/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) IFR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) AND (9) QDP (10) DependencyGraphProof [EQUIVALENT, 0 ms] (11) AND (12) QDP (13) MRRProof [EQUIVALENT, 63 ms] (14) QDP (15) PisEmptyProof [EQUIVALENT, 0 ms] (16) YES (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) IFR (EQUIVALENT) If Reductions: The following If expression "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" is transformed to "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); primDivNatS0 x y False = Zero; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="quot",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="quot vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="quot vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 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]; 274 -> 6[label="",style="solid", color="burlywood", weight=3]; 275[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 275[label="",style="solid", color="burlywood", weight=9]; 275 -> 7[label="",style="solid", color="burlywood", weight=3]; 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]; 276 -> 8[label="",style="solid", color="burlywood", weight=3]; 277[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];6 -> 277[label="",style="solid", color="burlywood", weight=9]; 277 -> 9[label="",style="solid", color="burlywood", weight=3]; 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]; 278 -> 10[label="",style="solid", color="burlywood", weight=3]; 279[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 279[label="",style="solid", color="burlywood", weight=9]; 279 -> 11[label="",style="solid", color="burlywood", weight=3]; 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]; 280 -> 12[label="",style="solid", color="burlywood", weight=3]; 281[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 281[label="",style="solid", color="burlywood", weight=9]; 281 -> 13[label="",style="solid", color="burlywood", weight=3]; 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]; 282 -> 14[label="",style="solid", color="burlywood", weight=3]; 283[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 283[label="",style="solid", color="burlywood", weight=9]; 283 -> 15[label="",style="solid", color="burlywood", weight=3]; 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]; 284 -> 16[label="",style="solid", color="burlywood", weight=3]; 285[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 285[label="",style="solid", color="burlywood", weight=9]; 285 -> 17[label="",style="solid", color="burlywood", weight=3]; 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]; 286 -> 18[label="",style="solid", color="burlywood", weight=3]; 287[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 287[label="",style="solid", color="burlywood", weight=9]; 287 -> 19[label="",style="solid", color="burlywood", weight=3]; 12[label="primQuotInt (Pos vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3]; 13[label="primQuotInt (Pos vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 14[label="primQuotInt (Pos vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 15[label="primQuotInt (Pos vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 16[label="primQuotInt (Neg vz30) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3]; 17[label="primQuotInt (Neg vz30) (Pos Zero)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3]; 18[label="primQuotInt (Neg vz30) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3]; 19[label="primQuotInt (Neg vz30) (Neg Zero)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3]; 20[label="Pos (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];20 -> 28[label="",style="dashed", color="green", weight=3]; 21[label="error []",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3]; 22[label="Neg (primDivNatS vz30 (Succ vz400))",fontsize=16,color="green",shape="box"];22 -> 30[label="",style="dashed", color="green", weight=3]; 23 -> 21[label="",style="dashed", color="red", weight=0]; 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]; 25 -> 21[label="",style="dashed", color="red", weight=0]; 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]; 27 -> 21[label="",style="dashed", color="red", weight=0]; 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]; 288 -> 33[label="",style="solid", color="burlywood", weight=3]; 289[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 289[label="",style="solid", color="burlywood", weight=9]; 289 -> 34[label="",style="solid", color="burlywood", weight=3]; 29[label="error []",fontsize=16,color="red",shape="box"];30 -> 28[label="",style="dashed", color="red", weight=0]; 30[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];30 -> 35[label="",style="dashed", color="magenta", weight=3]; 31 -> 28[label="",style="dashed", color="red", weight=0]; 31[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3]; 32 -> 28[label="",style="dashed", color="red", weight=0]; 32[label="primDivNatS vz30 (Succ vz400)",fontsize=16,color="magenta"];32 -> 37[label="",style="dashed", color="magenta", weight=3]; 32 -> 38[label="",style="dashed", color="magenta", weight=3]; 33[label="primDivNatS (Succ vz300) (Succ vz400)",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3]; 34[label="primDivNatS Zero (Succ vz400)",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3]; 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]; 290 -> 41[label="",style="solid", color="burlywood", weight=3]; 291[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];39 -> 291[label="",style="solid", color="burlywood", weight=9]; 291 -> 42[label="",style="solid", color="burlywood", weight=3]; 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]; 292 -> 43[label="",style="solid", color="burlywood", weight=3]; 293[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 293[label="",style="solid", color="burlywood", weight=9]; 293 -> 44[label="",style="solid", color="burlywood", weight=3]; 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]; 294 -> 45[label="",style="solid", color="burlywood", weight=3]; 295[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 295[label="",style="solid", color="burlywood", weight=9]; 295 -> 46[label="",style="solid", color="burlywood", weight=3]; 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]; 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]; 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]; 46[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3]; 47 -> 211[label="",style="dashed", color="red", weight=0]; 47[label="primDivNatS0 (Succ vz3000) (Succ vz4000) (primGEqNatS vz3000 vz4000)",fontsize=16,color="magenta"];47 -> 212[label="",style="dashed", color="magenta", weight=3]; 47 -> 213[label="",style="dashed", color="magenta", weight=3]; 47 -> 214[label="",style="dashed", color="magenta", weight=3]; 47 -> 215[label="",style="dashed", color="magenta", weight=3]; 48[label="primDivNatS0 (Succ vz3000) Zero True",fontsize=16,color="black",shape="box"];48 -> 53[label="",style="solid", color="black", weight=3]; 49[label="primDivNatS0 Zero (Succ vz4000) False",fontsize=16,color="black",shape="box"];49 -> 54[label="",style="solid", color="black", weight=3]; 50[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 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]; 296 -> 244[label="",style="solid", color="burlywood", weight=3]; 297[label="vz23/Zero",fontsize=10,color="white",style="solid",shape="box"];211 -> 297[label="",style="solid", color="burlywood", weight=9]; 297 -> 245[label="",style="solid", color="burlywood", weight=3]; 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]; 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]; 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]; 298 -> 246[label="",style="solid", color="burlywood", weight=3]; 299[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];244 -> 299[label="",style="solid", color="burlywood", weight=9]; 299 -> 247[label="",style="solid", color="burlywood", weight=3]; 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]; 300 -> 248[label="",style="solid", color="burlywood", weight=3]; 301[label="vz24/Zero",fontsize=10,color="white",style="solid",shape="box"];245 -> 301[label="",style="solid", color="burlywood", weight=9]; 301 -> 249[label="",style="solid", color="burlywood", weight=3]; 60 -> 28[label="",style="dashed", color="red", weight=0]; 60[label="primDivNatS (primMinusNatS (Succ vz3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];60 -> 66[label="",style="dashed", color="magenta", weight=3]; 60 -> 67[label="",style="dashed", color="magenta", weight=3]; 61 -> 28[label="",style="dashed", color="red", weight=0]; 61[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];61 -> 68[label="",style="dashed", color="magenta", weight=3]; 61 -> 69[label="",style="dashed", color="magenta", weight=3]; 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]; 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]; 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]; 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]; 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]; 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]; 250 -> 211[label="",style="dashed", color="red", weight=0]; 250[label="primDivNatS0 (Succ vz21) (Succ vz22) (primGEqNatS vz230 vz240)",fontsize=16,color="magenta"];250 -> 254[label="",style="dashed", color="magenta", weight=3]; 250 -> 255[label="",style="dashed", color="magenta", weight=3]; 251[label="primDivNatS0 (Succ vz21) (Succ vz22) True",fontsize=16,color="black",shape="triangle"];251 -> 256[label="",style="solid", color="black", weight=3]; 252[label="primDivNatS0 (Succ vz21) (Succ vz22) False",fontsize=16,color="black",shape="box"];252 -> 257[label="",style="solid", color="black", weight=3]; 253 -> 251[label="",style="dashed", color="red", weight=0]; 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]; 257[label="Zero",fontsize=16,color="green",shape="box"];258 -> 28[label="",style="dashed", color="red", weight=0]; 258[label="primDivNatS (primMinusNatS (Succ vz21) (Succ vz22)) (Succ (Succ vz22))",fontsize=16,color="magenta"];258 -> 259[label="",style="dashed", color="magenta", weight=3]; 258 -> 260[label="",style="dashed", color="magenta", weight=3]; 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]; 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]; 302 -> 262[label="",style="solid", color="burlywood", weight=3]; 303[label="vz21/Zero",fontsize=10,color="white",style="solid",shape="box"];261 -> 303[label="",style="solid", color="burlywood", weight=9]; 303 -> 263[label="",style="solid", color="burlywood", weight=3]; 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]; 304 -> 264[label="",style="solid", color="burlywood", weight=3]; 305[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];262 -> 305[label="",style="solid", color="burlywood", weight=9]; 305 -> 265[label="",style="solid", color="burlywood", weight=3]; 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]; 306 -> 266[label="",style="solid", color="burlywood", weight=3]; 307[label="vz22/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 307[label="",style="solid", color="burlywood", weight=9]; 307 -> 267[label="",style="solid", color="burlywood", weight=3]; 264[label="primMinusNatS (Succ vz210) (Succ vz220)",fontsize=16,color="black",shape="box"];264 -> 268[label="",style="solid", color="black", weight=3]; 265[label="primMinusNatS (Succ vz210) Zero",fontsize=16,color="black",shape="box"];265 -> 269[label="",style="solid", color="black", weight=3]; 266[label="primMinusNatS Zero (Succ vz220)",fontsize=16,color="black",shape="box"];266 -> 270[label="",style="solid", color="black", weight=3]; 267[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];267 -> 271[label="",style="solid", color="black", weight=3]; 268 -> 261[label="",style="dashed", color="red", weight=0]; 268[label="primMinusNatS vz210 vz220",fontsize=16,color="magenta"];268 -> 272[label="",style="dashed", color="magenta", weight=3]; 268 -> 273[label="",style="dashed", color="magenta", weight=3]; 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"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(vz220)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(vz3000) -> Succ(vz3000) new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS1(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (11) Complex Obligation (AND) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(vz220)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(vz3000) -> Succ(vz3000) new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS1(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) MRRProof (EQUIVALENT) 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. Strictly oriented dependency pairs: new_primDivNatS(Succ(Succ(vz3000)), Zero) -> new_primDivNatS(new_primMinusNatS1(vz3000), Zero) Strictly oriented rules of the TRS R: new_primMinusNatS0(Zero, Succ(vz220)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(vz3000) -> Succ(vz3000) new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) Used ordering: Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + 2*x_1 POL(Zero) = 1 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1 POL(new_primMinusNatS2) = 2 ---------------------------------------- (14) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS1(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (16) YES ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(vz220)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(vz3000) -> Succ(vz3000) new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS1(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPSizeChangeProof (EQUIVALENT) We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. Order:Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 1 POL(new_primMinusNatS0(x_1, x_2)) = x_1 From the DPs we obtained the following set of size-change graphs: *new_primDivNatS(Succ(Succ(vz3000)), Succ(vz4000)) -> new_primDivNatS0(vz3000, vz4000, vz3000, vz4000) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *new_primDivNatS0(vz21, vz22, Succ(vz230), Succ(vz240)) -> new_primDivNatS0(vz21, vz22, vz230, vz240) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(vz21, vz22, Succ(vz230), Zero) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS0(vz21, vz22, Zero, Zero) -> new_primDivNatS00(vz21, vz22) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS00(vz21, vz22) -> new_primDivNatS(new_primMinusNatS0(vz21, vz22), Succ(vz22)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS0(Zero, Succ(vz220)) -> Zero new_primMinusNatS0(Succ(vz210), Zero) -> Succ(vz210) new_primMinusNatS0(Succ(vz210), Succ(vz220)) -> new_primMinusNatS0(vz210, vz220) ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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_primMinusNatS(Succ(vz210), Succ(vz220)) -> new_primMinusNatS(vz210, vz220) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (22) YES