/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.hs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 0 ms] (4) HASKELL (5) Narrow [SOUND, 0 ms] (6) AND (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 29 ms] (15) YES (16) QDP (17) MRRProof [EQUIVALENT, 0 ms] (18) QDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; data Integer = Integer MyInt ; data MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; data Ratio a = CnPc a a ; data Tup2 a b = Tup2 a b ; error :: a; error = stop MyTrue; fromIntegerMyInt :: Integer -> MyInt; fromIntegerMyInt (Integer x) = x; fromIntegral = pt fromIntegerMyInt toIntegerMyInt; primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatS Main.Zero Main.Zero = Main.error; primDivNatS (Main.Succ x) Main.Zero = Main.error; primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); primDivNatS Main.Zero (Main.Succ x) = Main.Zero; primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); primDivNatS0 x y MyFalse = Main.Zero; primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; primGEqNatS (Main.Succ x) Main.Zero = MyTrue; primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; primGEqNatS Main.Zero (Main.Succ x) = MyFalse; primGEqNatS Main.Zero Main.Zero = MyTrue; primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; primMinusNatS x Main.Zero = x; primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; primModNatS Main.Zero Main.Zero = Main.error; primModNatS Main.Zero (Main.Succ x) = Main.Zero; primModNatS (Main.Succ x) Main.Zero = Main.error; primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); primModNatS0 x y MyFalse = Main.Succ x; primQrmInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y); primQuotInt :: MyInt -> MyInt -> MyInt; primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt ww wx = Main.error; primRemInt :: MyInt -> MyInt -> MyInt; primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt vy vz = Main.error; properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw); properFractionQ1 xv xw (Tup2 q vw) = q; properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw); properFractionR0 xv xw (Tup2 vx r) = r; properFractionRatio :: Ratio MyInt -> Tup2 MyInt (Ratio MyInt); properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y); properFractionVu30 xv xw = quotRemMyInt xv xw; pt :: (c -> a) -> (b -> c) -> b -> a; pt f g x = f (g x); quotRemMyInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; quotRemMyInt = primQrmInt; stop :: MyBool -> a; stop MyFalse = stop MyFalse; toIntegerMyInt :: MyInt -> Integer; toIntegerMyInt x = Integer x; truncateM xu = truncateM0 xu (truncateVu6 xu); truncateM0 xu (Tup2 m vv) = m; truncateRatio :: Ratio MyInt -> MyInt; truncateRatio x = truncateM x; truncateVu6 xu = properFractionRatio xu; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; data Integer = Integer MyInt ; data MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; data Ratio a = CnPc a a ; data Tup2 b a = Tup2 b a ; error :: a; error = stop MyTrue; fromIntegerMyInt :: Integer -> MyInt; fromIntegerMyInt (Integer x) = x; fromIntegral = pt fromIntegerMyInt toIntegerMyInt; primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatS Main.Zero Main.Zero = Main.error; primDivNatS (Main.Succ x) Main.Zero = Main.error; primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); primDivNatS Main.Zero (Main.Succ x) = Main.Zero; primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); primDivNatS0 x y MyFalse = Main.Zero; primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; primGEqNatS (Main.Succ x) Main.Zero = MyTrue; primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; primGEqNatS Main.Zero (Main.Succ x) = MyFalse; primGEqNatS Main.Zero Main.Zero = MyTrue; primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; primMinusNatS x Main.Zero = x; primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; primModNatS Main.Zero Main.Zero = Main.error; primModNatS Main.Zero (Main.Succ x) = Main.Zero; primModNatS (Main.Succ x) Main.Zero = Main.error; primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); primModNatS0 x y MyFalse = Main.Succ x; primQrmInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y); primQuotInt :: MyInt -> MyInt -> MyInt; primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt ww wx = Main.error; primRemInt :: MyInt -> MyInt -> MyInt; primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt vy vz = Main.error; properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw); properFractionQ1 xv xw (Tup2 q vw) = q; properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw); properFractionR0 xv xw (Tup2 vx r) = r; properFractionRatio :: Ratio MyInt -> Tup2 MyInt (Ratio MyInt); properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y); properFractionVu30 xv xw = quotRemMyInt xv xw; pt :: (c -> b) -> (a -> c) -> a -> b; pt f g x = f (g x); quotRemMyInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; quotRemMyInt = primQrmInt; stop :: MyBool -> a; stop MyFalse = stop MyFalse; toIntegerMyInt :: MyInt -> Integer; toIntegerMyInt x = Integer x; truncateM xu = truncateM0 xu (truncateVu6 xu); truncateM0 xu (Tup2 m vv) = m; truncateRatio :: Ratio MyInt -> MyInt; truncateRatio x = truncateM x; truncateVu6 xu = properFractionRatio xu; } ---------------------------------------- (3) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; data Integer = Integer MyInt ; data MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; data Ratio a = CnPc a a ; data Tup2 b a = Tup2 b a ; error :: a; error = stop MyTrue; fromIntegerMyInt :: Integer -> MyInt; fromIntegerMyInt (Integer x) = x; fromIntegral = pt fromIntegerMyInt toIntegerMyInt; primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatS Main.Zero Main.Zero = Main.error; primDivNatS (Main.Succ x) Main.Zero = Main.error; primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); primDivNatS Main.Zero (Main.Succ x) = Main.Zero; primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); primDivNatS0 x y MyFalse = Main.Zero; primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; primGEqNatS (Main.Succ x) Main.Zero = MyTrue; primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; primGEqNatS Main.Zero (Main.Succ x) = MyFalse; primGEqNatS Main.Zero Main.Zero = MyTrue; primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; primMinusNatS x Main.Zero = x; primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; primModNatS Main.Zero Main.Zero = Main.error; primModNatS Main.Zero (Main.Succ x) = Main.Zero; primModNatS (Main.Succ x) Main.Zero = Main.error; primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); primModNatS0 x y MyFalse = Main.Succ x; primQrmInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; primQrmInt x y = Tup2 (primQuotInt x y) (primRemInt x y); primQuotInt :: MyInt -> MyInt -> MyInt; primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primQuotInt ww wx = Main.error; primRemInt :: MyInt -> MyInt -> MyInt; primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); primRemInt vy vz = Main.error; properFractionQ xv xw = properFractionQ1 xv xw (properFractionVu30 xv xw); properFractionQ1 xv xw (Tup2 q vw) = q; properFractionR xv xw = properFractionR0 xv xw (properFractionVu30 xv xw); properFractionR0 xv xw (Tup2 vx r) = r; properFractionRatio :: Ratio MyInt -> Tup2 MyInt (Ratio MyInt); properFractionRatio (CnPc x y) = Tup2 (fromIntegral (properFractionQ x y)) (CnPc (properFractionR x y) y); properFractionVu30 xv xw = quotRemMyInt xv xw; pt :: (b -> c) -> (a -> b) -> a -> c; pt f g x = f (g x); quotRemMyInt :: MyInt -> MyInt -> Tup2 MyInt MyInt; quotRemMyInt = primQrmInt; stop :: MyBool -> a; stop MyFalse = stop MyFalse; toIntegerMyInt :: MyInt -> Integer; toIntegerMyInt x = Integer x; truncateM xu = truncateM0 xu (truncateVu6 xu); truncateM0 xu (Tup2 m vv) = m; truncateRatio :: Ratio MyInt -> MyInt; truncateRatio x = truncateM x; truncateVu6 xu = properFractionRatio xu; } ---------------------------------------- (5) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="truncateRatio",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 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34[label="error",fontsize=16,color="black",shape="triangle"];34 -> 42[label="",style="solid", color="black", weight=3]; 35[label="Neg (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];35 -> 43[label="",style="dashed", color="green", weight=3]; 36 -> 34[label="",style="dashed", color="red", weight=0]; 36[label="error",fontsize=16,color="magenta"];37[label="Neg (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];37 -> 44[label="",style="dashed", color="green", weight=3]; 38 -> 34[label="",style="dashed", color="red", weight=0]; 38[label="error",fontsize=16,color="magenta"];39[label="Pos (primDivNatS wy300 (Succ wy3100))",fontsize=16,color="green",shape="box"];39 -> 45[label="",style="dashed", color="green", weight=3]; 40 -> 34[label="",style="dashed", color="red", weight=0]; 40[label="error",fontsize=16,color="magenta"];41[label="primDivNatS wy300 (Succ wy3100)",fontsize=16,color="burlywood",shape="triangle"];303[label="wy300/Succ wy3000",fontsize=10,color="white",style="solid",shape="box"];41 -> 303[label="",style="solid", color="burlywood", weight=9]; 303 -> 46[label="",style="solid", color="burlywood", weight=3]; 304[label="wy300/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 304[label="",style="solid", color="burlywood", weight=9]; 304 -> 47[label="",style="solid", color="burlywood", weight=3]; 42[label="stop MyTrue",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3]; 43 -> 41[label="",style="dashed", color="red", weight=0]; 43[label="primDivNatS wy300 (Succ wy3100)",fontsize=16,color="magenta"];43 -> 49[label="",style="dashed", color="magenta", weight=3]; 44 -> 41[label="",style="dashed", color="red", weight=0]; 44[label="primDivNatS wy300 (Succ wy3100)",fontsize=16,color="magenta"];44 -> 50[label="",style="dashed", color="magenta", weight=3]; 45 -> 41[label="",style="dashed", color="red", weight=0]; 45[label="primDivNatS wy300 (Succ wy3100)",fontsize=16,color="magenta"];45 -> 51[label="",style="dashed", color="magenta", weight=3]; 45 -> 52[label="",style="dashed", color="magenta", weight=3]; 46[label="primDivNatS (Succ wy3000) (Succ wy3100)",fontsize=16,color="black",shape="box"];46 -> 53[label="",style="solid", color="black", weight=3]; 47[label="primDivNatS Zero (Succ wy3100)",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3]; 48[label="error []",fontsize=16,color="red",shape="box"];49[label="wy3100",fontsize=16,color="green",shape="box"];50[label="wy300",fontsize=16,color="green",shape="box"];51[label="wy3100",fontsize=16,color="green",shape="box"];52[label="wy300",fontsize=16,color="green",shape="box"];53[label="primDivNatS0 wy3000 wy3100 (primGEqNatS wy3000 wy3100)",fontsize=16,color="burlywood",shape="box"];305[label="wy3000/Succ wy30000",fontsize=10,color="white",style="solid",shape="box"];53 -> 305[label="",style="solid", color="burlywood", weight=9]; 305 -> 55[label="",style="solid", color="burlywood", weight=3]; 306[label="wy3000/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 306[label="",style="solid", color="burlywood", weight=9]; 306 -> 56[label="",style="solid", color="burlywood", weight=3]; 54[label="Zero",fontsize=16,color="green",shape="box"];55[label="primDivNatS0 (Succ wy30000) wy3100 (primGEqNatS (Succ wy30000) wy3100)",fontsize=16,color="burlywood",shape="box"];307[label="wy3100/Succ wy31000",fontsize=10,color="white",style="solid",shape="box"];55 -> 307[label="",style="solid", color="burlywood", weight=9]; 307 -> 57[label="",style="solid", color="burlywood", weight=3]; 308[label="wy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 308[label="",style="solid", color="burlywood", weight=9]; 308 -> 58[label="",style="solid", color="burlywood", weight=3]; 56[label="primDivNatS0 Zero wy3100 (primGEqNatS Zero wy3100)",fontsize=16,color="burlywood",shape="box"];309[label="wy3100/Succ wy31000",fontsize=10,color="white",style="solid",shape="box"];56 -> 309[label="",style="solid", color="burlywood", weight=9]; 309 -> 59[label="",style="solid", color="burlywood", weight=3]; 310[label="wy3100/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 310[label="",style="solid", color="burlywood", weight=9]; 310 -> 60[label="",style="solid", color="burlywood", weight=3]; 57[label="primDivNatS0 (Succ wy30000) (Succ wy31000) (primGEqNatS (Succ wy30000) (Succ wy31000))",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3]; 58[label="primDivNatS0 (Succ wy30000) Zero (primGEqNatS (Succ wy30000) Zero)",fontsize=16,color="black",shape="box"];58 -> 62[label="",style="solid", color="black", weight=3]; 59[label="primDivNatS0 Zero (Succ wy31000) (primGEqNatS Zero (Succ wy31000))",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3]; 60[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3]; 61 -> 225[label="",style="dashed", color="red", weight=0]; 61[label="primDivNatS0 (Succ wy30000) (Succ wy31000) (primGEqNatS wy30000 wy31000)",fontsize=16,color="magenta"];61 -> 226[label="",style="dashed", color="magenta", weight=3]; 61 -> 227[label="",style="dashed", color="magenta", weight=3]; 61 -> 228[label="",style="dashed", color="magenta", weight=3]; 61 -> 229[label="",style="dashed", color="magenta", weight=3]; 62[label="primDivNatS0 (Succ wy30000) Zero MyTrue",fontsize=16,color="black",shape="box"];62 -> 67[label="",style="solid", color="black", weight=3]; 63[label="primDivNatS0 Zero (Succ wy31000) MyFalse",fontsize=16,color="black",shape="box"];63 -> 68[label="",style="solid", color="black", weight=3]; 64[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];64 -> 69[label="",style="solid", color="black", weight=3]; 226[label="wy31000",fontsize=16,color="green",shape="box"];227[label="wy30000",fontsize=16,color="green",shape="box"];228[label="wy31000",fontsize=16,color="green",shape="box"];229[label="wy30000",fontsize=16,color="green",shape="box"];225[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS wy22 wy23)",fontsize=16,color="burlywood",shape="triangle"];311[label="wy22/Succ wy220",fontsize=10,color="white",style="solid",shape="box"];225 -> 311[label="",style="solid", color="burlywood", weight=9]; 311 -> 258[label="",style="solid", color="burlywood", weight=3]; 312[label="wy22/Zero",fontsize=10,color="white",style="solid",shape="box"];225 -> 312[label="",style="solid", color="burlywood", weight=9]; 312 -> 259[label="",style="solid", color="burlywood", weight=3]; 67[label="Succ (primDivNatS (primMinusNatS (Succ wy30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];67 -> 74[label="",style="dashed", color="green", weight=3]; 68[label="Zero",fontsize=16,color="green",shape="box"];69[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];69 -> 75[label="",style="dashed", color="green", weight=3]; 258[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS (Succ wy220) wy23)",fontsize=16,color="burlywood",shape="box"];313[label="wy23/Succ wy230",fontsize=10,color="white",style="solid",shape="box"];258 -> 313[label="",style="solid", color="burlywood", weight=9]; 313 -> 260[label="",style="solid", color="burlywood", weight=3]; 314[label="wy23/Zero",fontsize=10,color="white",style="solid",shape="box"];258 -> 314[label="",style="solid", color="burlywood", weight=9]; 314 -> 261[label="",style="solid", color="burlywood", weight=3]; 259[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS Zero wy23)",fontsize=16,color="burlywood",shape="box"];315[label="wy23/Succ wy230",fontsize=10,color="white",style="solid",shape="box"];259 -> 315[label="",style="solid", color="burlywood", weight=9]; 315 -> 262[label="",style="solid", color="burlywood", weight=3]; 316[label="wy23/Zero",fontsize=10,color="white",style="solid",shape="box"];259 -> 316[label="",style="solid", color="burlywood", weight=9]; 316 -> 263[label="",style="solid", color="burlywood", weight=3]; 74 -> 41[label="",style="dashed", color="red", weight=0]; 74[label="primDivNatS (primMinusNatS (Succ wy30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 80[label="",style="dashed", color="magenta", weight=3]; 74 -> 81[label="",style="dashed", color="magenta", weight=3]; 75 -> 41[label="",style="dashed", color="red", weight=0]; 75[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];75 -> 82[label="",style="dashed", color="magenta", weight=3]; 75 -> 83[label="",style="dashed", color="magenta", weight=3]; 260[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS (Succ wy220) (Succ wy230))",fontsize=16,color="black",shape="box"];260 -> 264[label="",style="solid", color="black", weight=3]; 261[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS (Succ wy220) Zero)",fontsize=16,color="black",shape="box"];261 -> 265[label="",style="solid", color="black", weight=3]; 262[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS Zero (Succ wy230))",fontsize=16,color="black",shape="box"];262 -> 266[label="",style="solid", color="black", weight=3]; 263[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];263 -> 267[label="",style="solid", color="black", weight=3]; 80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS (Succ wy30000) Zero",fontsize=16,color="black",shape="triangle"];81 -> 89[label="",style="solid", color="black", weight=3]; 82[label="Zero",fontsize=16,color="green",shape="box"];83[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];83 -> 90[label="",style="solid", color="black", weight=3]; 264 -> 225[label="",style="dashed", color="red", weight=0]; 264[label="primDivNatS0 (Succ wy20) (Succ wy21) (primGEqNatS wy220 wy230)",fontsize=16,color="magenta"];264 -> 268[label="",style="dashed", color="magenta", weight=3]; 264 -> 269[label="",style="dashed", color="magenta", weight=3]; 265[label="primDivNatS0 (Succ wy20) (Succ wy21) MyTrue",fontsize=16,color="black",shape="triangle"];265 -> 270[label="",style="solid", color="black", weight=3]; 266[label="primDivNatS0 (Succ wy20) (Succ wy21) MyFalse",fontsize=16,color="black",shape="box"];266 -> 271[label="",style="solid", color="black", weight=3]; 267 -> 265[label="",style="dashed", color="red", weight=0]; 267[label="primDivNatS0 (Succ wy20) (Succ wy21) MyTrue",fontsize=16,color="magenta"];89[label="Succ wy30000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];268[label="wy230",fontsize=16,color="green",shape="box"];269[label="wy220",fontsize=16,color="green",shape="box"];270[label="Succ (primDivNatS (primMinusNatS (Succ wy20) (Succ wy21)) (Succ (Succ wy21)))",fontsize=16,color="green",shape="box"];270 -> 272[label="",style="dashed", color="green", weight=3]; 271[label="Zero",fontsize=16,color="green",shape="box"];272 -> 41[label="",style="dashed", color="red", weight=0]; 272[label="primDivNatS (primMinusNatS (Succ wy20) (Succ wy21)) (Succ (Succ wy21))",fontsize=16,color="magenta"];272 -> 273[label="",style="dashed", color="magenta", weight=3]; 272 -> 274[label="",style="dashed", color="magenta", weight=3]; 273[label="Succ wy21",fontsize=16,color="green",shape="box"];274[label="primMinusNatS (Succ wy20) (Succ wy21)",fontsize=16,color="black",shape="box"];274 -> 275[label="",style="solid", color="black", weight=3]; 275[label="primMinusNatS wy20 wy21",fontsize=16,color="burlywood",shape="triangle"];317[label="wy20/Succ wy200",fontsize=10,color="white",style="solid",shape="box"];275 -> 317[label="",style="solid", color="burlywood", weight=9]; 317 -> 276[label="",style="solid", color="burlywood", weight=3]; 318[label="wy20/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 318[label="",style="solid", color="burlywood", weight=9]; 318 -> 277[label="",style="solid", color="burlywood", weight=3]; 276[label="primMinusNatS (Succ wy200) wy21",fontsize=16,color="burlywood",shape="box"];319[label="wy21/Succ wy210",fontsize=10,color="white",style="solid",shape="box"];276 -> 319[label="",style="solid", color="burlywood", weight=9]; 319 -> 278[label="",style="solid", color="burlywood", weight=3]; 320[label="wy21/Zero",fontsize=10,color="white",style="solid",shape="box"];276 -> 320[label="",style="solid", color="burlywood", weight=9]; 320 -> 279[label="",style="solid", color="burlywood", weight=3]; 277[label="primMinusNatS Zero wy21",fontsize=16,color="burlywood",shape="box"];321[label="wy21/Succ wy210",fontsize=10,color="white",style="solid",shape="box"];277 -> 321[label="",style="solid", color="burlywood", weight=9]; 321 -> 280[label="",style="solid", color="burlywood", weight=3]; 322[label="wy21/Zero",fontsize=10,color="white",style="solid",shape="box"];277 -> 322[label="",style="solid", color="burlywood", weight=9]; 322 -> 281[label="",style="solid", color="burlywood", weight=3]; 278[label="primMinusNatS (Succ wy200) (Succ wy210)",fontsize=16,color="black",shape="box"];278 -> 282[label="",style="solid", color="black", weight=3]; 279[label="primMinusNatS (Succ wy200) Zero",fontsize=16,color="black",shape="box"];279 -> 283[label="",style="solid", color="black", weight=3]; 280[label="primMinusNatS Zero (Succ wy210)",fontsize=16,color="black",shape="box"];280 -> 284[label="",style="solid", color="black", weight=3]; 281[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];281 -> 285[label="",style="solid", color="black", weight=3]; 282 -> 275[label="",style="dashed", color="red", weight=0]; 282[label="primMinusNatS wy200 wy210",fontsize=16,color="magenta"];282 -> 286[label="",style="dashed", color="magenta", weight=3]; 282 -> 287[label="",style="dashed", color="magenta", weight=3]; 283[label="Succ wy200",fontsize=16,color="green",shape="box"];284[label="Zero",fontsize=16,color="green",shape="box"];285[label="Zero",fontsize=16,color="green",shape="box"];286[label="wy200",fontsize=16,color="green",shape="box"];287[label="wy210",fontsize=16,color="green",shape="box"];} ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS(wy200, wy210) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_primMinusNatS(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS(wy200, wy210) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero) new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230) new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21) new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000) new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210) new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero new_primMinusNatS2 -> Main.Zero new_primMinusNatS1(wy30000) -> Main.Succ(wy30000) The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS2 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS1(x0) new_primMinusNatS0(Main.Succ(x0), Main.Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21) new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000) new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230) new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210) new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero new_primMinusNatS2 -> Main.Zero new_primMinusNatS1(wy30000) -> Main.Succ(wy30000) The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS2 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS1(x0) new_primMinusNatS0(Main.Succ(x0), Main.Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPSizeChangeProof (EQUIVALENT) We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. Order:Polynomial interpretation [POLO]: POL(Main.Succ(x_1)) = 1 + x_1 POL(Main.Zero) = 1 POL(new_primMinusNatS0(x_1, x_2)) = x_1 From the DPs we obtained the following set of size-change graphs: *new_primDivNatS00(wy20, wy21) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Succ(wy31000)) -> new_primDivNatS0(wy30000, wy31000, wy30000, wy31000) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Succ(wy230)) -> new_primDivNatS0(wy20, wy21, wy220, wy230) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(wy20, wy21, Main.Zero, Main.Zero) -> new_primDivNatS00(wy20, wy21) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS0(wy20, wy21, Main.Succ(wy220), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(wy20, wy21), Main.Succ(wy21)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200) new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210) ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210) new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero new_primMinusNatS2 -> Main.Zero new_primMinusNatS1(wy30000) -> Main.Succ(wy30000) The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS2 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS1(x0) new_primMinusNatS0(Main.Succ(x0), Main.Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: new_primDivNatS(Main.Succ(Main.Succ(wy30000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(wy30000), Main.Zero) Strictly oriented rules of the TRS R: new_primMinusNatS0(Main.Succ(wy200), Main.Succ(wy210)) -> new_primMinusNatS0(wy200, wy210) new_primMinusNatS0(Main.Succ(wy200), Main.Zero) -> Main.Succ(wy200) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(wy210)) -> Main.Zero Used ordering: Polynomial interpretation [POLO]: POL(Main.Succ(x_1)) = 1 + x_1 POL(Main.Zero) = 2 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(new_primMinusNatS1(x_1)) = 1 + x_1 POL(new_primMinusNatS2) = 2 ---------------------------------------- (18) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: new_primMinusNatS2 -> Main.Zero new_primMinusNatS1(wy30000) -> Main.Succ(wy30000) The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS2 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS1(x0) new_primMinusNatS0(Main.Succ(x0), Main.Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (20) YES