7.73/3.51 YES 9.59/4.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.59/4.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.59/4.02 9.59/4.02 9.59/4.02 H-Termination with start terms of the given HASKELL could be proven: 9.59/4.02 9.59/4.02 (0) HASKELL 9.59/4.02 (1) BR [EQUIVALENT, 0 ms] 9.59/4.02 (2) HASKELL 9.59/4.02 (3) COR [EQUIVALENT, 0 ms] 9.59/4.02 (4) HASKELL 9.59/4.02 (5) Narrow [SOUND, 0 ms] 9.59/4.02 (6) QDP 9.59/4.02 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.59/4.02 (8) YES 9.59/4.02 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (0) 9.59/4.02 Obligation: 9.59/4.02 mainModule Main 9.59/4.02 module Main where { 9.59/4.02 import qualified Prelude; 9.59/4.02 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.59/4.02 9.59/4.02 data Main.Nat = Succ Main.Nat | Zero ; 9.59/4.02 9.59/4.02 data Ordering = LT | EQ | GT ; 9.59/4.02 9.59/4.02 compareMyInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 compareMyInt = primCmpInt; 9.59/4.02 9.59/4.02 primCmpInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 9.59/4.02 9.59/4.02 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 9.59/4.02 primCmpNat Main.Zero Main.Zero = EQ; 9.59/4.02 primCmpNat Main.Zero (Main.Succ y) = LT; 9.59/4.02 primCmpNat (Main.Succ x) Main.Zero = GT; 9.59/4.02 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 9.59/4.02 9.59/4.02 } 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (1) BR (EQUIVALENT) 9.59/4.02 Replaced joker patterns by fresh variables and removed binding patterns. 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (2) 9.59/4.02 Obligation: 9.59/4.02 mainModule Main 9.59/4.02 module Main where { 9.59/4.02 import qualified Prelude; 9.59/4.02 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.59/4.02 9.59/4.02 data Main.Nat = Succ Main.Nat | Zero ; 9.59/4.02 9.59/4.02 data Ordering = LT | EQ | GT ; 9.59/4.02 9.59/4.02 compareMyInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 compareMyInt = primCmpInt; 9.59/4.02 9.59/4.02 primCmpInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 9.59/4.02 9.59/4.02 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 9.59/4.02 primCmpNat Main.Zero Main.Zero = EQ; 9.59/4.02 primCmpNat Main.Zero (Main.Succ y) = LT; 9.59/4.02 primCmpNat (Main.Succ x) Main.Zero = GT; 9.59/4.02 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 9.59/4.02 9.59/4.02 } 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (3) COR (EQUIVALENT) 9.59/4.02 Cond Reductions: 9.59/4.02 The following Function with conditions 9.59/4.02 "undefined |Falseundefined; 9.59/4.02 " 9.59/4.02 is transformed to 9.59/4.02 "undefined = undefined1; 9.59/4.02 " 9.59/4.02 "undefined0 True = undefined; 9.59/4.02 " 9.59/4.02 "undefined1 = undefined0 False; 9.59/4.02 " 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (4) 9.59/4.02 Obligation: 9.59/4.02 mainModule Main 9.59/4.02 module Main where { 9.59/4.02 import qualified Prelude; 9.59/4.02 data MyInt = Pos Main.Nat | Neg Main.Nat ; 9.59/4.02 9.59/4.02 data Main.Nat = Succ Main.Nat | Zero ; 9.59/4.02 9.59/4.02 data Ordering = LT | EQ | GT ; 9.59/4.02 9.59/4.02 compareMyInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 compareMyInt = primCmpInt; 9.59/4.02 9.59/4.02 primCmpInt :: MyInt -> MyInt -> Ordering; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y; 9.59/4.02 primCmpInt (Main.Pos x) (Main.Neg y) = GT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Pos y) = LT; 9.59/4.02 primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x; 9.59/4.02 9.59/4.02 primCmpNat :: Main.Nat -> Main.Nat -> Ordering; 9.59/4.02 primCmpNat Main.Zero Main.Zero = EQ; 9.59/4.02 primCmpNat Main.Zero (Main.Succ y) = LT; 9.59/4.02 primCmpNat (Main.Succ x) Main.Zero = GT; 9.59/4.02 primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y; 9.59/4.02 9.59/4.02 } 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (5) Narrow (SOUND) 9.59/4.02 Haskell To QDPs 9.59/4.02 9.59/4.02 digraph dp_graph { 9.59/4.02 node [outthreshold=100, inthreshold=100];1[label="compareMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.59/4.02 3[label="compareMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.59/4.02 4[label="compareMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.59/4.02 5[label="primCmpInt vx3 vx4",fontsize=16,color="burlywood",shape="box"];66[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 66[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 66 -> 6[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 67[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 67[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 67 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 6[label="primCmpInt (Pos vx30) vx4",fontsize=16,color="burlywood",shape="box"];68[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];6 -> 68[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 68 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 69[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];6 -> 69[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 69 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 7[label="primCmpInt (Neg vx30) vx4",fontsize=16,color="burlywood",shape="box"];70[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 70[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 70 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 71[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 71[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 71 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 8[label="primCmpInt (Pos (Succ vx300)) vx4",fontsize=16,color="burlywood",shape="box"];72[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];8 -> 72[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 72 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 73[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];8 -> 73[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 73 -> 13[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 9[label="primCmpInt (Pos Zero) vx4",fontsize=16,color="burlywood",shape="box"];74[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 74[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 74 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 75[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 75[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 75 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 10[label="primCmpInt (Neg (Succ vx300)) vx4",fontsize=16,color="burlywood",shape="box"];76[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 76[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 76 -> 16[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 77[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 77[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 77 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 11[label="primCmpInt (Neg Zero) vx4",fontsize=16,color="burlywood",shape="box"];78[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 78[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 78 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 79[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 79[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 79 -> 19[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 12[label="primCmpInt (Pos (Succ vx300)) (Pos vx40)",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 9.59/4.02 13[label="primCmpInt (Pos (Succ vx300)) (Neg vx40)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 9.59/4.02 14[label="primCmpInt (Pos Zero) (Pos vx40)",fontsize=16,color="burlywood",shape="box"];80[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 80 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 81[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 81[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 81 -> 23[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 15[label="primCmpInt (Pos Zero) (Neg vx40)",fontsize=16,color="burlywood",shape="box"];82[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 82[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 82 -> 24[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 83[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 83[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 83 -> 25[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 16[label="primCmpInt (Neg (Succ vx300)) (Pos vx40)",fontsize=16,color="black",shape="box"];16 -> 26[label="",style="solid", color="black", weight=3]; 9.59/4.02 17[label="primCmpInt (Neg (Succ vx300)) (Neg vx40)",fontsize=16,color="black",shape="box"];17 -> 27[label="",style="solid", color="black", weight=3]; 9.59/4.02 18[label="primCmpInt (Neg Zero) (Pos vx40)",fontsize=16,color="burlywood",shape="box"];84[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 84 -> 28[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 85[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 85[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 85 -> 29[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 19[label="primCmpInt (Neg Zero) (Neg vx40)",fontsize=16,color="burlywood",shape="box"];86[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 86 -> 30[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 87[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 87 -> 31[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 20[label="primCmpNat (Succ vx300) vx40",fontsize=16,color="burlywood",shape="triangle"];88[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 88[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 88 -> 32[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 89[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 89[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 89 -> 33[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 21[label="GT",fontsize=16,color="green",shape="box"];22[label="primCmpInt (Pos Zero) (Pos (Succ vx400))",fontsize=16,color="black",shape="box"];22 -> 34[label="",style="solid", color="black", weight=3]; 9.59/4.02 23[label="primCmpInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];23 -> 35[label="",style="solid", color="black", weight=3]; 9.59/4.02 24[label="primCmpInt (Pos Zero) (Neg (Succ vx400))",fontsize=16,color="black",shape="box"];24 -> 36[label="",style="solid", color="black", weight=3]; 9.59/4.02 25[label="primCmpInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];25 -> 37[label="",style="solid", color="black", weight=3]; 9.59/4.02 26[label="LT",fontsize=16,color="green",shape="box"];27[label="primCmpNat vx40 (Succ vx300)",fontsize=16,color="burlywood",shape="triangle"];90[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];27 -> 90[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 90 -> 38[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 91[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 91[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 91 -> 39[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 28[label="primCmpInt (Neg Zero) (Pos (Succ vx400))",fontsize=16,color="black",shape="box"];28 -> 40[label="",style="solid", color="black", weight=3]; 9.59/4.02 29[label="primCmpInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];29 -> 41[label="",style="solid", color="black", weight=3]; 9.59/4.02 30[label="primCmpInt (Neg Zero) (Neg (Succ vx400))",fontsize=16,color="black",shape="box"];30 -> 42[label="",style="solid", color="black", weight=3]; 9.59/4.02 31[label="primCmpInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3]; 9.59/4.02 32[label="primCmpNat (Succ vx300) (Succ vx400)",fontsize=16,color="black",shape="box"];32 -> 44[label="",style="solid", color="black", weight=3]; 9.59/4.02 33[label="primCmpNat (Succ vx300) Zero",fontsize=16,color="black",shape="box"];33 -> 45[label="",style="solid", color="black", weight=3]; 9.59/4.02 34 -> 27[label="",style="dashed", color="red", weight=0]; 9.59/4.02 34[label="primCmpNat Zero (Succ vx400)",fontsize=16,color="magenta"];34 -> 46[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 34 -> 47[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 35[label="EQ",fontsize=16,color="green",shape="box"];36[label="GT",fontsize=16,color="green",shape="box"];37[label="EQ",fontsize=16,color="green",shape="box"];38[label="primCmpNat (Succ vx400) (Succ vx300)",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3]; 9.59/4.02 39[label="primCmpNat Zero (Succ vx300)",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3]; 9.59/4.02 40[label="LT",fontsize=16,color="green",shape="box"];41[label="EQ",fontsize=16,color="green",shape="box"];42 -> 20[label="",style="dashed", color="red", weight=0]; 9.59/4.02 42[label="primCmpNat (Succ vx400) Zero",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 42 -> 51[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 43[label="EQ",fontsize=16,color="green",shape="box"];44[label="primCmpNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];92[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];44 -> 92[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 92 -> 52[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 93[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];44 -> 93[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 93 -> 53[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 45[label="GT",fontsize=16,color="green",shape="box"];46[label="vx400",fontsize=16,color="green",shape="box"];47[label="Zero",fontsize=16,color="green",shape="box"];48 -> 44[label="",style="dashed", color="red", weight=0]; 9.59/4.02 48[label="primCmpNat vx400 vx300",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 48 -> 55[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 49[label="LT",fontsize=16,color="green",shape="box"];50[label="Zero",fontsize=16,color="green",shape="box"];51[label="vx400",fontsize=16,color="green",shape="box"];52[label="primCmpNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];94[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];52 -> 94[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 94 -> 56[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 95[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 95[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 95 -> 57[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 53[label="primCmpNat Zero vx400",fontsize=16,color="burlywood",shape="box"];96[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];53 -> 96[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 96 -> 58[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 97[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 97[label="",style="solid", color="burlywood", weight=9]; 9.59/4.02 97 -> 59[label="",style="solid", color="burlywood", weight=3]; 9.59/4.02 54[label="vx400",fontsize=16,color="green",shape="box"];55[label="vx300",fontsize=16,color="green",shape="box"];56[label="primCmpNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 9.59/4.02 57[label="primCmpNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3]; 9.59/4.02 58[label="primCmpNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];58 -> 62[label="",style="solid", color="black", weight=3]; 9.59/4.02 59[label="primCmpNat Zero Zero",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3]; 9.59/4.02 60 -> 44[label="",style="dashed", color="red", weight=0]; 9.59/4.02 60[label="primCmpNat vx3000 vx4000",fontsize=16,color="magenta"];60 -> 64[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 60 -> 65[label="",style="dashed", color="magenta", weight=3]; 9.59/4.02 61[label="GT",fontsize=16,color="green",shape="box"];62[label="LT",fontsize=16,color="green",shape="box"];63[label="EQ",fontsize=16,color="green",shape="box"];64[label="vx3000",fontsize=16,color="green",shape="box"];65[label="vx4000",fontsize=16,color="green",shape="box"];} 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (6) 9.59/4.02 Obligation: 9.59/4.02 Q DP problem: 9.59/4.02 The TRS P consists of the following rules: 9.59/4.02 9.59/4.02 new_primCmpNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primCmpNat(vx3000, vx4000) 9.59/4.02 9.59/4.02 R is empty. 9.59/4.02 Q is empty. 9.59/4.02 We have to consider all minimal (P,Q,R)-chains. 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (7) QDPSizeChangeProof (EQUIVALENT) 9.59/4.02 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 9.59/4.02 9.59/4.02 From the DPs we obtained the following set of size-change graphs: 9.59/4.02 *new_primCmpNat(Main.Succ(vx3000), Main.Succ(vx4000)) -> new_primCmpNat(vx3000, vx4000) 9.59/4.02 The graph contains the following edges 1 > 1, 2 > 2 9.59/4.02 9.59/4.02 9.59/4.02 ---------------------------------------- 9.59/4.02 9.59/4.02 (8) 9.59/4.02 YES 9.76/4.05 EOF