8.59/3.74 YES 10.36/4.22 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.36/4.22 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.36/4.22 10.36/4.22 10.36/4.22 H-Termination with start terms of the given HASKELL could be proven: 10.36/4.22 10.36/4.22 (0) HASKELL 10.36/4.22 (1) IFR [EQUIVALENT, 0 ms] 10.36/4.22 (2) HASKELL 10.36/4.22 (3) BR [EQUIVALENT, 0 ms] 10.36/4.22 (4) HASKELL 10.36/4.22 (5) COR [EQUIVALENT, 0 ms] 10.36/4.22 (6) HASKELL 10.36/4.22 (7) Narrow [SOUND, 0 ms] 10.36/4.22 (8) AND 10.36/4.22 (9) QDP 10.36/4.22 (10) DependencyGraphProof [EQUIVALENT, 0 ms] 10.36/4.22 (11) AND 10.36/4.22 (12) QDP 10.36/4.22 (13) MRRProof [EQUIVALENT, 57 ms] 10.36/4.22 (14) QDP 10.36/4.22 (15) PisEmptyProof [EQUIVALENT, 0 ms] 10.36/4.22 (16) YES 10.36/4.22 (17) QDP 10.36/4.22 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.36/4.22 (19) YES 10.36/4.22 (20) QDP 10.36/4.22 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.36/4.22 (22) YES 10.36/4.22 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (0) 10.36/4.22 Obligation: 10.36/4.22 mainModule Main 10.36/4.22 module Main where { 10.36/4.22 import qualified Prelude; 10.36/4.22 } 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (1) IFR (EQUIVALENT) 10.36/4.22 If Reductions: 10.36/4.22 The following If expression 10.36/4.22 "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" 10.36/4.22 is transformed to 10.36/4.22 "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); 10.36/4.22 primDivNatS0 x y False = Zero; 10.36/4.22 " 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (2) 10.36/4.22 Obligation: 10.36/4.22 mainModule Main 10.36/4.22 module Main where { 10.36/4.22 import qualified Prelude; 10.36/4.22 } 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (3) BR (EQUIVALENT) 10.36/4.22 Replaced joker patterns by fresh variables and removed binding patterns. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (4) 10.36/4.22 Obligation: 10.36/4.22 mainModule Main 10.36/4.22 module Main where { 10.36/4.22 import qualified Prelude; 10.36/4.22 } 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (5) COR (EQUIVALENT) 10.36/4.22 Cond Reductions: 10.36/4.22 The following Function with conditions 10.36/4.22 "undefined |Falseundefined; 10.36/4.22 " 10.36/4.22 is transformed to 10.36/4.22 "undefined = undefined1; 10.36/4.22 " 10.36/4.22 "undefined0 True = undefined; 10.36/4.22 " 10.36/4.22 "undefined1 = undefined0 False; 10.36/4.22 " 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (6) 10.36/4.22 Obligation: 10.36/4.22 mainModule Main 10.36/4.22 module Main where { 10.36/4.22 import qualified Prelude; 10.36/4.22 } 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (7) Narrow (SOUND) 10.36/4.22 Haskell To QDPs 10.36/4.22 10.36/4.22 digraph dp_graph { 10.36/4.22 node [outthreshold=100, inthreshold=100];1[label="quot",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.36/4.22 3[label="quot vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.36/4.22 4[label="quot vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.36/4.22 5[label="primQuotInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];274[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 274[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 274 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 275[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 275[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 275 -> 7[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 6[label="primQuotInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];276[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 276[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 276 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 277[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 277[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 277 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 7[label="primQuotInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];278[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 278[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 278 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 279[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 279[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 279 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 8[label="primQuotInt (Pos vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];280[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 280[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 280 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 281[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 281[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 281 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 9[label="primQuotInt (Pos vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];282[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 282[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 282 -> 14[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 283[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 283[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 283 -> 15[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 10[label="primQuotInt (Neg vz30) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];284[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];10 -> 284[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 284 -> 16[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 285[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 285[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 285 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 11[label="primQuotInt (Neg vz30) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];286[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];11 -> 286[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 286 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 287[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 287[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 287 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 12[label="primQuotInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 10.36/4.22 13[label="primQuotInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 10.36/4.22 14[label="primQuotInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 10.36/4.22 15[label="primQuotInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 10.36/4.22 16[label="primQuotInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 10.36/4.22 17[label="primQuotInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 10.36/4.22 18[label="primQuotInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3]; 10.36/4.22 19[label="primQuotInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 10.36/4.22 20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3]; 10.36/4.22 21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3]; 10.36/4.22 22[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3]; 10.36/4.22 23 -> 21[label="",style="dashed", color="red", weight=0]; 10.36/4.22 23[label="error []",fontsize=16,color="magenta"];24[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];24 -> 31[label="",style="dashed", color="green", weight=3]; 10.36/4.22 25 -> 21[label="",style="dashed", color="red", weight=0]; 10.36/4.22 25[label="error []",fontsize=16,color="magenta"];26[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];26 -> 32[label="",style="dashed", color="green", weight=3]; 10.36/4.22 27 -> 21[label="",style="dashed", color="red", weight=0]; 10.36/4.22 27[label="error []",fontsize=16,color="magenta"];28[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="burlywood",shape="triangle"];288[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];28 -> 288[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 288 -> 33[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 289[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 289[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 289 -> 34[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 29[label="error []",fontsize=16,color="red",shape="box"];30 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 30[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 35[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 31 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 31[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 32 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 37[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 32 -> 38[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3]; 10.36/4.22 34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3]; 10.36/4.22 35[label="vz400",fontsize=16,color="green",shape="box"];36[label="vz30",fontsize=16,color="green",shape="box"];37[label="vz400",fontsize=16,color="green",shape="box"];38[label="vz30",fontsize=16,color="green",shape="box"];39[label="primDivNatS0 vz300 vz400 (primGEqNatS vz300 vz400)",fontsize=16,color="burlywood",shape="box"];290[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];39 -> 290[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 290 -> 41[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 291[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 291[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 291 -> 42[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 40[label="Zero",fontsize=16,color="green",shape="box"];41[label="primDivNatS0 (Succ vz3000) vz400 (primGEqNatS (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];292[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];41 -> 292[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 292 -> 43[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 293[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 293[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 293 -> 44[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 42[label="primDivNatS0 Zero vz400 (primGEqNatS Zero vz400)",fontsize=16,color="burlywood",shape="box"];294[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];42 -> 294[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 294 -> 45[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 295[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 295[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 295 -> 46[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 43[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3]; 10.36/4.22 44[label="primDivNatS0 (Succ vz3000) Zero (primGEqNatS (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3]; 10.36/4.22 45[label="primDivNatS0 Zero (Succ vz4000) (primGEqNatS Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3]; 10.36/4.22 46[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3]; 10.36/4.22 47 -> 211[label="",style="dashed", color="red", weight=0]; 10.36/4.22 47[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];47 -> 212[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 47 -> 213[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 47 -> 214[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 47 -> 215[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 48[label="primDivNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3]; 10.36/4.22 49[label="primDivNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3]; 10.36/4.22 50[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 10.36/4.22 212[label="vz4000",fontsize=16,color="green",shape="box"];213[label="vz3000",fontsize=16,color="green",shape="box"];214[label="vz4000",fontsize=16,color="green",shape="box"];215[label="vz3000",fontsize=16,color="green",shape="box"];211[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz23 vz24)",fontsize=16,color="burlywood",shape="triangle"];296[label="vz23/Succ vz230",fontsize=10,color="white",style="solid",shape="box"];211 -> 296[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 296 -> 244[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 297[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];211 -> 297[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 297 -> 245[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 53[label="Succ (primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];53 -> 60[label="",style="dashed", color="green", weight=3]; 10.36/4.22 54[label="Zero",fontsize=16,color="green",shape="box"];55[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];55 -> 61[label="",style="dashed", color="green", weight=3]; 10.36/4.22 244[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) vz24)",fontsize=16,color="burlywood",shape="box"];298[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];244 -> 298[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 298 -> 246[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 299[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];244 -> 299[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 299 -> 247[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 245[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero vz24)",fontsize=16,color="burlywood",shape="box"];300[label="vz24/Succ vz240",fontsize=10,color="white",style="solid",shape="box"];245 -> 300[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 300 -> 248[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 301[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];245 -> 301[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 301 -> 249[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 60 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 60[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];60 -> 66[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 60 -> 67[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 61 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 61[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];61 -> 68[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 61 -> 69[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 246[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) (Succ vz240))",fontsize=16,color="black",shape="box"];246 -> 250[label="",style="solid", color="black", weight=3]; 10.36/4.22 247[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS (Succ vz230) Zero)",fontsize=16,color="black",shape="box"];247 -> 251[label="",style="solid", color="black", weight=3]; 10.36/4.22 248[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero (Succ vz240))",fontsize=16,color="black",shape="box"];248 -> 252[label="",style="solid", color="black", weight=3]; 10.36/4.22 249[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];249 -> 253[label="",style="solid", color="black", weight=3]; 10.36/4.22 66[label="Zero",fontsize=16,color="green",shape="box"];67[label="primMinusNatS (Succ vz3000) Zero",fontsize=16,color="black",shape="triangle"];67 -> 75[label="",style="solid", color="black", weight=3]; 10.36/4.22 68[label="Zero",fontsize=16,color="green",shape="box"];69[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];69 -> 76[label="",style="solid", color="black", weight=3]; 10.36/4.22 250 -> 211[label="",style="dashed", color="red", weight=0]; 10.36/4.22 250[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];250 -> 254[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 250 -> 255[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 251[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="black",shape="triangle"];251 -> 256[label="",style="solid", color="black", weight=3]; 10.36/4.22 252[label="primDivNatS0 (Succ vz21) (Succ vz22) False",fontsize=16,color="black",shape="box"];252 -> 257[label="",style="solid", color="black", weight=3]; 10.36/4.22 253 -> 251[label="",style="dashed", color="red", weight=0]; 10.36/4.22 253[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="magenta"];75[label="Succ vz3000",fontsize=16,color="green",shape="box"];76[label="Zero",fontsize=16,color="green",shape="box"];254[label="vz240",fontsize=16,color="green",shape="box"];255[label="vz230",fontsize=16,color="green",shape="box"];256[label="Succ (primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22)))",fontsize=16,color="green",shape="box"];256 -> 258[label="",style="dashed", color="green", weight=3]; 10.36/4.22 257[label="Zero",fontsize=16,color="green",shape="box"];258 -> 28[label="",style="dashed", color="red", weight=0]; 10.36/4.22 258[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];258 -> 259[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 258 -> 260[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 259[label="Succ vz22",fontsize=16,color="green",shape="box"];260[label="primMinusNatS (Succ vz21) (Succ vz22)",fontsize=16,color="black",shape="box"];260 -> 261[label="",style="solid", color="black", weight=3]; 10.36/4.22 261[label="primMinusNatS vz21 vz22",fontsize=16,color="burlywood",shape="triangle"];302[label="vz21/Succ vz210",fontsize=10,color="white",style="solid",shape="box"];261 -> 302[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 302 -> 262[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 303[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];261 -> 303[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 303 -> 263[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 262[label="primMinusNatS (Succ vz210) vz22",fontsize=16,color="burlywood",shape="box"];304[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];262 -> 304[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 304 -> 264[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 305[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];262 -> 305[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 305 -> 265[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 263[label="primMinusNatS Zero vz22",fontsize=16,color="burlywood",shape="box"];306[label="vz22/Succ vz220",fontsize=10,color="white",style="solid",shape="box"];263 -> 306[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 306 -> 266[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 307[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 307[label="",style="solid", color="burlywood", weight=9]; 10.36/4.22 307 -> 267[label="",style="solid", color="burlywood", weight=3]; 10.36/4.22 264[label="primMinusNatS (Succ vz210) (Succ vz220)",fontsize=16,color="black",shape="box"];264 -> 268[label="",style="solid", color="black", weight=3]; 10.36/4.22 265[label="primMinusNatS (Succ vz210) Zero",fontsize=16,color="black",shape="box"];265 -> 269[label="",style="solid", color="black", weight=3]; 10.36/4.22 266[label="primMinusNatS Zero (Succ vz220)",fontsize=16,color="black",shape="box"];266 -> 270[label="",style="solid", color="black", weight=3]; 10.36/4.22 267[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];267 -> 271[label="",style="solid", color="black", weight=3]; 10.36/4.22 268 -> 261[label="",style="dashed", color="red", weight=0]; 10.36/4.22 268[label="primMinusNatS vz210 vz220",fontsize=16,color="magenta"];268 -> 272[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 268 -> 273[label="",style="dashed", color="magenta", weight=3]; 10.36/4.22 269[label="Succ vz210",fontsize=16,color="green",shape="box"];270[label="Zero",fontsize=16,color="green",shape="box"];271[label="Zero",fontsize=16,color="green",shape="box"];272[label="vz220",fontsize=16,color="green",shape="box"];273[label="vz210",fontsize=16,color="green",shape="box"];} 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (8) 10.36/4.22 Complex Obligation (AND) 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (9) 10.36/4.22 Obligation: 10.36/4.22 Q DP problem: 10.36/4.22 The TRS P consists of the following rules: 10.36/4.22 10.36/4.22 new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) 10.36/4.22 new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) 10.36/4.22 new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) 10.36/4.22 new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) 10.36/4.22 new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) 10.36/4.22 new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) 10.36/4.22 new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) 10.36/4.22 10.36/4.22 The TRS R consists of the following rules: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Zero, Succ(vz220)) -> Zero 10.36/4.22 new_primMinusNatS0(Zero, Zero) -> Zero 10.36/4.22 new_primMinusNatS1(vz3000) -> Succ(vz3000) 10.36/4.22 new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.36/4.22 new_primMinusNatS2 -> Zero 10.36/4.22 new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) 10.36/4.22 10.36/4.22 The set Q consists of the following terms: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Succ(x0), Zero) 10.36/4.22 new_primMinusNatS0(Zero, Zero) 10.36/4.22 new_primMinusNatS2 10.36/4.22 new_primMinusNatS0(Succ(x0), Succ(x1)) 10.36/4.22 new_primMinusNatS0(Zero, Succ(x0)) 10.36/4.22 new_primMinusNatS1(x0) 10.36/4.22 10.36/4.22 We have to consider all minimal (P,Q,R)-chains. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (10) DependencyGraphProof (EQUIVALENT) 10.36/4.22 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (11) 10.36/4.22 Complex Obligation (AND) 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (12) 10.36/4.22 Obligation: 10.36/4.22 Q DP problem: 10.36/4.22 The TRS P consists of the following rules: 10.36/4.22 10.36/4.22 new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) 10.36/4.22 10.36/4.22 The TRS R consists of the following rules: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Zero, Succ(vz220)) -> Zero 10.36/4.22 new_primMinusNatS0(Zero, Zero) -> Zero 10.36/4.22 new_primMinusNatS1(vz3000) -> Succ(vz3000) 10.36/4.22 new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.36/4.22 new_primMinusNatS2 -> Zero 10.36/4.22 new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) 10.36/4.22 10.36/4.22 The set Q consists of the following terms: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Succ(x0), Zero) 10.36/4.22 new_primMinusNatS0(Zero, Zero) 10.36/4.22 new_primMinusNatS2 10.36/4.22 new_primMinusNatS0(Succ(x0), Succ(x1)) 10.36/4.22 new_primMinusNatS0(Zero, Succ(x0)) 10.36/4.22 new_primMinusNatS1(x0) 10.36/4.22 10.36/4.22 We have to consider all minimal (P,Q,R)-chains. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (13) MRRProof (EQUIVALENT) 10.36/4.22 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. 10.36/4.22 10.36/4.22 Strictly oriented dependency pairs: 10.36/4.22 10.36/4.22 new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) 10.36/4.22 10.36/4.22 Strictly oriented rules of the TRS R: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Zero, Succ(vz220)) -> Zero 10.36/4.22 new_primMinusNatS0(Zero, Zero) -> Zero 10.36/4.22 new_primMinusNatS1(vz3000) -> Succ(vz3000) 10.36/4.22 new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.36/4.22 new_primMinusNatS2 -> Zero 10.36/4.22 new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) 10.36/4.22 10.36/4.22 Used ordering: Polynomial interpretation [POLO]: 10.36/4.22 10.36/4.22 POL(Succ(x_1)) = 1 + 2*x_1 10.36/4.22 POL(Zero) = 1 10.36/4.22 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 10.36/4.22 POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2 10.36/4.22 POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1 10.36/4.22 POL(new_primMinusNatS2) = 2 10.36/4.22 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (14) 10.36/4.22 Obligation: 10.36/4.22 Q DP problem: 10.36/4.22 P is empty. 10.36/4.22 R is empty. 10.36/4.22 The set Q consists of the following terms: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Succ(x0), Zero) 10.36/4.22 new_primMinusNatS0(Zero, Zero) 10.36/4.22 new_primMinusNatS2 10.36/4.22 new_primMinusNatS0(Succ(x0), Succ(x1)) 10.36/4.22 new_primMinusNatS0(Zero, Succ(x0)) 10.36/4.22 new_primMinusNatS1(x0) 10.36/4.22 10.36/4.22 We have to consider all minimal (P,Q,R)-chains. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (15) PisEmptyProof (EQUIVALENT) 10.36/4.22 The TRS P is empty. Hence, there is no (P,Q,R) chain. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (16) 10.36/4.22 YES 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (17) 10.36/4.22 Obligation: 10.36/4.22 Q DP problem: 10.36/4.22 The TRS P consists of the following rules: 10.36/4.22 10.36/4.22 new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) 10.36/4.22 new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) 10.36/4.22 new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) 10.36/4.22 new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) 10.36/4.22 new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) 10.36/4.22 10.36/4.22 The TRS R consists of the following rules: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Zero, Succ(vz220)) -> Zero 10.36/4.22 new_primMinusNatS0(Zero, Zero) -> Zero 10.36/4.22 new_primMinusNatS1(vz3000) -> Succ(vz3000) 10.36/4.22 new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.36/4.22 new_primMinusNatS2 -> Zero 10.36/4.22 new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) 10.36/4.22 10.36/4.22 The set Q consists of the following terms: 10.36/4.22 10.36/4.22 new_primMinusNatS0(Succ(x0), Zero) 10.36/4.22 new_primMinusNatS0(Zero, Zero) 10.36/4.22 new_primMinusNatS2 10.36/4.22 new_primMinusNatS0(Succ(x0), Succ(x1)) 10.36/4.22 new_primMinusNatS0(Zero, Succ(x0)) 10.36/4.22 new_primMinusNatS1(x0) 10.36/4.22 10.36/4.22 We have to consider all minimal (P,Q,R)-chains. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (18) QDPSizeChangeProof (EQUIVALENT) 10.36/4.22 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 10.36/4.22 10.36/4.22 Order:Polynomial interpretation [POLO]: 10.36/4.22 10.36/4.22 POL(Succ(x_1)) = 1 + x_1 10.36/4.22 POL(Zero) = 1 10.36/4.22 POL(new_primMinusNatS0(x_1, x_2)) = x_1 10.36/4.22 10.36/4.22 10.36/4.22 10.36/4.22 10.36/4.22 From the DPs we obtained the following set of size-change graphs: 10.36/4.22 *new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4}) 10.36/4.22 The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 10.36/4.22 10.36/4.22 10.36/4.22 *new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4}) 10.36/4.22 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 10.36/4.22 10.36/4.22 10.36/4.22 *new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2}) 10.36/4.22 The graph contains the following edges 1 >= 1 10.36/4.22 10.36/4.22 10.36/4.22 *new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2}) 10.36/4.22 The graph contains the following edges 1 >= 1, 2 >= 2 10.36/4.22 10.36/4.22 10.36/4.22 *new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2}) 10.36/4.22 The graph contains the following edges 1 >= 1 10.36/4.22 10.36/4.22 10.36/4.22 10.36/4.22 We oriented the following set of usable rules [AAECC05,FROCOS05]. 10.36/4.22 10.36/4.22 new_primMinusNatS0(Zero, Zero) -> Zero 10.36/4.22 new_primMinusNatS0(Zero, Succ(vz220)) -> Zero 10.36/4.22 new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) 10.36/4.22 new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (19) 10.36/4.22 YES 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (20) 10.36/4.22 Obligation: 10.36/4.22 Q DP problem: 10.36/4.22 The TRS P consists of the following rules: 10.36/4.22 10.36/4.22 new_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220) 10.36/4.22 10.36/4.22 R is empty. 10.36/4.22 Q is empty. 10.36/4.22 We have to consider all minimal (P,Q,R)-chains. 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (21) QDPSizeChangeProof (EQUIVALENT) 10.36/4.22 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. 10.36/4.22 10.36/4.22 From the DPs we obtained the following set of size-change graphs: 10.36/4.22 *new_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220) 10.36/4.22 The graph contains the following edges 1 > 1, 2 > 2 10.36/4.22 10.36/4.22 10.36/4.22 ---------------------------------------- 10.36/4.22 10.36/4.22 (22) 10.36/4.22 YES 10.54/4.34 EOF