/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 0 ms] (4)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, 0 ms] (15) YES (16) QDP (17) TransformationProof [EQUIVALENT, 0 ms] (18) QDP (19) TransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 7 ms] (22) YES (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; data MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; divMyInt :: MyInt -> MyInt -> MyInt; divMyInt = primDivInt; error :: a; error = stop MyTrue; primDivInt :: MyInt -> MyInt -> MyInt; primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt vv vw = Main.error; primDivNatP :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatP Main.Zero Main.Zero = Main.error; primDivNatP (Main.Succ x) Main.Zero = Main.error; primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y)); primDivNatP Main.Zero (Main.Succ x) = Main.Zero; 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; stop :: MyBool -> a; stop MyFalse = stop MyFalse; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; data MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; divMyInt :: MyInt -> MyInt -> MyInt; divMyInt = primDivInt; error :: a; error = stop MyTrue; primDivInt :: MyInt -> MyInt -> MyInt; primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt vv vw = Main.error; primDivNatP :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatP Main.Zero Main.Zero = Main.error; primDivNatP (Main.Succ x) Main.Zero = Main.error; primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y)); primDivNatP Main.Zero (Main.Succ x) = Main.Zero; 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; stop :: MyBool -> a; stop MyFalse = stop MyFalse; } ---------------------------------------- (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 MyBool = MyTrue | MyFalse ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; divMyInt :: MyInt -> MyInt -> MyInt; divMyInt = primDivInt; error :: a; error = stop MyTrue; primDivInt :: MyInt -> MyInt -> MyInt; primDivInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatP x (Main.Succ y)); primDivInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); primDivInt vv vw = Main.error; primDivNatP :: Main.Nat -> Main.Nat -> Main.Nat; primDivNatP Main.Zero Main.Zero = Main.error; primDivNatP (Main.Succ x) Main.Zero = Main.error; primDivNatP (Main.Succ x) (Main.Succ y) = Main.Succ (primDivNatP (primMinusNatS x y) (Main.Succ y)); primDivNatP Main.Zero (Main.Succ x) = Main.Zero; 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; stop :: MyBool -> a; stop MyFalse = stop MyFalse; } ---------------------------------------- (5) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="divMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="divMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="divMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="primDivInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];289[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 289[label="",style="solid", color="burlywood", weight=9]; 289 -> 6[label="",style="solid", color="burlywood", weight=3]; 290[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 290[label="",style="solid", color="burlywood", weight=9]; 290 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="primDivInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];291[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 291[label="",style="solid", color="burlywood", weight=9]; 291 -> 8[label="",style="solid", color="burlywood", weight=3]; 292[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 292[label="",style="solid", color="burlywood", weight=9]; 292 -> 9[label="",style="solid", color="burlywood", weight=3]; 7[label="primDivInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];293[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 293[label="",style="solid", color="burlywood", weight=9]; 293 -> 10[label="",style="solid", color="burlywood", weight=3]; 294[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 294[label="",style="solid", color="burlywood", weight=9]; 294 -> 11[label="",style="solid", color="burlywood", weight=3]; 8[label="primDivInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];295[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 295[label="",style="solid", color="burlywood", weight=9]; 295 -> 12[label="",style="solid", color="burlywood", weight=3]; 296[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 296[label="",style="solid", color="burlywood", weight=9]; 296 -> 13[label="",style="solid", color="burlywood", weight=3]; 9[label="primDivInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];297[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 297[label="",style="solid", color="burlywood", weight=9]; 297 -> 14[label="",style="solid", color="burlywood", weight=3]; 298[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 298[label="",style="solid", color="burlywood", weight=9]; 298 -> 15[label="",style="solid", color="burlywood", weight=3]; 10[label="primDivInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];299[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 299[label="",style="solid", color="burlywood", weight=9]; 299 -> 16[label="",style="solid", color="burlywood", weight=3]; 300[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 300[label="",style="solid", color="burlywood", weight=9]; 300 -> 17[label="",style="solid", color="burlywood", weight=3]; 11[label="primDivInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];301[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 301[label="",style="solid", color="burlywood", weight=9]; 301 -> 18[label="",style="solid", color="burlywood", weight=3]; 302[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 302[label="",style="solid", color="burlywood", weight=9]; 302 -> 19[label="",style="solid", color="burlywood", weight=3]; 12[label="primDivInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 13[label="primDivInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 14[label="primDivInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 15[label="primDivInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 16[label="primDivInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 17[label="primDivInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 18[label="primDivInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3]; 19[label="primDivInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3]; 21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3]; 22[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3]; 23 -> 21[label="",style="dashed", color="red", weight=0]; 23[label="error",fontsize=16,color="magenta"];24[label="Neg (primDivNatP vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3]; 25 -> 21[label="",style="dashed", color="red", weight=0]; 25[label="error",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3]; 27 -> 21[label="",style="dashed", color="red", weight=0]; 27[label="error",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];303[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 303[label="",style="solid", color="burlywood", weight=9]; 303 -> 33[label="",style="solid", color="burlywood", weight=3]; 304[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 304[label="",style="solid", color="burlywood", weight=9]; 304 -> 34[label="",style="solid", color="burlywood", weight=3]; 29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3]; 30[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];305[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 305[label="",style="solid", color="burlywood", weight=9]; 305 -> 36[label="",style="solid", color="burlywood", weight=3]; 306[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 306[label="",style="solid", color="burlywood", weight=9]; 306 -> 37[label="",style="solid", color="burlywood", weight=3]; 31 -> 30[label="",style="dashed", color="red", weight=0]; 31[label="primDivNatP vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 38[label="",style="dashed", color="magenta", weight=3]; 31 -> 39[label="",style="dashed", color="magenta", weight=3]; 32 -> 28[label="",style="dashed", color="red", weight=0]; 32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 40[label="",style="dashed", color="magenta", weight=3]; 32 -> 41[label="",style="dashed", color="magenta", weight=3]; 33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3]; 34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3]; 35[label="error []",fontsize=16,color="red",shape="box"];36[label="primDivNatP (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3]; 37[label="primDivNatP Zero (Succ vz400)",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3]; 38[label="vz30",fontsize=16,color="green",shape="box"];39[label="vz400",fontsize=16,color="green",shape="box"];40[label="vz30",fontsize=16,color="green",shape="box"];41[label="vz400",fontsize=16,color="green",shape="box"];42[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];307[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 307[label="",style="solid", color="burlywood", weight=9]; 307 -> 46[label="",style="solid", color="burlywood", weight=3]; 308[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 308[label="",style="solid", color="burlywood", weight=9]; 308 -> 47[label="",style="solid", color="burlywood", weight=3]; 43[label="Zero",fontsize=16,color="green",shape="box"];44[label="Succ (primDivNatP (primMinusNatS vz300 vz400) (Succ vz400))",fontsize=16,color="green",shape="box"];44 -> 48[label="",style="dashed", color="green", weight=3]; 45[label="Zero",fontsize=16,color="green",shape="box"];46[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];309[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 309[label="",style="solid", color="burlywood", weight=9]; 309 -> 49[label="",style="solid", color="burlywood", weight=3]; 310[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 310[label="",style="solid", color="burlywood", weight=9]; 310 -> 50[label="",style="solid", color="burlywood", weight=3]; 47[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];311[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 311[label="",style="solid", color="burlywood", weight=9]; 311 -> 51[label="",style="solid", color="burlywood", weight=3]; 312[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 312[label="",style="solid", color="burlywood", weight=9]; 312 -> 52[label="",style="solid", color="burlywood", weight=3]; 48 -> 30[label="",style="dashed", color="red", weight=0]; 48[label="primDivNatP (primMinusNatS vz300 vz400) (Succ vz400)",fontsize=16,color="magenta"];48 -> 53[label="",style="dashed", color="magenta", weight=3]; 49[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3]; 50[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 51[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3]; 52[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3]; 53[label="primMinusNatS vz300 vz400",fontsize=16,color="burlywood",shape="triangle"];313[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];53 -> 313[label="",style="solid", color="burlywood", weight=9]; 313 -> 58[label="",style="solid", color="burlywood", weight=3]; 314[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 314[label="",style="solid", color="burlywood", weight=9]; 314 -> 59[label="",style="solid", color="burlywood", weight=3]; 54 -> 237[label="",style="dashed", color="red", weight=0]; 54[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];54 -> 238[label="",style="dashed", color="magenta", weight=3]; 54 -> 239[label="",style="dashed", color="magenta", weight=3]; 54 -> 240[label="",style="dashed", color="magenta", weight=3]; 54 -> 241[label="",style="dashed", color="magenta", weight=3]; 55[label="primDivNatS0 (Succ vz3000) Zero MyTrue",fontsize=16,color="black",shape="box"];55 -> 62[label="",style="solid", color="black", weight=3]; 56[label="primDivNatS0 Zero (Succ vz4000) MyFalse",fontsize=16,color="black",shape="box"];56 -> 63[label="",style="solid", color="black", weight=3]; 57[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];57 -> 64[label="",style="solid", color="black", weight=3]; 58[label="primMinusNatS (Succ vz3000) vz400",fontsize=16,color="burlywood",shape="box"];315[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 315[label="",style="solid", color="burlywood", weight=9]; 315 -> 65[label="",style="solid", color="burlywood", weight=3]; 316[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 316[label="",style="solid", color="burlywood", weight=9]; 316 -> 66[label="",style="solid", color="burlywood", weight=3]; 59[label="primMinusNatS Zero vz400",fontsize=16,color="burlywood",shape="box"];317[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];59 -> 317[label="",style="solid", color="burlywood", weight=9]; 317 -> 67[label="",style="solid", color="burlywood", weight=3]; 318[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 318[label="",style="solid", color="burlywood", weight=9]; 318 -> 68[label="",style="solid", color="burlywood", weight=3]; 238[label="vz4000",fontsize=16,color="green",shape="box"];239[label="vz4000",fontsize=16,color="green",shape="box"];240[label="vz3000",fontsize=16,color="green",shape="box"];241[label="vz3000",fontsize=16,color="green",shape="box"];237[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];319[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];237 -> 319[label="",style="solid", color="burlywood", weight=9]; 319 -> 270[label="",style="solid", color="burlywood", weight=3]; 320[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];237 -> 320[label="",style="solid", color="burlywood", weight=9]; 320 -> 271[label="",style="solid", color="burlywood", weight=3]; 62[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 73[label="",style="dashed", color="green", weight=3]; 63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 74[label="",style="dashed", color="green", weight=3]; 65[label="primMinusNatS (Succ vz3000) (Succ vz4000)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3]; 66[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3]; 67[label="primMinusNatS Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3]; 68[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3]; 270[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];321[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];270 -> 321[label="",style="solid", color="burlywood", weight=9]; 321 -> 272[label="",style="solid", color="burlywood", weight=3]; 322[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 322[label="",style="solid", color="burlywood", weight=9]; 322 -> 273[label="",style="solid", color="burlywood", weight=3]; 271[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];323[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];271 -> 323[label="",style="solid", color="burlywood", weight=9]; 323 -> 274[label="",style="solid", color="burlywood", weight=3]; 324[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];271 -> 324[label="",style="solid", color="burlywood", weight=9]; 324 -> 275[label="",style="solid", color="burlywood", weight=3]; 73 -> 28[label="",style="dashed", color="red", weight=0]; 73[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];73 -> 83[label="",style="dashed", color="magenta", weight=3]; 73 -> 84[label="",style="dashed", color="magenta", weight=3]; 74 -> 28[label="",style="dashed", color="red", weight=0]; 74[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];74 -> 85[label="",style="dashed", color="magenta", weight=3]; 74 -> 86[label="",style="dashed", color="magenta", weight=3]; 75 -> 53[label="",style="dashed", color="red", weight=0]; 75[label="primMinusNatS vz3000 vz4000",fontsize=16,color="magenta"];75 -> 87[label="",style="dashed", color="magenta", weight=3]; 75 -> 88[label="",style="dashed", color="magenta", weight=3]; 76[label="Succ vz3000",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];78[label="Zero",fontsize=16,color="green",shape="box"];272[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3]; 273[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3]; 274[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3]; 275[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];275 -> 279[label="",style="solid", color="black", weight=3]; 83 -> 53[label="",style="dashed", color="red", weight=0]; 83[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="magenta"];83 -> 94[label="",style="dashed", color="magenta", weight=3]; 83 -> 95[label="",style="dashed", color="magenta", weight=3]; 84[label="Zero",fontsize=16,color="green",shape="box"];85 -> 53[label="",style="dashed", color="red", weight=0]; 85[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];85 -> 96[label="",style="dashed", color="magenta", weight=3]; 85 -> 97[label="",style="dashed", color="magenta", weight=3]; 86[label="Zero",fontsize=16,color="green",shape="box"];87[label="vz4000",fontsize=16,color="green",shape="box"];88[label="vz3000",fontsize=16,color="green",shape="box"];276 -> 237[label="",style="dashed", color="red", weight=0]; 276[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];276 -> 280[label="",style="dashed", color="magenta", weight=3]; 276 -> 281[label="",style="dashed", color="magenta", weight=3]; 277[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="black",shape="triangle"];277 -> 282[label="",style="solid", color="black", weight=3]; 278[label="primDivNatS0 (Succ vz21) (Succ vz22) MyFalse",fontsize=16,color="black",shape="box"];278 -> 283[label="",style="solid", color="black", weight=3]; 279 -> 277[label="",style="dashed", color="red", weight=0]; 279[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="magenta"];94[label="Zero",fontsize=16,color="green",shape="box"];95[label="Succ vz3000",fontsize=16,color="green",shape="box"];96[label="Zero",fontsize=16,color="green",shape="box"];97[label="Zero",fontsize=16,color="green",shape="box"];280[label="vz240",fontsize=16,color="green",shape="box"];281[label="vz230",fontsize=16,color="green",shape="box"];282[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];282 -> 284[label="",style="dashed", color="green", weight=3]; 283[label="Zero",fontsize=16,color="green",shape="box"];284 -> 28[label="",style="dashed", color="red", weight=0]; 284[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];284 -> 285[label="",style="dashed", color="magenta", weight=3]; 284 -> 286[label="",style="dashed", color="magenta", weight=3]; 285 -> 53[label="",style="dashed", color="red", weight=0]; 285[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="magenta"];285 -> 287[label="",style="dashed", color="magenta", weight=3]; 285 -> 288[label="",style="dashed", color="magenta", weight=3]; 286[label="Succ vz22",fontsize=16,color="green",shape="box"];287[label="Succ vz22",fontsize=16,color="green",shape="box"];288[label="Succ vz21",fontsize=16,color="green",shape="box"];} ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatP(Main.Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) 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) = 0 POL(new_primMinusNatS0(x_1, x_2)) = x_1 From the DPs we obtained the following set of size-change graphs: *new_primDivNatP(Main.Succ(vz300), vz400) -> new_primDivNatP(new_primMinusNatS0(vz300, vz400), vz400) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 > 1, 2 >= 2 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero) new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Zero, Main.Zero), Main.Zero) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 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_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 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_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz3000), Main.Zero), Main.Zero) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 > 1, 2 >= 2 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]: (new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)),new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(Main.Succ(vz21), Main.Succ(vz22)), Main.Succ(vz22)) at position [0] we obtained the following new rules [LPAR04]: (new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)),new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) The TRS R consists of the following rules: new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero The set Q consists of the following terms: new_primMinusNatS0(Main.Zero, Main.Zero) new_primMinusNatS0(Main.Zero, Main.Succ(x0)) new_primMinusNatS0(Main.Succ(x0), Main.Zero) new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero new_primMinusNatS0(Main.Zero, Main.Succ(vz4000)) -> Main.Zero new_primMinusNatS0(Main.Succ(vz3000), Main.Zero) -> Main.Succ(vz3000) new_primMinusNatS0(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS0(vz3000, vz4000) ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Main.Succ(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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(vz3000), Main.Succ(vz4000)) -> new_primMinusNatS(vz3000, vz4000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (25) YES