/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) IFR [EQUIVALENT, 0 ms] (4) HASKELL (5) BR [EQUIVALENT, 0 ms] (6) HASKELL (7) COR [EQUIVALENT, 0 ms] (8) HASKELL (9) LetRed [EQUIVALENT, 0 ms] (10) HASKELL (11) NumRed [SOUND, 0 ms] (12) HASKELL (13) Narrow [SOUND, 0 ms] (14) AND (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) AND (18) QDP (19) MRRProof [EQUIVALENT, 0 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) QDPSizeChangeProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\(q,_)->q" is transformed to "q1 (q,_) = q; " The following Lambda expression "\(_,r)->r" is transformed to "r0 (_,r) = r; " The following Lambda expression "\(m,_)->m" is transformed to "m0 (m,_) = m; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) 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; " The following If expression "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" is transformed to "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); primModNatS0 x y False = Succ x; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. Binding Reductions: The bind variable of the following binding Pattern "frac@(Float vv vw)" is replaced by the following term "Float vv vw" The bind variable of the following binding Pattern "frac@(Double ww wx)" is replaced by the following term "Double ww wx" ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (8) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (9) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "(fromIntegral q,r :% y) where { q = q1 vu30; ; q1 (q,wy) = q; ; r = r0 vu30; ; r0 (wz,r) = r; ; vu30 = quotRem x y; } " are unpacked to the following functions on top level "properFractionVu30 xw xx = quotRem xw xx; " "properFractionQ1 xw xx (q,wy) = q; " "properFractionR xw xx = properFractionR0 xw xx (properFractionVu30 xw xx); " "properFractionQ xw xx = properFractionQ1 xw xx (properFractionVu30 xw xx); " "properFractionR0 xw xx (wz,r) = r; " The bindings of the following Let/Where expression "m where { m = m0 vu6; ; m0 (m,xv) = m; ; vu6 = properFraction x; } " are unpacked to the following functions on top level "truncateM xy = truncateM0 xy (truncateVu6 xy); " "truncateVu6 xy = properFraction xy; " "truncateM0 xy (m,xv) = m; " ---------------------------------------- (10) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (11) NumRed (SOUND) Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. ---------------------------------------- (12) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (13) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="truncate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="truncate xz3",fontsize=16,color="blue",shape="box"];326[label="truncate :: Float -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 326[label="",style="solid", color="blue", weight=9]; 326 -> 4[label="",style="solid", color="blue", weight=3]; 327[label="truncate :: (Ratio a) -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 327[label="",style="solid", color="blue", weight=9]; 327 -> 5[label="",style="solid", color="blue", weight=3]; 328[label="truncate :: Double -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 328[label="",style="solid", color="blue", weight=9]; 328 -> 6[label="",style="solid", color="blue", weight=3]; 4[label="truncate xz3",fontsize=16,color="black",shape="box"];4 -> 7[label="",style="solid", color="black", weight=3]; 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79[label="fromInteger (properFractionQ1 (Integer xz300) (Integer xz310) (Integer (primQuotInt xz300 xz310),Integer (primRemInt xz300 xz310)))",fontsize=16,color="magenta"];79 -> 85[label="",style="dashed", color="magenta", weight=3]; 80[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];352[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];80 -> 352[label="",style="solid", color="burlywood", weight=9]; 352 -> 86[label="",style="solid", color="burlywood", weight=3]; 353[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];80 -> 353[label="",style="solid", color="burlywood", weight=9]; 353 -> 87[label="",style="solid", color="burlywood", weight=3]; 81[label="Zero",fontsize=16,color="green",shape="box"];83 -> 32[label="",style="dashed", color="red", weight=0]; 83[label="primQuotInt xz30 xz31",fontsize=16,color="magenta"];83 -> 88[label="",style="dashed", color="magenta", weight=3]; 83 -> 89[label="",style="dashed", color="magenta", weight=3]; 82[label="properFractionQ1 xz30 xz31 (xz6,primRemInt xz30 xz31)",fontsize=16,color="black",shape="triangle"];82 -> 90[label="",style="solid", color="black", weight=3]; 85 -> 32[label="",style="dashed", color="red", weight=0]; 85[label="primQuotInt xz300 xz310",fontsize=16,color="magenta"];85 -> 91[label="",style="dashed", color="magenta", weight=3]; 85 -> 92[label="",style="dashed", color="magenta", weight=3]; 84[label="fromInteger (properFractionQ1 (Integer xz300) (Integer xz310) (Integer xz7,Integer (primRemInt xz300 xz310)))",fontsize=16,color="black",shape="triangle"];84 -> 93[label="",style="solid", color="black", weight=3]; 86[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];354[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];86 -> 354[label="",style="solid", color="burlywood", weight=9]; 354 -> 94[label="",style="solid", color="burlywood", weight=3]; 355[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 355[label="",style="solid", color="burlywood", weight=9]; 355 -> 95[label="",style="solid", color="burlywood", weight=3]; 87[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];356[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];87 -> 356[label="",style="solid", color="burlywood", weight=9]; 356 -> 96[label="",style="solid", color="burlywood", weight=3]; 357[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];87 -> 357[label="",style="solid", color="burlywood", weight=9]; 357 -> 97[label="",style="solid", color="burlywood", weight=3]; 88[label="xz30",fontsize=16,color="green",shape="box"];89[label="xz31",fontsize=16,color="green",shape="box"];90[label="xz6",fontsize=16,color="green",shape="box"];91[label="xz300",fontsize=16,color="green",shape="box"];92[label="xz310",fontsize=16,color="green",shape="box"];93[label="fromInteger (Integer xz7)",fontsize=16,color="black",shape="box"];93 -> 98[label="",style="solid", color="black", weight=3]; 94[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];94 -> 99[label="",style="solid", color="black", weight=3]; 95[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];95 -> 100[label="",style="solid", color="black", weight=3]; 96[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];96 -> 101[label="",style="solid", color="black", weight=3]; 97[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];97 -> 102[label="",style="solid", color="black", weight=3]; 98[label="xz7",fontsize=16,color="green",shape="box"];99 -> 263[label="",style="dashed", color="red", weight=0]; 99[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];99 -> 264[label="",style="dashed", color="magenta", weight=3]; 99 -> 265[label="",style="dashed", color="magenta", weight=3]; 99 -> 266[label="",style="dashed", color="magenta", weight=3]; 99 -> 267[label="",style="dashed", color="magenta", weight=3]; 100[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];100 -> 105[label="",style="solid", color="black", weight=3]; 101[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];101 -> 106[label="",style="solid", color="black", weight=3]; 102[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];102 -> 107[label="",style="solid", color="black", weight=3]; 264[label="xz31000",fontsize=16,color="green",shape="box"];265[label="xz30000",fontsize=16,color="green",shape="box"];266[label="xz30000",fontsize=16,color="green",shape="box"];267[label="xz31000",fontsize=16,color="green",shape="box"];263[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz26 xz27)",fontsize=16,color="burlywood",shape="triangle"];358[label="xz26/Succ xz260",fontsize=10,color="white",style="solid",shape="box"];263 -> 358[label="",style="solid", color="burlywood", weight=9]; 358 -> 296[label="",style="solid", color="burlywood", weight=3]; 359[label="xz26/Zero",fontsize=10,color="white",style="solid",shape="box"];263 -> 359[label="",style="solid", color="burlywood", weight=9]; 359 -> 297[label="",style="solid", color="burlywood", weight=3]; 105[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];105 -> 112[label="",style="dashed", color="green", weight=3]; 106[label="Zero",fontsize=16,color="green",shape="box"];107[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];107 -> 113[label="",style="dashed", color="green", weight=3]; 296[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) xz27)",fontsize=16,color="burlywood",shape="box"];360[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];296 -> 360[label="",style="solid", color="burlywood", weight=9]; 360 -> 298[label="",style="solid", color="burlywood", weight=3]; 361[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];296 -> 361[label="",style="solid", color="burlywood", weight=9]; 361 -> 299[label="",style="solid", color="burlywood", weight=3]; 297[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero xz27)",fontsize=16,color="burlywood",shape="box"];362[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];297 -> 362[label="",style="solid", color="burlywood", weight=9]; 362 -> 300[label="",style="solid", color="burlywood", weight=3]; 363[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];297 -> 363[label="",style="solid", color="burlywood", weight=9]; 363 -> 301[label="",style="solid", color="burlywood", weight=3]; 112 -> 65[label="",style="dashed", color="red", weight=0]; 112[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];112 -> 118[label="",style="dashed", color="magenta", weight=3]; 112 -> 119[label="",style="dashed", color="magenta", weight=3]; 113 -> 65[label="",style="dashed", color="red", weight=0]; 113[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];113 -> 120[label="",style="dashed", color="magenta", weight=3]; 113 -> 121[label="",style="dashed", color="magenta", weight=3]; 298[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) (Succ xz270))",fontsize=16,color="black",shape="box"];298 -> 302[label="",style="solid", color="black", weight=3]; 299[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) Zero)",fontsize=16,color="black",shape="box"];299 -> 303[label="",style="solid", color="black", weight=3]; 300[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero (Succ xz270))",fontsize=16,color="black",shape="box"];300 -> 304[label="",style="solid", color="black", weight=3]; 301[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];301 -> 305[label="",style="solid", color="black", weight=3]; 118[label="Zero",fontsize=16,color="green",shape="box"];119[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];119 -> 127[label="",style="solid", color="black", weight=3]; 120[label="Zero",fontsize=16,color="green",shape="box"];121[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];121 -> 128[label="",style="solid", color="black", weight=3]; 302 -> 263[label="",style="dashed", color="red", weight=0]; 302[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz260 xz270)",fontsize=16,color="magenta"];302 -> 306[label="",style="dashed", color="magenta", weight=3]; 302 -> 307[label="",style="dashed", color="magenta", weight=3]; 303[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="black",shape="triangle"];303 -> 308[label="",style="solid", color="black", weight=3]; 304[label="primDivNatS0 (Succ xz24) (Succ xz25) False",fontsize=16,color="black",shape="box"];304 -> 309[label="",style="solid", color="black", weight=3]; 305 -> 303[label="",style="dashed", color="red", weight=0]; 305[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="magenta"];127[label="Succ xz30000",fontsize=16,color="green",shape="box"];128[label="Zero",fontsize=16,color="green",shape="box"];306[label="xz270",fontsize=16,color="green",shape="box"];307[label="xz260",fontsize=16,color="green",shape="box"];308[label="Succ (primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25)))",fontsize=16,color="green",shape="box"];308 -> 310[label="",style="dashed", color="green", weight=3]; 309[label="Zero",fontsize=16,color="green",shape="box"];310 -> 65[label="",style="dashed", color="red", weight=0]; 310[label="primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25))",fontsize=16,color="magenta"];310 -> 311[label="",style="dashed", color="magenta", weight=3]; 310 -> 312[label="",style="dashed", color="magenta", weight=3]; 311[label="Succ xz25",fontsize=16,color="green",shape="box"];312[label="primMinusNatS (Succ xz24) (Succ xz25)",fontsize=16,color="black",shape="box"];312 -> 313[label="",style="solid", color="black", weight=3]; 313[label="primMinusNatS xz24 xz25",fontsize=16,color="burlywood",shape="triangle"];364[label="xz24/Succ xz240",fontsize=10,color="white",style="solid",shape="box"];313 -> 364[label="",style="solid", color="burlywood", weight=9]; 364 -> 314[label="",style="solid", color="burlywood", weight=3]; 365[label="xz24/Zero",fontsize=10,color="white",style="solid",shape="box"];313 -> 365[label="",style="solid", color="burlywood", weight=9]; 365 -> 315[label="",style="solid", color="burlywood", weight=3]; 314[label="primMinusNatS (Succ xz240) xz25",fontsize=16,color="burlywood",shape="box"];366[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];314 -> 366[label="",style="solid", color="burlywood", weight=9]; 366 -> 316[label="",style="solid", color="burlywood", weight=3]; 367[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];314 -> 367[label="",style="solid", color="burlywood", weight=9]; 367 -> 317[label="",style="solid", color="burlywood", weight=3]; 315[label="primMinusNatS Zero xz25",fontsize=16,color="burlywood",shape="box"];368[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];315 -> 368[label="",style="solid", color="burlywood", weight=9]; 368 -> 318[label="",style="solid", color="burlywood", weight=3]; 369[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];315 -> 369[label="",style="solid", color="burlywood", weight=9]; 369 -> 319[label="",style="solid", color="burlywood", weight=3]; 316[label="primMinusNatS (Succ xz240) (Succ xz250)",fontsize=16,color="black",shape="box"];316 -> 320[label="",style="solid", color="black", weight=3]; 317[label="primMinusNatS (Succ xz240) Zero",fontsize=16,color="black",shape="box"];317 -> 321[label="",style="solid", color="black", weight=3]; 318[label="primMinusNatS Zero (Succ xz250)",fontsize=16,color="black",shape="box"];318 -> 322[label="",style="solid", color="black", weight=3]; 319[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];319 -> 323[label="",style="solid", color="black", weight=3]; 320 -> 313[label="",style="dashed", color="red", weight=0]; 320[label="primMinusNatS xz240 xz250",fontsize=16,color="magenta"];320 -> 324[label="",style="dashed", color="magenta", weight=3]; 320 -> 325[label="",style="dashed", color="magenta", weight=3]; 321[label="Succ xz240",fontsize=16,color="green",shape="box"];322[label="Zero",fontsize=16,color="green",shape="box"];323[label="Zero",fontsize=16,color="green",shape="box"];324[label="xz240",fontsize=16,color="green",shape="box"];325[label="xz250",fontsize=16,color="green",shape="box"];} ---------------------------------------- (14) Complex Obligation (AND) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (17) Complex Obligation (AND) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) Strictly oriented rules of the TRS R: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) 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 ---------------------------------------- (20) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) (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(xz250)) -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(xz240), Succ(xz250)) -> new_primMinusNatS(xz240, xz250) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) 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(xz240), Succ(xz250)) -> new_primMinusNatS(xz240, xz250) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (28) YES