8.39/3.70 YES 10.54/4.24 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.54/4.24 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.54/4.24 10.54/4.24 10.54/4.24 H-Termination with start terms of the given HASKELL could be proven: 10.54/4.24 10.54/4.24 (0) HASKELL 10.54/4.24 (1) BR [EQUIVALENT, 0 ms] 10.54/4.24 (2) HASKELL 10.54/4.24 (3) COR [EQUIVALENT, 0 ms] 10.54/4.24 (4) HASKELL 10.54/4.24 (5) Narrow [SOUND, 0 ms] 10.54/4.24 (6) AND 10.54/4.24 (7) QDP 10.54/4.24 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.54/4.24 (9) YES 10.54/4.24 (10) QDP 10.54/4.24 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 10.54/4.24 (12) AND 10.54/4.24 (13) QDP 10.54/4.24 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.54/4.24 (15) YES 10.54/4.24 (16) QDP 10.54/4.24 (17) MRRProof [EQUIVALENT, 25 ms] 10.54/4.24 (18) QDP 10.54/4.24 (19) PisEmptyProof [EQUIVALENT, 0 ms] 10.54/4.24 (20) YES 10.54/4.24 10.54/4.24 10.54/4.24 ---------------------------------------- 10.54/4.24 10.54/4.24 (0) 10.54/4.24 Obligation: 10.54/4.24 mainModule Main 10.54/4.24 module Main where { 10.54/4.24 import qualified Prelude; 10.54/4.24 data MyBool = MyTrue | MyFalse ; 10.54/4.24 10.54/4.24 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.54/4.24 10.54/4.24 data Main.Nat = Succ Main.Nat | Zero ; 10.54/4.24 10.54/4.24 error :: a; 10.54/4.24 error = stop MyTrue; 10.54/4.24 10.54/4.24 primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.24 primDivNatS Main.Zero Main.Zero = Main.error; 10.54/4.24 primDivNatS (Main.Succ x) Main.Zero = Main.error; 10.54/4.24 primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); 10.54/4.24 primDivNatS Main.Zero (Main.Succ x) = Main.Zero; 10.54/4.24 10.54/4.24 primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); 10.54/4.24 primDivNatS0 x y MyFalse = Main.Zero; 10.54/4.24 10.54/4.24 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.54/4.24 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.54/4.24 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.54/4.24 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.54/4.24 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.54/4.24 10.54/4.24 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.24 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.54/4.24 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.54/4.24 primMinusNatS x Main.Zero = x; 10.54/4.24 10.54/4.24 primQuotInt :: MyInt -> MyInt -> MyInt; 10.54/4.24 primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.24 primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.24 primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.24 primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.24 primQuotInt vv vw = Main.error; 10.54/4.24 10.54/4.24 quotMyInt :: MyInt -> MyInt -> MyInt; 10.54/4.24 quotMyInt = primQuotInt; 10.54/4.24 10.54/4.24 stop :: MyBool -> a; 10.54/4.24 stop MyFalse = stop MyFalse; 10.54/4.24 10.54/4.24 } 10.54/4.24 10.54/4.24 ---------------------------------------- 10.54/4.24 10.54/4.24 (1) BR (EQUIVALENT) 10.54/4.24 Replaced joker patterns by fresh variables and removed binding patterns. 10.54/4.24 ---------------------------------------- 10.54/4.24 10.54/4.24 (2) 10.54/4.24 Obligation: 10.54/4.24 mainModule Main 10.54/4.24 module Main where { 10.54/4.24 import qualified Prelude; 10.54/4.24 data MyBool = MyTrue | MyFalse ; 10.54/4.24 10.54/4.24 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.54/4.24 10.54/4.24 data Main.Nat = Succ Main.Nat | Zero ; 10.54/4.24 10.54/4.24 error :: a; 10.54/4.24 error = stop MyTrue; 10.54/4.24 10.54/4.24 primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.24 primDivNatS Main.Zero Main.Zero = Main.error; 10.54/4.24 primDivNatS (Main.Succ x) Main.Zero = Main.error; 10.54/4.24 primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); 10.54/4.25 primDivNatS Main.Zero (Main.Succ x) = Main.Zero; 10.54/4.25 10.54/4.25 primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); 10.54/4.25 primDivNatS0 x y MyFalse = Main.Zero; 10.54/4.25 10.54/4.25 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.54/4.25 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.54/4.25 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.54/4.25 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.54/4.25 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.54/4.25 10.54/4.25 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.25 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.54/4.25 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.54/4.25 primMinusNatS x Main.Zero = x; 10.54/4.25 10.54/4.25 primQuotInt :: MyInt -> MyInt -> MyInt; 10.54/4.25 primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt vv vw = Main.error; 10.54/4.25 10.54/4.25 quotMyInt :: MyInt -> MyInt -> MyInt; 10.54/4.25 quotMyInt = primQuotInt; 10.54/4.25 10.54/4.25 stop :: MyBool -> a; 10.54/4.25 stop MyFalse = stop MyFalse; 10.54/4.25 10.54/4.25 } 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (3) COR (EQUIVALENT) 10.54/4.25 Cond Reductions: 10.54/4.25 The following Function with conditions 10.54/4.25 "undefined |Falseundefined; 10.54/4.25 " 10.54/4.25 is transformed to 10.54/4.25 "undefined = undefined1; 10.54/4.25 " 10.54/4.25 "undefined0 True = undefined; 10.54/4.25 " 10.54/4.25 "undefined1 = undefined0 False; 10.54/4.25 " 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (4) 10.54/4.25 Obligation: 10.54/4.25 mainModule Main 10.54/4.25 module Main where { 10.54/4.25 import qualified Prelude; 10.54/4.25 data MyBool = MyTrue | MyFalse ; 10.54/4.25 10.54/4.25 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.54/4.25 10.54/4.25 data Main.Nat = Succ Main.Nat | Zero ; 10.54/4.25 10.54/4.25 error :: a; 10.54/4.25 error = stop MyTrue; 10.54/4.25 10.54/4.25 primDivNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.25 primDivNatS Main.Zero Main.Zero = Main.error; 10.54/4.25 primDivNatS (Main.Succ x) Main.Zero = Main.error; 10.54/4.25 primDivNatS (Main.Succ x) (Main.Succ y) = primDivNatS0 x y (primGEqNatS x y); 10.54/4.25 primDivNatS Main.Zero (Main.Succ x) = Main.Zero; 10.54/4.25 10.54/4.25 primDivNatS0 x y MyTrue = Main.Succ (primDivNatS (primMinusNatS x y) (Main.Succ y)); 10.54/4.25 primDivNatS0 x y MyFalse = Main.Zero; 10.54/4.25 10.54/4.25 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.54/4.25 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.54/4.25 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.54/4.25 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.54/4.25 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.54/4.25 10.54/4.25 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.54/4.25 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.54/4.25 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.54/4.25 primMinusNatS x Main.Zero = x; 10.54/4.25 10.54/4.25 primQuotInt :: MyInt -> MyInt -> MyInt; 10.54/4.25 primQuotInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Pos (primDivNatS x (Main.Succ y)); 10.54/4.25 primQuotInt vv vw = Main.error; 10.54/4.25 10.54/4.25 quotMyInt :: MyInt -> MyInt -> MyInt; 10.54/4.25 quotMyInt = primQuotInt; 10.54/4.25 10.54/4.25 stop :: MyBool -> a; 10.54/4.25 stop MyFalse = stop MyFalse; 10.54/4.25 10.54/4.25 } 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (5) Narrow (SOUND) 10.54/4.25 Haskell To QDPs 10.54/4.25 10.54/4.25 digraph dp_graph { 10.54/4.25 node [outthreshold=100, inthreshold=100];1[label="quotMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.54/4.25 3[label="quotMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.54/4.25 4[label="quotMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.54/4.25 5[label="primQuotInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];275[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 275[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 275 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 276[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 276[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 276 -> 7[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 6[label="primQuotInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];277[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 277[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 277 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 278[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 278[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 278 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 7[label="primQuotInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];279[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 279[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 279 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 280[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 280[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 280 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 8[label="primQuotInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];281[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 281[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 281 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 282[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 282[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 282 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 9[label="primQuotInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];283[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 283[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 283 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 284[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 284[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 284 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 10[label="primQuotInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];285[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 285[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 285 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 286[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 286[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 286 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 11[label="primQuotInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];287[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 287[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 287 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 288[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 288[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 288 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 12[label="primQuotInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 10.54/4.25 13[label="primQuotInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 10.54/4.25 14[label="primQuotInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 10.54/4.25 15[label="primQuotInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 10.54/4.25 16[label="primQuotInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 10.54/4.25 17[label="primQuotInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 10.54/4.25 18[label="primQuotInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3]; 10.54/4.25 19[label="primQuotInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 10.54/4.25 20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3]; 10.54/4.25 21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3]; 10.54/4.25 22[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3]; 10.54/4.25 23 -> 21[label="",style="dashed", color="red", weight=0]; 10.54/4.25 23[label="error",fontsize=16,color="magenta"];24[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3]; 10.54/4.25 25 -> 21[label="",style="dashed", color="red", weight=0]; 10.54/4.25 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]; 10.54/4.25 27 -> 21[label="",style="dashed", color="red", weight=0]; 10.54/4.25 27[label="error",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];289[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 289[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 289 -> 33[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 290[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 290[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 290 -> 34[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3]; 10.54/4.25 30 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 30[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 31 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 31[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 32 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 38[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 32 -> 39[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3]; 10.54/4.25 34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3]; 10.54/4.25 35[label="error []",fontsize=16,color="red",shape="box"];36[label="vz400",fontsize=16,color="green",shape="box"];37[label="vz30",fontsize=16,color="green",shape="box"];38[label="vz30",fontsize=16,color="green",shape="box"];39[label="vz400",fontsize=16,color="green",shape="box"];40[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];291[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];40 -> 291[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 291 -> 42[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 292[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];40 -> 292[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 292 -> 43[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 41[label="Zero",fontsize=16,color="green",shape="box"];42[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];293[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];42 -> 293[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 293 -> 44[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 294[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 294[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 294 -> 45[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 43[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];295[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];43 -> 295[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 295 -> 46[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 296[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 296[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 296 -> 47[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 44[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3]; 10.54/4.25 45[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3]; 10.54/4.25 46[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3]; 10.54/4.25 47[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3]; 10.54/4.25 48 -> 212[label="",style="dashed", color="red", weight=0]; 10.54/4.25 48[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];48 -> 213[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 48 -> 214[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 48 -> 215[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 48 -> 216[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 49[label="primDivNatS0 (Succ vz3000) Zero MyTrue",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3]; 10.54/4.25 50[label="primDivNatS0 Zero (Succ vz4000) MyFalse",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 10.54/4.25 51[label="primDivNatS0 Zero Zero MyTrue",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3]; 10.54/4.25 213[label="vz3000",fontsize=16,color="green",shape="box"];214[label="vz3000",fontsize=16,color="green",shape="box"];215[label="vz4000",fontsize=16,color="green",shape="box"];216[label="vz4000",fontsize=16,color="green",shape="box"];212[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];297[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];212 -> 297[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 297 -> 245[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 298[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];212 -> 298[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 298 -> 246[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 54[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];54 -> 61[label="",style="dashed", color="green", weight=3]; 10.54/4.25 55[label="Zero",fontsize=16,color="green",shape="box"];56[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];56 -> 62[label="",style="dashed", color="green", weight=3]; 10.54/4.25 245[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];299[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];245 -> 299[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 299 -> 247[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 300[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];245 -> 300[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 300 -> 248[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 246[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];301[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];246 -> 301[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 301 -> 249[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 302[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];246 -> 302[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 302 -> 250[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 61 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 61[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];61 -> 67[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 61 -> 68[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 62 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 62[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];62 -> 69[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 62 -> 70[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 247[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];247 -> 251[label="",style="solid", color="black", weight=3]; 10.54/4.25 248[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];248 -> 252[label="",style="solid", color="black", weight=3]; 10.54/4.25 249[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];249 -> 253[label="",style="solid", color="black", weight=3]; 10.54/4.25 250[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];250 -> 254[label="",style="solid", color="black", weight=3]; 10.54/4.25 67[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="triangle"];67 -> 76[label="",style="solid", color="black", weight=3]; 10.54/4.25 68[label="Zero",fontsize=16,color="green",shape="box"];69[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];69 -> 77[label="",style="solid", color="black", weight=3]; 10.54/4.25 70[label="Zero",fontsize=16,color="green",shape="box"];251 -> 212[label="",style="dashed", color="red", weight=0]; 10.54/4.25 251[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];251 -> 255[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 251 -> 256[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 252[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="black",shape="triangle"];252 -> 257[label="",style="solid", color="black", weight=3]; 10.54/4.25 253[label="primDivNatS0 (Succ vz21) (Succ vz22) MyFalse",fontsize=16,color="black",shape="box"];253 -> 258[label="",style="solid", color="black", weight=3]; 10.54/4.25 254 -> 252[label="",style="dashed", color="red", weight=0]; 10.54/4.25 254[label="primDivNatS0 (Succ vz21) (Succ vz22) MyTrue",fontsize=16,color="magenta"];76[label="Succ vz3000",fontsize=16,color="green",shape="box"];77[label="Zero",fontsize=16,color="green",shape="box"];255[label="vz230",fontsize=16,color="green",shape="box"];256[label="vz240",fontsize=16,color="green",shape="box"];257[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];257 -> 259[label="",style="dashed", color="green", weight=3]; 10.54/4.25 258[label="Zero",fontsize=16,color="green",shape="box"];259 -> 28[label="",style="dashed", color="red", weight=0]; 10.54/4.25 259[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];259 -> 260[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 259 -> 261[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 260[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="black",shape="box"];260 -> 262[label="",style="solid", color="black", weight=3]; 10.54/4.25 261[label="Succ vz22",fontsize=16,color="green",shape="box"];262[label="primMinusNatS vz21 vz22",fontsize=16,color="burlywood",shape="triangle"];303[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];262 -> 303[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 303 -> 263[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 304[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];262 -> 304[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 304 -> 264[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 263[label="primMinusNatS (Succ vz210) vz22",fontsize=16,color="burlywood",shape="box"];305[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];263 -> 305[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 305 -> 265[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 306[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 306[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 306 -> 266[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 264[label="primMinusNatS Zero vz22",fontsize=16,color="burlywood",shape="box"];307[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];264 -> 307[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 307 -> 267[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 308[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];264 -> 308[label="",style="solid", color="burlywood", weight=9]; 10.54/4.25 308 -> 268[label="",style="solid", color="burlywood", weight=3]; 10.54/4.25 265[label="primMinusNatS (Succ vz210) (Succ vz220)",fontsize=16,color="black",shape="box"];265 -> 269[label="",style="solid", color="black", weight=3]; 10.54/4.25 266[label="primMinusNatS (Succ vz210) Zero",fontsize=16,color="black",shape="box"];266 -> 270[label="",style="solid", color="black", weight=3]; 10.54/4.25 267[label="primMinusNatS Zero (Succ vz220)",fontsize=16,color="black",shape="box"];267 -> 271[label="",style="solid", color="black", weight=3]; 10.54/4.25 268[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];268 -> 272[label="",style="solid", color="black", weight=3]; 10.54/4.25 269 -> 262[label="",style="dashed", color="red", weight=0]; 10.54/4.25 269[label="primMinusNatS vz210 vz220",fontsize=16,color="magenta"];269 -> 273[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 269 -> 274[label="",style="dashed", color="magenta", weight=3]; 10.54/4.25 270[label="Succ vz210",fontsize=16,color="green",shape="box"];271[label="Zero",fontsize=16,color="green",shape="box"];272[label="Zero",fontsize=16,color="green",shape="box"];273[label="vz210",fontsize=16,color="green",shape="box"];274[label="vz220",fontsize=16,color="green",shape="box"];} 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (6) 10.54/4.25 Complex Obligation (AND) 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (7) 10.54/4.25 Obligation: 10.54/4.25 Q DP problem: 10.54/4.25 The TRS P consists of the following rules: 10.54/4.25 10.54/4.25 new_primMinusNatS(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS(vz210, vz220) 10.54/4.25 10.54/4.25 R is empty. 10.54/4.25 Q is empty. 10.54/4.25 We have to consider all minimal (P,Q,R)-chains. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (8) QDPSizeChangeProof (EQUIVALENT) 10.54/4.25 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. 10.54/4.25 10.54/4.25 From the DPs we obtained the following set of size-change graphs: 10.54/4.25 *new_primMinusNatS(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS(vz210, vz220) 10.54/4.25 The graph contains the following edges 1 > 1, 2 > 2 10.54/4.25 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (9) 10.54/4.25 YES 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (10) 10.54/4.25 Obligation: 10.54/4.25 Q DP problem: 10.54/4.25 The TRS P consists of the following rules: 10.54/4.25 10.54/4.25 new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero) 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) 10.54/4.25 new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) 10.54/4.25 new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) 10.54/4.25 new_primDivNatS(Main.Succ(Main.Zero), Main.Zero) -> new_primDivNatS(new_primMinusNatS2, Main.Zero) 10.54/4.25 10.54/4.25 The TRS R consists of the following rules: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero 10.54/4.25 new_primMinusNatS1(vz3000) -> Main.Succ(vz3000) 10.54/4.25 new_primMinusNatS2 -> Main.Zero 10.54/4.25 10.54/4.25 The set Q consists of the following terms: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Zero) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) 10.54/4.25 new_primMinusNatS2 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(x0)) 10.54/4.25 new_primMinusNatS1(x0) 10.54/4.25 10.54/4.25 We have to consider all minimal (P,Q,R)-chains. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (11) DependencyGraphProof (EQUIVALENT) 10.54/4.25 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (12) 10.54/4.25 Complex Obligation (AND) 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (13) 10.54/4.25 Obligation: 10.54/4.25 Q DP problem: 10.54/4.25 The TRS P consists of the following rules: 10.54/4.25 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) 10.54/4.25 new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) 10.54/4.25 new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) 10.54/4.25 new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) 10.54/4.25 10.54/4.25 The TRS R consists of the following rules: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero 10.54/4.25 new_primMinusNatS1(vz3000) -> Main.Succ(vz3000) 10.54/4.25 new_primMinusNatS2 -> Main.Zero 10.54/4.25 10.54/4.25 The set Q consists of the following terms: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Zero) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) 10.54/4.25 new_primMinusNatS2 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(x0)) 10.54/4.25 new_primMinusNatS1(x0) 10.54/4.25 10.54/4.25 We have to consider all minimal (P,Q,R)-chains. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (14) QDPSizeChangeProof (EQUIVALENT) 10.54/4.25 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 10.54/4.25 10.54/4.25 Order:Polynomial interpretation [POLO]: 10.54/4.25 10.54/4.25 POL(Main.Succ(x_1)) = 1 + x_1 10.54/4.25 POL(Main.Zero) = 1 10.54/4.25 POL(new_primMinusNatS0(x_1, x_2)) = x_1 10.54/4.25 10.54/4.25 10.54/4.25 10.54/4.25 10.54/4.25 From the DPs we obtained the following set of size-change graphs: 10.54/4.25 *new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2}) 10.54/4.25 The graph contains the following edges 1 >= 1 10.54/4.25 10.54/4.25 10.54/4.25 *new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4}) 10.54/4.25 The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 10.54/4.25 10.54/4.25 10.54/4.25 *new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4}) 10.54/4.25 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.54/4.25 10.54/4.25 10.54/4.25 *new_primDivNatS0(vz21, vz22, Main.Zero, Main.Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2}) 10.54/4.25 The graph contains the following edges 1 >= 1, 2 >= 2 10.54/4.25 10.54/4.25 10.54/4.25 *new_primDivNatS0(vz21, vz22, Main.Succ(vz230), Main.Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Main.Succ(vz22)) (allowed arguments on rhs = {1, 2}) 10.54/4.25 The graph contains the following edges 1 >= 1 10.54/4.25 10.54/4.25 10.54/4.25 10.54/4.25 We oriented the following set of usable rules [AAECC05,FROCOS05]. 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210) 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (15) 10.54/4.25 YES 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (16) 10.54/4.25 Obligation: 10.54/4.25 Q DP problem: 10.54/4.25 The TRS P consists of the following rules: 10.54/4.25 10.54/4.25 new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero) 10.54/4.25 10.54/4.25 The TRS R consists of the following rules: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero 10.54/4.25 new_primMinusNatS1(vz3000) -> Main.Succ(vz3000) 10.54/4.25 new_primMinusNatS2 -> Main.Zero 10.54/4.25 10.54/4.25 The set Q consists of the following terms: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Zero) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) 10.54/4.25 new_primMinusNatS2 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(x0)) 10.54/4.25 new_primMinusNatS1(x0) 10.54/4.25 10.54/4.25 We have to consider all minimal (P,Q,R)-chains. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (17) MRRProof (EQUIVALENT) 10.54/4.25 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. 10.54/4.25 10.54/4.25 Strictly oriented dependency pairs: 10.54/4.25 10.54/4.25 new_primDivNatS(Main.Succ(Main.Succ(vz3000)), Main.Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Main.Zero) 10.54/4.25 10.54/4.25 Strictly oriented rules of the TRS R: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.54/4.25 new_primMinusNatS0(Main.Succ(vz210), Main.Zero) -> Main.Succ(vz210) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) -> Main.Zero 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(vz220)) -> Main.Zero 10.54/4.25 10.54/4.25 Used ordering: Polynomial interpretation [POLO]: 10.54/4.25 10.54/4.25 POL(Main.Succ(x_1)) = 1 + x_1 10.54/4.25 POL(Main.Zero) = 2 10.54/4.25 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 10.54/4.25 POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 10.54/4.25 POL(new_primMinusNatS1(x_1)) = 1 + x_1 10.54/4.25 POL(new_primMinusNatS2) = 2 10.54/4.25 10.54/4.25 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (18) 10.54/4.25 Obligation: 10.54/4.25 Q DP problem: 10.54/4.25 P is empty. 10.54/4.25 The TRS R consists of the following rules: 10.54/4.25 10.54/4.25 new_primMinusNatS1(vz3000) -> Main.Succ(vz3000) 10.54/4.25 new_primMinusNatS2 -> Main.Zero 10.54/4.25 10.54/4.25 The set Q consists of the following terms: 10.54/4.25 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Zero) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Zero) 10.54/4.25 new_primMinusNatS2 10.54/4.25 new_primMinusNatS0(Main.Succ(x0), Main.Succ(x1)) 10.54/4.25 new_primMinusNatS0(Main.Zero, Main.Succ(x0)) 10.54/4.25 new_primMinusNatS1(x0) 10.54/4.25 10.54/4.25 We have to consider all minimal (P,Q,R)-chains. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (19) PisEmptyProof (EQUIVALENT) 10.54/4.25 The TRS P is empty. Hence, there is no (P,Q,R) chain. 10.54/4.25 ---------------------------------------- 10.54/4.25 10.54/4.25 (20) 10.54/4.25 YES 10.54/4.31 EOF