8.61/3.76 YES 10.31/4.26 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.31/4.26 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.31/4.26 10.31/4.26 10.31/4.26 H-Termination with start terms of the given HASKELL could be proven: 10.31/4.26 10.31/4.26 (0) HASKELL 10.31/4.26 (1) BR [EQUIVALENT, 0 ms] 10.31/4.26 (2) HASKELL 10.31/4.26 (3) COR [EQUIVALENT, 0 ms] 10.31/4.26 (4) HASKELL 10.31/4.26 (5) Narrow [SOUND, 0 ms] 10.31/4.26 (6) AND 10.31/4.26 (7) QDP 10.31/4.26 (8) DependencyGraphProof [EQUIVALENT, 0 ms] 10.31/4.26 (9) AND 10.31/4.26 (10) QDP 10.31/4.26 (11) MRRProof [EQUIVALENT, 61 ms] 10.31/4.26 (12) QDP 10.31/4.26 (13) PisEmptyProof [EQUIVALENT, 0 ms] 10.31/4.26 (14) YES 10.31/4.26 (15) QDP 10.31/4.26 (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.31/4.26 (17) YES 10.31/4.26 (18) QDP 10.31/4.26 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.31/4.26 (20) YES 10.31/4.26 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (0) 10.31/4.26 Obligation: 10.31/4.26 mainModule Main 10.31/4.26 module Main where { 10.31/4.26 import qualified Prelude; 10.31/4.26 data MyBool = MyTrue | MyFalse ; 10.31/4.26 10.31/4.26 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.31/4.26 10.31/4.26 data Main.Nat = Succ Main.Nat | Zero ; 10.31/4.26 10.31/4.26 error :: a; 10.31/4.26 error = stop MyTrue; 10.31/4.26 10.31/4.26 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.31/4.26 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.31/4.26 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.31/4.26 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.31/4.26 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.31/4.26 10.31/4.26 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.31/4.26 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.31/4.26 primMinusNatS x Main.Zero = x; 10.31/4.26 10.31/4.26 primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primModNatS Main.Zero Main.Zero = Main.error; 10.31/4.26 primModNatS Main.Zero (Main.Succ x) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) Main.Zero = Main.error; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); 10.31/4.26 10.31/4.26 primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); 10.31/4.26 primModNatS0 x y MyFalse = Main.Succ x; 10.31/4.26 10.31/4.26 primRemInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt vv vw = Main.error; 10.31/4.26 10.31/4.26 remMyInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 remMyInt = primRemInt; 10.31/4.26 10.31/4.26 stop :: MyBool -> a; 10.31/4.26 stop MyFalse = stop MyFalse; 10.31/4.26 10.31/4.26 } 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (1) BR (EQUIVALENT) 10.31/4.26 Replaced joker patterns by fresh variables and removed binding patterns. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (2) 10.31/4.26 Obligation: 10.31/4.26 mainModule Main 10.31/4.26 module Main where { 10.31/4.26 import qualified Prelude; 10.31/4.26 data MyBool = MyTrue | MyFalse ; 10.31/4.26 10.31/4.26 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.31/4.26 10.31/4.26 data Main.Nat = Succ Main.Nat | Zero ; 10.31/4.26 10.31/4.26 error :: a; 10.31/4.26 error = stop MyTrue; 10.31/4.26 10.31/4.26 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.31/4.26 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.31/4.26 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.31/4.26 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.31/4.26 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.31/4.26 10.31/4.26 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.31/4.26 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.31/4.26 primMinusNatS x Main.Zero = x; 10.31/4.26 10.31/4.26 primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primModNatS Main.Zero Main.Zero = Main.error; 10.31/4.26 primModNatS Main.Zero (Main.Succ x) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) Main.Zero = Main.error; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); 10.31/4.26 10.31/4.26 primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); 10.31/4.26 primModNatS0 x y MyFalse = Main.Succ x; 10.31/4.26 10.31/4.26 primRemInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt vv vw = Main.error; 10.31/4.26 10.31/4.26 remMyInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 remMyInt = primRemInt; 10.31/4.26 10.31/4.26 stop :: MyBool -> a; 10.31/4.26 stop MyFalse = stop MyFalse; 10.31/4.26 10.31/4.26 } 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (3) COR (EQUIVALENT) 10.31/4.26 Cond Reductions: 10.31/4.26 The following Function with conditions 10.31/4.26 "undefined |Falseundefined; 10.31/4.26 " 10.31/4.26 is transformed to 10.31/4.26 "undefined = undefined1; 10.31/4.26 " 10.31/4.26 "undefined0 True = undefined; 10.31/4.26 " 10.31/4.26 "undefined1 = undefined0 False; 10.31/4.26 " 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (4) 10.31/4.26 Obligation: 10.31/4.26 mainModule Main 10.31/4.26 module Main where { 10.31/4.26 import qualified Prelude; 10.31/4.26 data MyBool = MyTrue | MyFalse ; 10.31/4.26 10.31/4.26 data MyInt = Pos Main.Nat | Neg Main.Nat ; 10.31/4.26 10.31/4.26 data Main.Nat = Succ Main.Nat | Zero ; 10.31/4.26 10.31/4.26 error :: a; 10.31/4.26 error = stop MyTrue; 10.31/4.26 10.31/4.26 primGEqNatS :: Main.Nat -> Main.Nat -> MyBool; 10.31/4.26 primGEqNatS (Main.Succ x) Main.Zero = MyTrue; 10.31/4.26 primGEqNatS (Main.Succ x) (Main.Succ y) = primGEqNatS x y; 10.31/4.26 primGEqNatS Main.Zero (Main.Succ x) = MyFalse; 10.31/4.26 primGEqNatS Main.Zero Main.Zero = MyTrue; 10.31/4.26 10.31/4.26 primMinusNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primMinusNatS (Main.Succ x) (Main.Succ y) = primMinusNatS x y; 10.31/4.26 primMinusNatS Main.Zero (Main.Succ y) = Main.Zero; 10.31/4.26 primMinusNatS x Main.Zero = x; 10.31/4.26 10.31/4.26 primModNatS :: Main.Nat -> Main.Nat -> Main.Nat; 10.31/4.26 primModNatS Main.Zero Main.Zero = Main.error; 10.31/4.26 primModNatS Main.Zero (Main.Succ x) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) Main.Zero = Main.error; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ Main.Zero) = Main.Zero; 10.31/4.26 primModNatS (Main.Succ x) (Main.Succ (Main.Succ y)) = primModNatS0 x y (primGEqNatS x (Main.Succ y)); 10.31/4.26 10.31/4.26 primModNatS0 x y MyTrue = primModNatS (primMinusNatS x (Main.Succ y)) (Main.Succ (Main.Succ y)); 10.31/4.26 primModNatS0 x y MyFalse = Main.Succ x; 10.31/4.26 10.31/4.26 primRemInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 primRemInt (Main.Pos x) (Main.Pos (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Pos x) (Main.Neg (Main.Succ y)) = Main.Pos (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Pos (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt (Main.Neg x) (Main.Neg (Main.Succ y)) = Main.Neg (primModNatS x (Main.Succ y)); 10.31/4.26 primRemInt vv vw = Main.error; 10.31/4.26 10.31/4.26 remMyInt :: MyInt -> MyInt -> MyInt; 10.31/4.26 remMyInt = primRemInt; 10.31/4.26 10.31/4.26 stop :: MyBool -> a; 10.31/4.26 stop MyFalse = stop MyFalse; 10.31/4.26 10.31/4.26 } 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (5) Narrow (SOUND) 10.31/4.26 Haskell To QDPs 10.31/4.26 10.31/4.26 digraph dp_graph { 10.31/4.26 node [outthreshold=100, inthreshold=100];1[label="remMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.31/4.26 3[label="remMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.31/4.26 4[label="remMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.31/4.26 5[label="primRemInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];299[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 299[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 299 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 300[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 300[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 300 -> 7[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 6[label="primRemInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];301[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 301[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 301 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 302[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 302[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 302 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 7[label="primRemInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];303[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 303[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 303 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 304[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 304[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 304 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 8[label="primRemInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];305[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 305[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 305 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 306[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 306[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 306 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 9[label="primRemInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];307[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 307[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 307 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 308[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 308[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 308 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 10[label="primRemInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];309[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 309[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 309 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 310[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 310[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 310 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 11[label="primRemInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];311[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 311[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 311 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 312[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 312[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 312 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 12[label="primRemInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 10.31/4.26 13[label="primRemInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 10.31/4.26 14[label="primRemInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 10.31/4.26 15[label="primRemInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 10.31/4.26 16[label="primRemInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 10.31/4.26 17[label="primRemInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 10.31/4.26 18[label="primRemInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3]; 10.31/4.26 19[label="primRemInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 10.31/4.26 20[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3]; 10.31/4.26 21[label="error",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3]; 10.31/4.26 22[label="Pos (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3]; 10.31/4.26 23 -> 21[label="",style="dashed", color="red", weight=0]; 10.31/4.26 23[label="error",fontsize=16,color="magenta"];24[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3]; 10.31/4.26 25 -> 21[label="",style="dashed", color="red", weight=0]; 10.31/4.26 25[label="error",fontsize=16,color="magenta"];26[label="Neg (primModNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3]; 10.31/4.26 27 -> 21[label="",style="dashed", color="red", weight=0]; 10.31/4.26 27[label="error",fontsize=16,color="magenta"];28[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];313[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 313[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 313 -> 33[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 314[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 314[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 314 -> 34[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 29[label="stop MyTrue",fontsize=16,color="black",shape="box"];29 -> 35[label="",style="solid", color="black", weight=3]; 10.31/4.26 30 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 30[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 31 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 31[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 32 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 32[label="primModNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 38[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 32 -> 39[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 33[label="primModNatS (Succ vz300) (Succ vz400)",fontsize=16,color="burlywood",shape="box"];315[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 315[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 315 -> 40[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 316[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 316[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 316 -> 41[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 34[label="primModNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3]; 10.31/4.26 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="vz400",fontsize=16,color="green",shape="box"];39[label="vz30",fontsize=16,color="green",shape="box"];40[label="primModNatS (Succ vz300) (Succ (Succ vz4000))",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3]; 10.31/4.26 41[label="primModNatS (Succ vz300) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 44[label="",style="solid", color="black", weight=3]; 10.31/4.26 42[label="Zero",fontsize=16,color="green",shape="box"];43[label="primModNatS0 vz300 vz4000 (primGEqNatS vz300 (Succ vz4000))",fontsize=16,color="burlywood",shape="box"];317[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];43 -> 317[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 317 -> 45[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 318[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];43 -> 318[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 318 -> 46[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 44[label="Zero",fontsize=16,color="green",shape="box"];45[label="primModNatS0 (Succ vz3000) vz4000 (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3]; 10.31/4.26 46[label="primModNatS0 Zero vz4000 (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3]; 10.31/4.26 47[label="primModNatS0 (Succ vz3000) vz4000 (primGEqNatS vz3000 vz4000)",fontsize=16,color="burlywood",shape="box"];319[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];47 -> 319[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 319 -> 49[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 320[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 320[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 320 -> 50[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 48[label="primModNatS0 Zero vz4000 MyFalse",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3]; 10.31/4.26 49[label="primModNatS0 (Succ (Succ vz30000)) vz4000 (primGEqNatS (Succ vz30000) vz4000)",fontsize=16,color="burlywood",shape="box"];321[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];49 -> 321[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 321 -> 52[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 322[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 322[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 322 -> 53[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 50[label="primModNatS0 (Succ Zero) vz4000 (primGEqNatS Zero vz4000)",fontsize=16,color="burlywood",shape="box"];323[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];50 -> 323[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 323 -> 54[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 324[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 324[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 324 -> 55[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 51[label="Succ Zero",fontsize=16,color="green",shape="box"];52[label="primModNatS0 (Succ (Succ vz30000)) (Succ vz40000) (primGEqNatS (Succ vz30000) (Succ vz40000))",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3]; 10.31/4.26 53[label="primModNatS0 (Succ (Succ vz30000)) Zero (primGEqNatS (Succ vz30000) Zero)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3]; 10.31/4.26 54[label="primModNatS0 (Succ Zero) (Succ vz40000) (primGEqNatS Zero (Succ vz40000))",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3]; 10.31/4.26 55[label="primModNatS0 (Succ Zero) Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 10.31/4.26 56 -> 229[label="",style="dashed", color="red", weight=0]; 10.31/4.26 56[label="primModNatS0 (Succ (Succ vz30000)) (Succ vz40000) (primGEqNatS vz30000 vz40000)",fontsize=16,color="magenta"];56 -> 230[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 56 -> 231[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 56 -> 232[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 56 -> 233[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 57[label="primModNatS0 (Succ (Succ vz30000)) Zero MyTrue",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3]; 10.31/4.26 58[label="primModNatS0 (Succ Zero) (Succ vz40000) MyFalse",fontsize=16,color="black",shape="box"];58 -> 63[label="",style="solid", color="black", weight=3]; 10.31/4.26 59[label="primModNatS0 (Succ Zero) Zero MyTrue",fontsize=16,color="black",shape="box"];59 -> 64[label="",style="solid", color="black", weight=3]; 10.31/4.26 230[label="vz40000",fontsize=16,color="green",shape="box"];231[label="vz40000",fontsize=16,color="green",shape="box"];232[label="vz30000",fontsize=16,color="green",shape="box"];233[label="Succ vz30000",fontsize=16,color="green",shape="box"];229[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS vz20 vz21)",fontsize=16,color="burlywood",shape="triangle"];325[label="vz20/Succ vz200",fontsize=10,color="white",style="solid",shape="box"];229 -> 325[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 325 -> 266[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 326[label="vz20/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 326[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 326 -> 267[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 62 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 62[label="primModNatS (primMinusNatS (Succ (Succ vz30000)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];62 -> 69[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 62 -> 70[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 63[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];64 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 64[label="primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];64 -> 71[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 64 -> 72[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 266[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS (Succ vz200) vz21)",fontsize=16,color="burlywood",shape="box"];327[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];266 -> 327[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 327 -> 268[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 328[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];266 -> 328[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 328 -> 269[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 267[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS Zero vz21)",fontsize=16,color="burlywood",shape="box"];329[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];267 -> 329[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 329 -> 270[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 330[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];267 -> 330[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 330 -> 271[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 69[label="Succ Zero",fontsize=16,color="green",shape="box"];70[label="primMinusNatS (Succ (Succ vz30000)) (Succ Zero)",fontsize=16,color="black",shape="triangle"];70 -> 77[label="",style="solid", color="black", weight=3]; 10.31/4.26 71[label="Succ Zero",fontsize=16,color="green",shape="box"];72[label="primMinusNatS (Succ Zero) (Succ Zero)",fontsize=16,color="black",shape="triangle"];72 -> 78[label="",style="solid", color="black", weight=3]; 10.31/4.26 268[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS (Succ vz200) (Succ vz210))",fontsize=16,color="black",shape="box"];268 -> 272[label="",style="solid", color="black", weight=3]; 10.31/4.26 269[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS (Succ vz200) Zero)",fontsize=16,color="black",shape="box"];269 -> 273[label="",style="solid", color="black", weight=3]; 10.31/4.26 270[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS Zero (Succ vz210))",fontsize=16,color="black",shape="box"];270 -> 274[label="",style="solid", color="black", weight=3]; 10.31/4.26 271[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3]; 10.31/4.26 77[label="primMinusNatS (Succ vz30000) Zero",fontsize=16,color="black",shape="box"];77 -> 84[label="",style="solid", color="black", weight=3]; 10.31/4.26 78[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];78 -> 85[label="",style="solid", color="black", weight=3]; 10.31/4.26 272 -> 229[label="",style="dashed", color="red", weight=0]; 10.31/4.26 272[label="primModNatS0 (Succ vz18) (Succ vz19) (primGEqNatS vz200 vz210)",fontsize=16,color="magenta"];272 -> 276[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 272 -> 277[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 273[label="primModNatS0 (Succ vz18) (Succ vz19) MyTrue",fontsize=16,color="black",shape="triangle"];273 -> 278[label="",style="solid", color="black", weight=3]; 10.31/4.26 274[label="primModNatS0 (Succ vz18) (Succ vz19) MyFalse",fontsize=16,color="black",shape="box"];274 -> 279[label="",style="solid", color="black", weight=3]; 10.31/4.26 275 -> 273[label="",style="dashed", color="red", weight=0]; 10.31/4.26 275[label="primModNatS0 (Succ vz18) (Succ vz19) MyTrue",fontsize=16,color="magenta"];84[label="Succ vz30000",fontsize=16,color="green",shape="box"];85[label="Zero",fontsize=16,color="green",shape="box"];276[label="vz210",fontsize=16,color="green",shape="box"];277[label="vz200",fontsize=16,color="green",shape="box"];278 -> 28[label="",style="dashed", color="red", weight=0]; 10.31/4.26 278[label="primModNatS (primMinusNatS (Succ vz18) (Succ (Succ vz19))) (Succ (Succ (Succ vz19)))",fontsize=16,color="magenta"];278 -> 280[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 278 -> 281[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 279[label="Succ (Succ vz18)",fontsize=16,color="green",shape="box"];280[label="Succ (Succ vz19)",fontsize=16,color="green",shape="box"];281[label="primMinusNatS (Succ vz18) (Succ (Succ vz19))",fontsize=16,color="black",shape="box"];281 -> 282[label="",style="solid", color="black", weight=3]; 10.31/4.26 282[label="primMinusNatS vz18 (Succ vz19)",fontsize=16,color="burlywood",shape="box"];331[label="vz18/Succ vz180",fontsize=10,color="white",style="solid",shape="box"];282 -> 331[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 331 -> 283[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 332[label="vz18/Zero",fontsize=10,color="white",style="solid",shape="box"];282 -> 332[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 332 -> 284[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 283[label="primMinusNatS (Succ vz180) (Succ vz19)",fontsize=16,color="black",shape="box"];283 -> 285[label="",style="solid", color="black", weight=3]; 10.31/4.26 284[label="primMinusNatS Zero (Succ vz19)",fontsize=16,color="black",shape="box"];284 -> 286[label="",style="solid", color="black", weight=3]; 10.31/4.26 285[label="primMinusNatS vz180 vz19",fontsize=16,color="burlywood",shape="triangle"];333[label="vz180/Succ vz1800",fontsize=10,color="white",style="solid",shape="box"];285 -> 333[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 333 -> 287[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 334[label="vz180/Zero",fontsize=10,color="white",style="solid",shape="box"];285 -> 334[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 334 -> 288[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 286[label="Zero",fontsize=16,color="green",shape="box"];287[label="primMinusNatS (Succ vz1800) vz19",fontsize=16,color="burlywood",shape="box"];335[label="vz19/Succ vz190",fontsize=10,color="white",style="solid",shape="box"];287 -> 335[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 335 -> 289[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 336[label="vz19/Zero",fontsize=10,color="white",style="solid",shape="box"];287 -> 336[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 336 -> 290[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 288[label="primMinusNatS Zero vz19",fontsize=16,color="burlywood",shape="box"];337[label="vz19/Succ vz190",fontsize=10,color="white",style="solid",shape="box"];288 -> 337[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 337 -> 291[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 338[label="vz19/Zero",fontsize=10,color="white",style="solid",shape="box"];288 -> 338[label="",style="solid", color="burlywood", weight=9]; 10.31/4.26 338 -> 292[label="",style="solid", color="burlywood", weight=3]; 10.31/4.26 289[label="primMinusNatS (Succ vz1800) (Succ vz190)",fontsize=16,color="black",shape="box"];289 -> 293[label="",style="solid", color="black", weight=3]; 10.31/4.26 290[label="primMinusNatS (Succ vz1800) Zero",fontsize=16,color="black",shape="box"];290 -> 294[label="",style="solid", color="black", weight=3]; 10.31/4.26 291[label="primMinusNatS Zero (Succ vz190)",fontsize=16,color="black",shape="box"];291 -> 295[label="",style="solid", color="black", weight=3]; 10.31/4.26 292[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];292 -> 296[label="",style="solid", color="black", weight=3]; 10.31/4.26 293 -> 285[label="",style="dashed", color="red", weight=0]; 10.31/4.26 293[label="primMinusNatS vz1800 vz190",fontsize=16,color="magenta"];293 -> 297[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 293 -> 298[label="",style="dashed", color="magenta", weight=3]; 10.31/4.26 294[label="Succ vz1800",fontsize=16,color="green",shape="box"];295[label="Zero",fontsize=16,color="green",shape="box"];296[label="Zero",fontsize=16,color="green",shape="box"];297[label="vz190",fontsize=16,color="green",shape="box"];298[label="vz1800",fontsize=16,color="green",shape="box"];} 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (6) 10.31/4.26 Complex Obligation (AND) 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (7) 10.31/4.26 Obligation: 10.31/4.26 Q DP problem: 10.31/4.26 The TRS P consists of the following rules: 10.31/4.26 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Zero, Main.Zero) -> new_primModNatS00(vz18, vz19) 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1(vz30000), Main.Succ(Main.Zero)) 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Succ(vz210)) -> new_primModNatS0(vz18, vz19, vz200, vz210) 10.31/4.26 new_primModNatS00(vz18, vz19) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000) 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Zero) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Zero)), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS2, Main.Succ(Main.Zero)) 10.31/4.26 10.31/4.26 The TRS R consists of the following rules: 10.31/4.26 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(vz190)) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS3(vz1800, vz190) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Zero) -> Main.Succ(vz1800) 10.31/4.26 new_primMinusNatS0(Main.Succ(vz180), vz19) -> new_primMinusNatS3(vz180, vz19) 10.31/4.26 new_primMinusNatS0(Main.Zero, vz19) -> Main.Zero 10.31/4.26 new_primMinusNatS2 -> Main.Zero 10.31/4.26 new_primMinusNatS1(vz30000) -> Main.Succ(vz30000) 10.31/4.26 10.31/4.26 The set Q consists of the following terms: 10.31/4.26 10.31/4.26 new_primMinusNatS1(x0) 10.31/4.26 new_primMinusNatS2 10.31/4.26 new_primMinusNatS0(Main.Succ(x0), x1) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Zero) 10.31/4.26 new_primMinusNatS0(Main.Zero, x0) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(x0)) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1)) 10.31/4.26 10.31/4.26 We have to consider all minimal (P,Q,R)-chains. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (8) DependencyGraphProof (EQUIVALENT) 10.31/4.26 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (9) 10.31/4.26 Complex Obligation (AND) 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (10) 10.31/4.26 Obligation: 10.31/4.26 Q DP problem: 10.31/4.26 The TRS P consists of the following rules: 10.31/4.26 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1(vz30000), Main.Succ(Main.Zero)) 10.31/4.26 10.31/4.26 The TRS R consists of the following rules: 10.31/4.26 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(vz190)) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS3(vz1800, vz190) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Zero) -> Main.Succ(vz1800) 10.31/4.26 new_primMinusNatS0(Main.Succ(vz180), vz19) -> new_primMinusNatS3(vz180, vz19) 10.31/4.26 new_primMinusNatS0(Main.Zero, vz19) -> Main.Zero 10.31/4.26 new_primMinusNatS2 -> Main.Zero 10.31/4.26 new_primMinusNatS1(vz30000) -> Main.Succ(vz30000) 10.31/4.26 10.31/4.26 The set Q consists of the following terms: 10.31/4.26 10.31/4.26 new_primMinusNatS1(x0) 10.31/4.26 new_primMinusNatS2 10.31/4.26 new_primMinusNatS0(Main.Succ(x0), x1) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Zero) 10.31/4.26 new_primMinusNatS0(Main.Zero, x0) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(x0)) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1)) 10.31/4.26 10.31/4.26 We have to consider all minimal (P,Q,R)-chains. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (11) MRRProof (EQUIVALENT) 10.31/4.26 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. 10.31/4.26 10.31/4.26 Strictly oriented dependency pairs: 10.31/4.26 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Zero)) -> new_primModNatS(new_primMinusNatS1(vz30000), Main.Succ(Main.Zero)) 10.31/4.26 10.31/4.26 Strictly oriented rules of the TRS R: 10.31/4.26 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(vz190)) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS3(vz1800, vz190) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Zero) -> Main.Succ(vz1800) 10.31/4.26 new_primMinusNatS0(Main.Succ(vz180), vz19) -> new_primMinusNatS3(vz180, vz19) 10.31/4.26 new_primMinusNatS0(Main.Zero, vz19) -> Main.Zero 10.31/4.26 new_primMinusNatS1(vz30000) -> Main.Succ(vz30000) 10.31/4.26 10.31/4.26 Used ordering: Polynomial interpretation [POLO]: 10.31/4.26 10.31/4.26 POL(Main.Succ(x_1)) = 1 + x_1 10.31/4.26 POL(Main.Zero) = 2 10.31/4.26 POL(new_primMinusNatS0(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 10.31/4.26 POL(new_primMinusNatS1(x_1)) = 2 + x_1 10.31/4.26 POL(new_primMinusNatS2) = 2 10.31/4.26 POL(new_primMinusNatS3(x_1, x_2)) = 2*x_1 + x_2 10.31/4.26 POL(new_primModNatS(x_1, x_2)) = x_1 + x_2 10.31/4.26 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (12) 10.31/4.26 Obligation: 10.31/4.26 Q DP problem: 10.31/4.26 P is empty. 10.31/4.26 The TRS R consists of the following rules: 10.31/4.26 10.31/4.26 new_primMinusNatS2 -> Main.Zero 10.31/4.26 10.31/4.26 The set Q consists of the following terms: 10.31/4.26 10.31/4.26 new_primMinusNatS1(x0) 10.31/4.26 new_primMinusNatS2 10.31/4.26 new_primMinusNatS0(Main.Succ(x0), x1) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Zero) 10.31/4.26 new_primMinusNatS0(Main.Zero, x0) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(x0)) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1)) 10.31/4.26 10.31/4.26 We have to consider all minimal (P,Q,R)-chains. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (13) PisEmptyProof (EQUIVALENT) 10.31/4.26 The TRS P is empty. Hence, there is no (P,Q,R) chain. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (14) 10.31/4.26 YES 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (15) 10.31/4.26 Obligation: 10.31/4.26 Q DP problem: 10.31/4.26 The TRS P consists of the following rules: 10.31/4.26 10.31/4.26 new_primModNatS00(vz18, vz19) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) 10.31/4.26 new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000) 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Zero, Main.Zero) -> new_primModNatS00(vz18, vz19) 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Succ(vz210)) -> new_primModNatS0(vz18, vz19, vz200, vz210) 10.31/4.26 new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Zero) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) 10.31/4.26 10.31/4.26 The TRS R consists of the following rules: 10.31/4.26 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(vz190)) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS3(vz1800, vz190) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Zero) -> Main.Succ(vz1800) 10.31/4.26 new_primMinusNatS0(Main.Succ(vz180), vz19) -> new_primMinusNatS3(vz180, vz19) 10.31/4.26 new_primMinusNatS0(Main.Zero, vz19) -> Main.Zero 10.31/4.26 new_primMinusNatS2 -> Main.Zero 10.31/4.26 new_primMinusNatS1(vz30000) -> Main.Succ(vz30000) 10.31/4.26 10.31/4.26 The set Q consists of the following terms: 10.31/4.26 10.31/4.26 new_primMinusNatS1(x0) 10.31/4.26 new_primMinusNatS2 10.31/4.26 new_primMinusNatS0(Main.Succ(x0), x1) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Zero) 10.31/4.26 new_primMinusNatS0(Main.Zero, x0) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(x0)) 10.31/4.26 new_primMinusNatS3(Main.Succ(x0), Main.Succ(x1)) 10.31/4.26 10.31/4.26 We have to consider all minimal (P,Q,R)-chains. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (16) QDPSizeChangeProof (EQUIVALENT) 10.31/4.26 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 10.31/4.26 10.31/4.26 Order:Polynomial interpretation [POLO]: 10.31/4.26 10.31/4.26 POL(Main.Succ(x_1)) = 1 + x_1 10.31/4.26 POL(Main.Zero) = 1 10.31/4.26 POL(new_primMinusNatS0(x_1, x_2)) = x_1 10.31/4.26 POL(new_primMinusNatS3(x_1, x_2)) = x_1 10.31/4.26 10.31/4.26 10.31/4.26 10.31/4.26 10.31/4.26 From the DPs we obtained the following set of size-change graphs: 10.31/4.26 *new_primModNatS(Main.Succ(Main.Succ(Main.Succ(vz30000))), Main.Succ(Main.Succ(vz40000))) -> new_primModNatS0(Main.Succ(vz30000), vz40000, vz30000, vz40000) (allowed arguments on rhs = {1, 2, 3, 4}) 10.31/4.26 The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 10.31/4.26 10.31/4.26 10.31/4.26 *new_primModNatS0(vz18, vz19, Main.Zero, Main.Zero) -> new_primModNatS00(vz18, vz19) (allowed arguments on rhs = {1, 2}) 10.31/4.26 The graph contains the following edges 1 >= 1, 2 >= 2 10.31/4.26 10.31/4.26 10.31/4.26 *new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Zero) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) (allowed arguments on rhs = {1, 2}) 10.31/4.26 The graph contains the following edges 1 >= 1 10.31/4.26 10.31/4.26 10.31/4.26 *new_primModNatS0(vz18, vz19, Main.Succ(vz200), Main.Succ(vz210)) -> new_primModNatS0(vz18, vz19, vz200, vz210) (allowed arguments on rhs = {1, 2, 3, 4}) 10.31/4.26 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.31/4.26 10.31/4.26 10.31/4.26 *new_primModNatS00(vz18, vz19) -> new_primModNatS(new_primMinusNatS0(vz18, vz19), Main.Succ(Main.Succ(vz19))) (allowed arguments on rhs = {1, 2}) 10.31/4.26 The graph contains the following edges 1 >= 1 10.31/4.26 10.31/4.26 10.31/4.26 10.31/4.26 We oriented the following set of usable rules [AAECC05,FROCOS05]. 10.31/4.26 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Zero) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Zero, Main.Succ(vz190)) -> Main.Zero 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Zero) -> Main.Succ(vz1800) 10.31/4.26 new_primMinusNatS3(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS3(vz1800, vz190) 10.31/4.26 new_primMinusNatS0(Main.Zero, vz19) -> Main.Zero 10.31/4.26 new_primMinusNatS0(Main.Succ(vz180), vz19) -> new_primMinusNatS3(vz180, vz19) 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (17) 10.31/4.26 YES 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (18) 10.31/4.26 Obligation: 10.31/4.26 Q DP problem: 10.31/4.26 The TRS P consists of the following rules: 10.31/4.26 10.31/4.26 new_primMinusNatS(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS(vz1800, vz190) 10.31/4.26 10.31/4.26 R is empty. 10.31/4.26 Q is empty. 10.31/4.26 We have to consider all minimal (P,Q,R)-chains. 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (19) QDPSizeChangeProof (EQUIVALENT) 10.31/4.26 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. 10.31/4.26 10.31/4.26 From the DPs we obtained the following set of size-change graphs: 10.31/4.26 *new_primMinusNatS(Main.Succ(vz1800), Main.Succ(vz190)) -> new_primMinusNatS(vz1800, vz190) 10.31/4.26 The graph contains the following edges 1 > 1, 2 > 2 10.31/4.26 10.31/4.26 10.31/4.26 ---------------------------------------- 10.31/4.26 10.31/4.26 (20) 10.31/4.26 YES 10.58/4.29 EOF