/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.hs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 0 ms] (4) HASKELL (5) Narrow [SOUND, 0 ms] (6) AND (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; data Float = Float MyInt MyInt ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; flip :: (c -> b -> a) -> b -> c -> a; flip f x y = f y x; msFloat :: Float -> Float -> Float; msFloat = primMinusFloat; msMyInt :: MyInt -> MyInt -> MyInt; msMyInt = primMinusInt; primMinusFloat :: Float -> Float -> Float; primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2); primMinusInt :: MyInt -> MyInt -> MyInt; primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; primMinusNat :: Main.Nat -> Main.Nat -> MyInt; primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; primMulInt :: MyInt -> MyInt -> MyInt; primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y); primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y); primMulNat :: Main.Nat -> Main.Nat -> Main.Nat; primMulNat Main.Zero Main.Zero = Main.Zero; primMulNat Main.Zero (Main.Succ y) = Main.Zero; primMulNat (Main.Succ x) Main.Zero = Main.Zero; primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y); primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; primPlusNat Main.Zero Main.Zero = Main.Zero; primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); srMyInt :: MyInt -> MyInt -> MyInt; srMyInt = primMulInt; subtractFloat :: Float -> Float -> Float; subtractFloat = flip msFloat; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; data Float = Float MyInt MyInt ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; flip :: (c -> b -> a) -> b -> c -> a; flip f x y = f y x; msFloat :: Float -> Float -> Float; msFloat = primMinusFloat; msMyInt :: MyInt -> MyInt -> MyInt; msMyInt = primMinusInt; primMinusFloat :: Float -> Float -> Float; primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2); primMinusInt :: MyInt -> MyInt -> MyInt; primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; primMinusNat :: Main.Nat -> Main.Nat -> MyInt; primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; primMulInt :: MyInt -> MyInt -> MyInt; primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y); primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y); primMulNat :: Main.Nat -> Main.Nat -> Main.Nat; primMulNat Main.Zero Main.Zero = Main.Zero; primMulNat Main.Zero (Main.Succ y) = Main.Zero; primMulNat (Main.Succ x) Main.Zero = Main.Zero; primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y); primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; primPlusNat Main.Zero Main.Zero = Main.Zero; primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); srMyInt :: MyInt -> MyInt -> MyInt; srMyInt = primMulInt; subtractFloat :: Float -> Float -> Float; subtractFloat = flip msFloat; } ---------------------------------------- (3) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; data Float = Float MyInt MyInt ; data MyInt = Pos Main.Nat | Neg Main.Nat ; data Main.Nat = Succ Main.Nat | Zero ; flip :: (a -> c -> b) -> c -> a -> b; flip f x y = f y x; msFloat :: Float -> Float -> Float; msFloat = primMinusFloat; msMyInt :: MyInt -> MyInt -> MyInt; msMyInt = primMinusInt; primMinusFloat :: Float -> Float -> Float; primMinusFloat (Float x1 x2) (Float y1 y2) = Float (msMyInt x1 y1) (srMyInt x2 y2); primMinusInt :: MyInt -> MyInt -> MyInt; primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y); primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x; primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y; primMinusNat :: Main.Nat -> Main.Nat -> MyInt; primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero; primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y); primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x); primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y; primMulInt :: MyInt -> MyInt -> MyInt; primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y); primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y); primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y); primMulNat :: Main.Nat -> Main.Nat -> Main.Nat; primMulNat Main.Zero Main.Zero = Main.Zero; primMulNat Main.Zero (Main.Succ y) = Main.Zero; primMulNat (Main.Succ x) Main.Zero = Main.Zero; primMulNat (Main.Succ x) (Main.Succ y) = primPlusNat (primMulNat x (Main.Succ y)) (Main.Succ y); primPlusNat :: Main.Nat -> Main.Nat -> Main.Nat; primPlusNat Main.Zero Main.Zero = Main.Zero; primPlusNat Main.Zero (Main.Succ y) = Main.Succ y; primPlusNat (Main.Succ x) Main.Zero = Main.Succ x; primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y)); srMyInt :: MyInt -> MyInt -> MyInt; srMyInt = primMulInt; subtractFloat :: Float -> Float -> Float; subtractFloat = flip msFloat; } ---------------------------------------- (5) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="subtractFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="subtractFloat vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="subtractFloat vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="flip msFloat vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="msFloat vx4 vx3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="primMinusFloat vx4 vx3",fontsize=16,color="burlywood",shape="box"];88[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 88[label="",style="solid", color="burlywood", weight=9]; 88 -> 8[label="",style="solid", color="burlywood", weight=3]; 8[label="primMinusFloat (Float vx40 vx41) vx3",fontsize=16,color="burlywood",shape="box"];89[label="vx3/Float vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];8 -> 89[label="",style="solid", color="burlywood", weight=9]; 89 -> 9[label="",style="solid", color="burlywood", weight=3]; 9[label="primMinusFloat (Float vx40 vx41) (Float vx30 vx31)",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 10[label="Float (msMyInt vx40 vx30) (srMyInt vx41 vx31)",fontsize=16,color="green",shape="box"];10 -> 11[label="",style="dashed", color="green", weight=3]; 10 -> 12[label="",style="dashed", color="green", weight=3]; 11[label="msMyInt vx40 vx30",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 12[label="srMyInt vx41 vx31",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 13[label="primMinusInt vx40 vx30",fontsize=16,color="burlywood",shape="box"];90[label="vx40/Pos vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 90[label="",style="solid", color="burlywood", weight=9]; 90 -> 15[label="",style="solid", color="burlywood", weight=3]; 91[label="vx40/Neg vx400",fontsize=10,color="white",style="solid",shape="box"];13 -> 91[label="",style="solid", color="burlywood", weight=9]; 91 -> 16[label="",style="solid", color="burlywood", weight=3]; 14[label="primMulInt vx41 vx31",fontsize=16,color="burlywood",shape="box"];92[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 92[label="",style="solid", color="burlywood", weight=9]; 92 -> 17[label="",style="solid", color="burlywood", weight=3]; 93[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];14 -> 93[label="",style="solid", color="burlywood", weight=9]; 93 -> 18[label="",style="solid", color="burlywood", weight=3]; 15[label="primMinusInt (Pos vx400) vx30",fontsize=16,color="burlywood",shape="box"];94[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 94[label="",style="solid", color="burlywood", weight=9]; 94 -> 19[label="",style="solid", color="burlywood", weight=3]; 95[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];15 -> 95[label="",style="solid", color="burlywood", weight=9]; 95 -> 20[label="",style="solid", color="burlywood", weight=3]; 16[label="primMinusInt (Neg vx400) vx30",fontsize=16,color="burlywood",shape="box"];96[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 96[label="",style="solid", color="burlywood", weight=9]; 96 -> 21[label="",style="solid", color="burlywood", weight=3]; 97[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 97[label="",style="solid", color="burlywood", weight=9]; 97 -> 22[label="",style="solid", color="burlywood", weight=3]; 17[label="primMulInt (Pos vx410) vx31",fontsize=16,color="burlywood",shape="box"];98[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 98[label="",style="solid", color="burlywood", weight=9]; 98 -> 23[label="",style="solid", color="burlywood", weight=3]; 99[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];17 -> 99[label="",style="solid", color="burlywood", weight=9]; 99 -> 24[label="",style="solid", color="burlywood", weight=3]; 18[label="primMulInt (Neg vx410) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx31/Pos vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 100[label="",style="solid", color="burlywood", weight=9]; 100 -> 25[label="",style="solid", color="burlywood", weight=3]; 101[label="vx31/Neg vx310",fontsize=10,color="white",style="solid",shape="box"];18 -> 101[label="",style="solid", color="burlywood", weight=9]; 101 -> 26[label="",style="solid", color="burlywood", weight=3]; 19[label="primMinusInt (Pos vx400) (Pos vx300)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 20[label="primMinusInt (Pos vx400) (Neg vx300)",fontsize=16,color="black",shape="box"];20 -> 28[label="",style="solid", color="black", weight=3]; 21[label="primMinusInt (Neg vx400) (Pos vx300)",fontsize=16,color="black",shape="box"];21 -> 29[label="",style="solid", color="black", weight=3]; 22[label="primMinusInt (Neg vx400) (Neg vx300)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3]; 23[label="primMulInt (Pos vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3]; 24[label="primMulInt (Pos vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3]; 25[label="primMulInt (Neg vx410) (Pos vx310)",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3]; 26[label="primMulInt (Neg vx410) (Neg vx310)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3]; 27[label="primMinusNat vx400 vx300",fontsize=16,color="burlywood",shape="triangle"];102[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 102[label="",style="solid", color="burlywood", weight=9]; 102 -> 35[label="",style="solid", color="burlywood", weight=3]; 103[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 103[label="",style="solid", color="burlywood", weight=9]; 103 -> 36[label="",style="solid", color="burlywood", weight=3]; 28[label="Pos (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];28 -> 37[label="",style="dashed", color="green", weight=3]; 29[label="Neg (primPlusNat vx400 vx300)",fontsize=16,color="green",shape="box"];29 -> 38[label="",style="dashed", color="green", weight=3]; 30 -> 27[label="",style="dashed", color="red", weight=0]; 30[label="primMinusNat vx300 vx400",fontsize=16,color="magenta"];30 -> 39[label="",style="dashed", color="magenta", weight=3]; 30 -> 40[label="",style="dashed", color="magenta", weight=3]; 31[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];31 -> 41[label="",style="dashed", color="green", weight=3]; 32[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];32 -> 42[label="",style="dashed", color="green", weight=3]; 33[label="Neg (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];33 -> 43[label="",style="dashed", color="green", weight=3]; 34[label="Pos (primMulNat vx410 vx310)",fontsize=16,color="green",shape="box"];34 -> 44[label="",style="dashed", color="green", weight=3]; 35[label="primMinusNat (Succ vx4000) vx300",fontsize=16,color="burlywood",shape="box"];104[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 104[label="",style="solid", color="burlywood", weight=9]; 104 -> 45[label="",style="solid", color="burlywood", weight=3]; 105[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 105[label="",style="solid", color="burlywood", weight=9]; 105 -> 46[label="",style="solid", color="burlywood", weight=3]; 36[label="primMinusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];106[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];36 -> 106[label="",style="solid", color="burlywood", weight=9]; 106 -> 47[label="",style="solid", color="burlywood", weight=3]; 107[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 107[label="",style="solid", color="burlywood", weight=9]; 107 -> 48[label="",style="solid", color="burlywood", weight=3]; 37[label="primPlusNat vx400 vx300",fontsize=16,color="burlywood",shape="triangle"];108[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];37 -> 108[label="",style="solid", color="burlywood", weight=9]; 108 -> 49[label="",style="solid", color="burlywood", weight=3]; 109[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9]; 109 -> 50[label="",style="solid", color="burlywood", weight=3]; 38 -> 37[label="",style="dashed", color="red", weight=0]; 38[label="primPlusNat vx400 vx300",fontsize=16,color="magenta"];38 -> 51[label="",style="dashed", color="magenta", weight=3]; 38 -> 52[label="",style="dashed", color="magenta", weight=3]; 39[label="vx400",fontsize=16,color="green",shape="box"];40[label="vx300",fontsize=16,color="green",shape="box"];41[label="primMulNat vx410 vx310",fontsize=16,color="burlywood",shape="triangle"];110[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];41 -> 110[label="",style="solid", color="burlywood", weight=9]; 110 -> 53[label="",style="solid", color="burlywood", weight=3]; 111[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 111[label="",style="solid", color="burlywood", weight=9]; 111 -> 54[label="",style="solid", color="burlywood", weight=3]; 42 -> 41[label="",style="dashed", color="red", weight=0]; 42[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];42 -> 55[label="",style="dashed", color="magenta", weight=3]; 43 -> 41[label="",style="dashed", color="red", weight=0]; 43[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];43 -> 56[label="",style="dashed", color="magenta", weight=3]; 44 -> 41[label="",style="dashed", color="red", weight=0]; 44[label="primMulNat vx410 vx310",fontsize=16,color="magenta"];44 -> 57[label="",style="dashed", color="magenta", weight=3]; 44 -> 58[label="",style="dashed", color="magenta", weight=3]; 45[label="primMinusNat (Succ vx4000) (Succ vx3000)",fontsize=16,color="black",shape="box"];45 -> 59[label="",style="solid", color="black", weight=3]; 46[label="primMinusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];46 -> 60[label="",style="solid", color="black", weight=3]; 47[label="primMinusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];47 -> 61[label="",style="solid", color="black", weight=3]; 48[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];48 -> 62[label="",style="solid", color="black", weight=3]; 49[label="primPlusNat (Succ vx4000) vx300",fontsize=16,color="burlywood",shape="box"];112[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];49 -> 112[label="",style="solid", color="burlywood", weight=9]; 112 -> 63[label="",style="solid", color="burlywood", weight=3]; 113[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 113[label="",style="solid", color="burlywood", weight=9]; 113 -> 64[label="",style="solid", color="burlywood", weight=3]; 50[label="primPlusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];114[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];50 -> 114[label="",style="solid", color="burlywood", weight=9]; 114 -> 65[label="",style="solid", color="burlywood", weight=3]; 115[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];50 -> 115[label="",style="solid", color="burlywood", weight=9]; 115 -> 66[label="",style="solid", color="burlywood", weight=3]; 51[label="vx400",fontsize=16,color="green",shape="box"];52[label="vx300",fontsize=16,color="green",shape="box"];53[label="primMulNat (Succ vx4100) vx310",fontsize=16,color="burlywood",shape="box"];116[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];53 -> 116[label="",style="solid", color="burlywood", weight=9]; 116 -> 67[label="",style="solid", color="burlywood", weight=3]; 117[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 117[label="",style="solid", color="burlywood", weight=9]; 117 -> 68[label="",style="solid", color="burlywood", weight=3]; 54[label="primMulNat Zero vx310",fontsize=16,color="burlywood",shape="box"];118[label="vx310/Succ vx3100",fontsize=10,color="white",style="solid",shape="box"];54 -> 118[label="",style="solid", color="burlywood", weight=9]; 118 -> 69[label="",style="solid", color="burlywood", weight=3]; 119[label="vx310/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 119[label="",style="solid", color="burlywood", weight=9]; 119 -> 70[label="",style="solid", color="burlywood", weight=3]; 55[label="vx310",fontsize=16,color="green",shape="box"];56[label="vx410",fontsize=16,color="green",shape="box"];57[label="vx410",fontsize=16,color="green",shape="box"];58[label="vx310",fontsize=16,color="green",shape="box"];59 -> 27[label="",style="dashed", color="red", weight=0]; 59[label="primMinusNat vx4000 vx3000",fontsize=16,color="magenta"];59 -> 71[label="",style="dashed", color="magenta", weight=3]; 59 -> 72[label="",style="dashed", color="magenta", weight=3]; 60[label="Pos (Succ vx4000)",fontsize=16,color="green",shape="box"];61[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];62[label="Pos Zero",fontsize=16,color="green",shape="box"];63[label="primPlusNat (Succ vx4000) (Succ vx3000)",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3]; 64[label="primPlusNat (Succ vx4000) Zero",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3]; 65[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3]; 66[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3]; 67[label="primMulNat (Succ vx4100) (Succ vx3100)",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3]; 68[label="primMulNat (Succ vx4100) Zero",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3]; 69[label="primMulNat Zero (Succ vx3100)",fontsize=16,color="black",shape="box"];69 -> 79[label="",style="solid", color="black", weight=3]; 70[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];70 -> 80[label="",style="solid", color="black", weight=3]; 71[label="vx3000",fontsize=16,color="green",shape="box"];72[label="vx4000",fontsize=16,color="green",shape="box"];73[label="Succ (Succ (primPlusNat vx4000 vx3000))",fontsize=16,color="green",shape="box"];73 -> 81[label="",style="dashed", color="green", weight=3]; 74[label="Succ vx4000",fontsize=16,color="green",shape="box"];75[label="Succ vx3000",fontsize=16,color="green",shape="box"];76[label="Zero",fontsize=16,color="green",shape="box"];77 -> 37[label="",style="dashed", color="red", weight=0]; 77[label="primPlusNat (primMulNat vx4100 (Succ vx3100)) (Succ vx3100)",fontsize=16,color="magenta"];77 -> 82[label="",style="dashed", color="magenta", weight=3]; 77 -> 83[label="",style="dashed", color="magenta", weight=3]; 78[label="Zero",fontsize=16,color="green",shape="box"];79[label="Zero",fontsize=16,color="green",shape="box"];80[label="Zero",fontsize=16,color="green",shape="box"];81 -> 37[label="",style="dashed", color="red", weight=0]; 81[label="primPlusNat vx4000 vx3000",fontsize=16,color="magenta"];81 -> 84[label="",style="dashed", color="magenta", weight=3]; 81 -> 85[label="",style="dashed", color="magenta", weight=3]; 82 -> 41[label="",style="dashed", color="red", weight=0]; 82[label="primMulNat vx4100 (Succ vx3100)",fontsize=16,color="magenta"];82 -> 86[label="",style="dashed", color="magenta", weight=3]; 82 -> 87[label="",style="dashed", color="magenta", weight=3]; 83[label="Succ vx3100",fontsize=16,color="green",shape="box"];84[label="vx4000",fontsize=16,color="green",shape="box"];85[label="vx3000",fontsize=16,color="green",shape="box"];86[label="vx4100",fontsize=16,color="green",shape="box"];87[label="Succ vx3100",fontsize=16,color="green",shape="box"];} ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMulNat(Main.Succ(vx4100), Main.Succ(vx3100)) -> new_primMulNat(vx4100, Main.Succ(vx3100)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primMulNat(Main.Succ(vx4100), Main.Succ(vx3100)) -> new_primMulNat(vx4100, Main.Succ(vx3100)) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_primPlusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primPlusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primPlusNat(vx4000, vx3000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (12) YES ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primMinusNat(vx4000, vx3000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primMinusNat(Main.Succ(vx4000), Main.Succ(vx3000)) -> new_primMinusNat(vx4000, vx3000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (15) YES