.93/4.16 9.93/4.16 esEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 esEsChar = primEqChar; 9.93/4.16 9.93/4.16 foldr :: (b -> a -> a) -> a -> List b -> a; 9.93/4.16 foldr f z Nil = z; 9.93/4.16 foldr f z (Cons x xs) = f x (foldr f z xs); 9.93/4.16 9.93/4.16 fsEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 fsEsChar x y = not (esEsChar x y); 9.93/4.16 9.93/4.16 map :: (a -> b) -> List a -> List b; 9.93/4.16 map f Nil = Nil; 9.93/4.16 map f (Cons x xs) = Cons (f x) (map f xs); 9.93/4.16 9.93/4.16 not :: MyBool -> MyBool; 9.93/4.16 not MyTrue = MyFalse; 9.93/4.16 not MyFalse = MyTrue; 9.93/4.16 9.93/4.16 notElemChar :: Main.Char -> List Main.Char -> MyBool; 9.93/4.16 notElemChar = pt all fsEsChar; 9.93/4.16 9.93/4.16 primEqChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 primEqChar (Main.Char x) (Main.Char y) = primEqInt x y; 9.93/4.16 9.93/4.16 primEqInt :: MyInt -> MyInt -> MyBool; 9.93/4.16 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt vv vw = MyFalse; 9.93/4.16 9.93/4.16 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.93/4.16 primEqNat Main.Zero Main.Zero = MyTrue; 9.93/4.16 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.93/4.16 9.93/4.16 pt :: (c -> b) -> (a -> c) -> a -> b; 9.93/4.16 pt f g x = f (g x); 9.93/4.16 9.93/4.16 } 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (1) BR (EQUIVALENT) 9.93/4.16 Replaced joker patterns by fresh variables and removed binding patterns. 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (2) 9.93/4.16 Obligation: 9.93/4.16 mainModule Main 9.93/4.16 module Main where { 9.93/4.16 import qualified Prelude; 9.93/4.16 data Main.Char = Char MyInt ; 9.93/4.16 9.93/4.16 data List a = Cons a (List a) | Nil ; 9.93/4.16 9.93/4.16 data MyBool = MyTrue | MyFalse ; 9.93/4.16 9.93/4.16 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.93/4.16 9.93/4.16 data Main.Nat = Succ Main.Nat | Zero ; 9.93/4.16 9.93/4.16 all :: (a -> MyBool) -> List a -> MyBool; 9.93/4.16 all p = pt and (map p); 9.93/4.16 9.93/4.16 and :: List MyBool -> MyBool; 9.93/4.16 and = foldr asAs MyTrue; 9.93/4.16 9.93/4.16 asAs :: MyBool -> MyBool -> MyBool; 9.93/4.16 asAs MyFalse x = MyFalse; 9.93/4.16 asAs MyTrue x = x; 9.93/4.16 9.93/4.16 esEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 esEsChar = primEqChar; 9.93/4.16 9.93/4.16 foldr :: (a -> b -> b) -> b -> List a -> b; 9.93/4.16 foldr f z Nil = z; 9.93/4.16 foldr f z (Cons x xs) = f x (foldr f z xs); 9.93/4.16 9.93/4.16 fsEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 fsEsChar x y = not (esEsChar x y); 9.93/4.16 9.93/4.16 map :: (b -> a) -> List b -> List a; 9.93/4.16 map f Nil = Nil; 9.93/4.16 map f (Cons x xs) = Cons (f x) (map f xs); 9.93/4.16 9.93/4.16 not :: MyBool -> MyBool; 9.93/4.16 not MyTrue = MyFalse; 9.93/4.16 not MyFalse = MyTrue; 9.93/4.16 9.93/4.16 notElemChar :: Main.Char -> List Main.Char -> MyBool; 9.93/4.16 notElemChar = pt all fsEsChar; 9.93/4.16 9.93/4.16 primEqChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 primEqChar (Main.Char x) (Main.Char y) = primEqInt x y; 9.93/4.16 9.93/4.16 primEqInt :: MyInt -> MyInt -> MyBool; 9.93/4.16 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt vv vw = MyFalse; 9.93/4.16 9.93/4.16 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.93/4.16 primEqNat Main.Zero Main.Zero = MyTrue; 9.93/4.16 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.93/4.16 9.93/4.16 pt :: (b -> a) -> (c -> b) -> c -> a; 9.93/4.16 pt f g x = f (g x); 9.93/4.16 9.93/4.16 } 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (3) COR (EQUIVALENT) 9.93/4.16 Cond Reductions: 9.93/4.16 The following Function with conditions 9.93/4.16 "undefined |Falseundefined; 9.93/4.16 " 9.93/4.16 is transformed to 9.93/4.16 "undefined = undefined1; 9.93/4.16 " 9.93/4.16 "undefined0 True = undefined; 9.93/4.16 " 9.93/4.16 "undefined1 = undefined0 False; 9.93/4.16 " 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (4) 9.93/4.16 Obligation: 9.93/4.16 mainModule Main 9.93/4.16 module Main where { 9.93/4.16 import qualified Prelude; 9.93/4.16 data Main.Char = Char MyInt ; 9.93/4.16 9.93/4.16 data List a = Cons a (List a) | Nil ; 9.93/4.16 9.93/4.16 data MyBool = MyTrue | MyFalse ; 9.93/4.16 9.93/4.16 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.93/4.16 9.93/4.16 data Main.Nat = Succ Main.Nat | Zero ; 9.93/4.16 9.93/4.16 all :: (a -> MyBool) -> List a -> MyBool; 9.93/4.16 all p = pt and (map p); 9.93/4.16 9.93/4.16 and :: List MyBool -> MyBool; 9.93/4.16 and = foldr asAs MyTrue; 9.93/4.16 9.93/4.16 asAs :: MyBool -> MyBool -> MyBool; 9.93/4.16 asAs MyFalse x = MyFalse; 9.93/4.16 asAs MyTrue x = x; 9.93/4.16 9.93/4.16 esEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 esEsChar = primEqChar; 9.93/4.16 9.93/4.16 foldr :: (a -> b -> b) -> b -> List a -> b; 9.93/4.16 foldr f z Nil = z; 9.93/4.16 foldr f z (Cons x xs) = f x (foldr f z xs); 9.93/4.16 9.93/4.16 fsEsChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 fsEsChar x y = not (esEsChar x y); 9.93/4.16 9.93/4.16 map :: (b -> a) -> List b -> List a; 9.93/4.16 map f Nil = Nil; 9.93/4.16 map f (Cons x xs) = Cons (f x) (map f xs); 9.93/4.16 9.93/4.16 not :: MyBool -> MyBool; 9.93/4.16 not MyTrue = MyFalse; 9.93/4.16 not MyFalse = MyTrue; 9.93/4.16 9.93/4.16 notElemChar :: Main.Char -> List Main.Char -> MyBool; 9.93/4.16 notElemChar = pt all fsEsChar; 9.93/4.16 9.93/4.16 primEqChar :: Main.Char -> Main.Char -> MyBool; 9.93/4.16 primEqChar (Main.Char x) (Main.Char y) = primEqInt x y; 9.93/4.16 9.93/4.16 primEqInt :: MyInt -> MyInt -> MyBool; 9.93/4.16 primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue; 9.93/4.16 primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue; 9.93/4.16 primEqInt vv vw = MyFalse; 9.93/4.16 9.93/4.16 primEqNat :: Main.Nat -> Main.Nat -> MyBool; 9.93/4.16 primEqNat Main.Zero Main.Zero = MyTrue; 9.93/4.16 primEqNat Main.Zero (Main.Succ y) = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) Main.Zero = MyFalse; 9.93/4.16 primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y; 9.93/4.16 9.93/4.16 pt :: (b -> a) -> (c -> b) -> c -> a; 9.93/4.16 pt f g x = f (g x); 9.93/4.16 9.93/4.16 } 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (5) Narrow (SOUND) 9.93/4.16 Haskell To QDPs 9.93/4.16 9.93/4.16 digraph dp_graph { 9.93/4.16 node [outthreshold=100, inthreshold=100];1[label="notElemChar",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.93/4.16 3[label="notElemChar vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.93/4.16 4[label="notElemChar vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.93/4.16 5[label="pt all fsEsChar vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.93/4.16 6[label="all (fsEsChar vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 9.93/4.16 7[label="pt and (map (fsEsChar vz3)) vz4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 9.93/4.16 8[label="and (map (fsEsChar vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9.93/4.16 9[label="foldr asAs MyTrue (map (fsEsChar vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];82[label="vz4/Cons vz40 vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 82 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 83[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 83 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 10[label="foldr asAs MyTrue (map (fsEsChar vz3) (Cons vz40 vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.93/4.16 11[label="foldr asAs MyTrue (map (fsEsChar vz3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 9.93/4.16 12[label="foldr asAs MyTrue (Cons (fsEsChar vz3 vz40) (map (fsEsChar vz3) vz41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 9.93/4.16 13[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3]; 9.93/4.16 14 -> 16[label="",style="dashed", color="red", weight=0]; 9.93/4.16 14[label="asAs (fsEsChar vz3 vz40) (foldr asAs MyTrue (map (fsEsChar vz3) vz41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0]; 9.93/4.16 17[label="foldr asAs MyTrue (map (fsEsChar vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 16[label="asAs (fsEsChar vz3 vz40) vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3]; 9.93/4.16 18[label="vz41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsChar vz3 vz40)) vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 9.93/4.16 20[label="asAs (not (primEqChar vz3 vz40)) vz5",fontsize=16,color="burlywood",shape="box"];84[label="vz3/Char vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 84[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 84 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 21[label="asAs (not (primEqChar (Char vz30) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];85[label="vz40/Char vz400",fontsize=10,color="white",style="solid",shape="box"];21 -> 85[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 85 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 22[label="asAs (not (primEqChar (Char vz30) (Char vz400))) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3]; 9.93/4.16 23[label="asAs (not (primEqInt vz30 vz400)) vz5",fontsize=16,color="burlywood",shape="box"];86[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 86[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 86 -> 24[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 87[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];23 -> 87[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 87 -> 25[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 24[label="asAs (not (primEqInt (Pos vz300) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];88[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 88 -> 26[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 89[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 89 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 25[label="asAs (not (primEqInt (Neg vz300) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];90[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];25 -> 90[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 90 -> 28[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 91[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 91[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 91 -> 29[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 26[label="asAs (not (primEqInt (Pos (Succ vz3000)) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];92[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 92[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 92 -> 30[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 93[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 93[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 93 -> 31[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 27[label="asAs (not (primEqInt (Pos Zero) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];94[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 94[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 94 -> 32[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 95[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 95[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 95 -> 33[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 28[label="asAs (not (primEqInt (Neg (Succ vz3000)) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];96[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 96[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 96 -> 34[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 97[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 97[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 97 -> 35[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 29[label="asAs (not (primEqInt (Neg Zero) vz400)) vz5",fontsize=16,color="burlywood",shape="box"];98[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 98 -> 36[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 99[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 99[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 99 -> 37[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 30[label="asAs (not (primEqInt (Pos (Succ vz3000)) (Pos vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];100[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];30 -> 100[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 100 -> 38[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 101[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 101[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 101 -> 39[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 31[label="asAs (not (primEqInt (Pos (Succ vz3000)) (Neg vz4000))) vz5",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3]; 9.93/4.16 32[label="asAs (not (primEqInt (Pos Zero) (Pos vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];102[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];32 -> 102[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 102 -> 41[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 103[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 103[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 103 -> 42[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 33[label="asAs (not (primEqInt (Pos Zero) (Neg vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];104[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];33 -> 104[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 104 -> 43[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 105[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 105[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 105 -> 44[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 34[label="asAs (not (primEqInt (Neg (Succ vz3000)) (Pos vz4000))) vz5",fontsize=16,color="black",shape="box"];34 -> 45[label="",style="solid", color="black", weight=3]; 9.93/4.16 35[label="asAs (not (primEqInt (Neg (Succ vz3000)) (Neg vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];106[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];35 -> 106[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 106 -> 46[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 107[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 107[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 107 -> 47[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 36[label="asAs (not (primEqInt (Neg Zero) (Pos vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];108[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];36 -> 108[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 108 -> 48[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 109[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 109[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 109 -> 49[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 37[label="asAs (not (primEqInt (Neg Zero) (Neg vz4000))) vz5",fontsize=16,color="burlywood",shape="box"];110[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];37 -> 110[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 110 -> 50[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 111[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 111[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 111 -> 51[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 38[label="asAs (not (primEqInt (Pos (Succ vz3000)) (Pos (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 9.93/4.16 39[label="asAs (not (primEqInt (Pos (Succ vz3000)) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3]; 9.93/4.16 40[label="asAs (not MyFalse) vz5",fontsize=16,color="black",shape="triangle"];40 -> 54[label="",style="solid", color="black", weight=3]; 9.93/4.16 41[label="asAs (not (primEqInt (Pos Zero) (Pos (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3]; 9.93/4.16 42[label="asAs (not (primEqInt (Pos Zero) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3]; 9.93/4.16 43[label="asAs (not (primEqInt (Pos Zero) (Neg (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3]; 9.93/4.16 44[label="asAs (not (primEqInt (Pos Zero) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];44 -> 58[label="",style="solid", color="black", weight=3]; 9.93/4.16 45 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 45[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];46[label="asAs (not (primEqInt (Neg (Succ vz3000)) (Neg (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3]; 9.93/4.16 47[label="asAs (not (primEqInt (Neg (Succ vz3000)) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3]; 9.93/4.16 48[label="asAs (not (primEqInt (Neg Zero) (Pos (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3]; 9.93/4.16 49[label="asAs (not (primEqInt (Neg Zero) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];49 -> 62[label="",style="solid", color="black", weight=3]; 9.93/4.16 50[label="asAs (not (primEqInt (Neg Zero) (Neg (Succ vz40000)))) vz5",fontsize=16,color="black",shape="box"];50 -> 63[label="",style="solid", color="black", weight=3]; 9.93/4.16 51[label="asAs (not (primEqInt (Neg Zero) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];51 -> 64[label="",style="solid", color="black", weight=3]; 9.93/4.16 52[label="asAs (not (primEqNat vz3000 vz40000)) vz5",fontsize=16,color="burlywood",shape="triangle"];112[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];52 -> 112[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 112 -> 65[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 113[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 113[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 113 -> 66[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 53 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 53[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];54[label="asAs MyTrue vz5",fontsize=16,color="black",shape="box"];54 -> 67[label="",style="solid", color="black", weight=3]; 9.93/4.16 55 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 55[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];56[label="asAs (not MyTrue) vz5",fontsize=16,color="black",shape="triangle"];56 -> 68[label="",style="solid", color="black", weight=3]; 9.93/4.16 57 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 57[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];58 -> 56[label="",style="dashed", color="red", weight=0]; 9.93/4.16 58[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];59 -> 52[label="",style="dashed", color="red", weight=0]; 9.93/4.16 59[label="asAs (not (primEqNat vz3000 vz40000)) vz5",fontsize=16,color="magenta"];59 -> 69[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 59 -> 70[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 60 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 60[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];61 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 61[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];62 -> 56[label="",style="dashed", color="red", weight=0]; 9.93/4.16 62[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];63 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 63[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];64 -> 56[label="",style="dashed", color="red", weight=0]; 9.93/4.16 64[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];65[label="asAs (not (primEqNat (Succ vz30000) vz40000)) vz5",fontsize=16,color="burlywood",shape="box"];114[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];65 -> 114[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 114 -> 71[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 115[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 115[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 115 -> 72[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 66[label="asAs (not (primEqNat Zero vz40000)) vz5",fontsize=16,color="burlywood",shape="box"];116[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];66 -> 116[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 116 -> 73[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 117[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];66 -> 117[label="",style="solid", color="burlywood", weight=9]; 9.93/4.16 117 -> 74[label="",style="solid", color="burlywood", weight=3]; 9.93/4.16 67[label="vz5",fontsize=16,color="green",shape="box"];68[label="asAs MyFalse vz5",fontsize=16,color="black",shape="box"];68 -> 75[label="",style="solid", color="black", weight=3]; 9.93/4.16 69[label="vz40000",fontsize=16,color="green",shape="box"];70[label="vz3000",fontsize=16,color="green",shape="box"];71[label="asAs (not (primEqNat (Succ vz30000) (Succ vz400000))) vz5",fontsize=16,color="black",shape="box"];71 -> 76[label="",style="solid", color="black", weight=3]; 9.93/4.16 72[label="asAs (not (primEqNat (Succ vz30000) Zero)) vz5",fontsize=16,color="black",shape="box"];72 -> 77[label="",style="solid", color="black", weight=3]; 9.93/4.16 73[label="asAs (not (primEqNat Zero (Succ vz400000))) vz5",fontsize=16,color="black",shape="box"];73 -> 78[label="",style="solid", color="black", weight=3]; 9.93/4.16 74[label="asAs (not (primEqNat Zero Zero)) vz5",fontsize=16,color="black",shape="box"];74 -> 79[label="",style="solid", color="black", weight=3]; 9.93/4.16 75[label="MyFalse",fontsize=16,color="green",shape="box"];76 -> 52[label="",style="dashed", color="red", weight=0]; 9.93/4.16 76[label="asAs (not (primEqNat vz30000 vz400000)) vz5",fontsize=16,color="magenta"];76 -> 80[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 76 -> 81[label="",style="dashed", color="magenta", weight=3]; 9.93/4.16 77 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 77[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];78 -> 40[label="",style="dashed", color="red", weight=0]; 9.93/4.16 78[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];79 -> 56[label="",style="dashed", color="red", weight=0]; 9.93/4.16 79[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];80[label="vz400000",fontsize=16,color="green",shape="box"];81[label="vz30000",fontsize=16,color="green",shape="box"];} 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (6) 9.93/4.16 Complex Obligation (AND) 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (7) 9.93/4.16 Obligation: 9.93/4.16 Q DP problem: 9.93/4.16 The TRS P consists of the following rules: 9.93/4.16 9.93/4.16 new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41) 9.93/4.16 9.93/4.16 R is empty. 9.93/4.16 Q is empty. 9.93/4.16 We have to consider all minimal (P,Q,R)-chains. 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (8) QDPSizeChangeProof (EQUIVALENT) 9.93/4.16 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. 9.93/4.16 9.93/4.16 From the DPs we obtained the following set of size-change graphs: 9.93/4.16 *new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41) 9.93/4.16 The graph contains the following edges 1 >= 1, 2 > 2 9.93/4.16 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (9) 9.93/4.16 YES 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (10) 9.93/4.16 Obligation: 9.93/4.16 Q DP problem: 9.93/4.16 The TRS P consists of the following rules: 9.93/4.16 9.93/4.16 new_asAs(Main.Succ(vz30000), Main.Succ(vz400000), vz5) -> new_asAs(vz30000, vz400000, vz5) 9.93/4.16 9.93/4.16 R is empty. 9.93/4.16 Q is empty. 9.93/4.16 We have to consider all minimal (P,Q,R)-chains. 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (11) QDPSizeChangeProof (EQUIVALENT) 9.93/4.16 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. 9.93/4.16 9.93/4.16 From the DPs we obtained the following set of size-change graphs: 9.93/4.16 *new_asAs(Main.Succ(vz30000), Main.Succ(vz400000), vz5) -> new_asAs(vz30000, vz400000, vz5) 9.93/4.16 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 9.93/4.16 9.93/4.16 9.93/4.16 ---------------------------------------- 9.93/4.16 9.93/4.16 (12) 9.93/4.16 YES 10.40/4.22 EOF