/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) IFR [EQUIVALENT, 0 ms] (4)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, 8 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, 21 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 "\(m,_)->m" is transformed to "m0 (m,_) = m; " The following Lambda expression "\(q,_)->q" is transformed to "q1 (q,_) = q; " The following Lambda expression "\(_,r)->r" is transformed to "r0 (_,r) = r; " ---------------------------------------- (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 vz wu)" is replaced by the following term "Float vz wu" The bind variable of the following binding Pattern "frac@(Double xu xv)" is replaced by the following term "Double xu xv" ---------------------------------------- (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 "m where { m = m0 vu6; ; m0 (m,vv) = m; ; vu6 = properFraction x; } " are unpacked to the following functions on top level "truncateVu6 xw = properFraction xw; " "truncateM xw = truncateM0 xw (truncateVu6 xw); " "truncateM0 xw (m,vv) = m; " The bindings of the following Let/Where expression "(fromIntegral q,r :% y) where { q = q1 vu30; ; q1 (q,vw) = q; ; r = r0 vu30; ; r0 (vx,r) = r; ; vu30 = quotRem x y; } " are unpacked to the following functions on top level "properFractionR0 xx xy (vx,r) = r; " "properFractionQ xx xy = properFractionQ1 xx xy (properFractionVu30 xx xy); " "properFractionQ1 xx xy (q,vw) = q; " "properFractionVu30 xx xy = quotRem xx xy; " "properFractionR xx xy = properFractionR0 xx xy (properFractionVu30 xx xy); " ---------------------------------------- (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="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 4[label="truncateM xz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="truncateM0 xz3 (floatProperFractionFloat xz3)",fontsize=16,color="burlywood",shape="box"];281[label="xz3/Float xz30 xz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 281[label="",style="solid", color="burlywood", weight=9]; 281 -> 8[label="",style="solid", color="burlywood", weight=3]; 8[label="truncateM0 (Float xz30 xz31) (floatProperFractionFloat (Float xz30 xz31))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 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296 -> 40[label="",style="solid", color="burlywood", weight=3]; 297[label="xz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 297[label="",style="solid", color="burlywood", weight=9]; 297 -> 41[label="",style="solid", color="burlywood", weight=3]; 36[label="error []",fontsize=16,color="red",shape="box"];37 -> 35[label="",style="dashed", color="red", weight=0]; 37[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3]; 38 -> 35[label="",style="dashed", color="red", weight=0]; 38[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];38 -> 43[label="",style="dashed", color="magenta", weight=3]; 39 -> 35[label="",style="dashed", color="red", weight=0]; 39[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];39 -> 44[label="",style="dashed", color="magenta", weight=3]; 39 -> 45[label="",style="dashed", color="magenta", weight=3]; 40[label="primDivNatS (Succ xz3000) (Succ xz3100)",fontsize=16,color="black",shape="box"];40 -> 46[label="",style="solid", color="black", weight=3]; 41[label="primDivNatS Zero (Succ xz3100)",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3]; 42[label="xz3100",fontsize=16,color="green",shape="box"];43[label="xz300",fontsize=16,color="green",shape="box"];44[label="xz300",fontsize=16,color="green",shape="box"];45[label="xz3100",fontsize=16,color="green",shape="box"];46[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];298[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];46 -> 298[label="",style="solid", color="burlywood", weight=9]; 298 -> 48[label="",style="solid", color="burlywood", weight=3]; 299[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 299[label="",style="solid", color="burlywood", weight=9]; 299 -> 49[label="",style="solid", color="burlywood", weight=3]; 47[label="Zero",fontsize=16,color="green",shape="box"];48[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];300[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];48 -> 300[label="",style="solid", color="burlywood", weight=9]; 300 -> 50[label="",style="solid", color="burlywood", weight=3]; 301[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 301[label="",style="solid", color="burlywood", weight=9]; 301 -> 51[label="",style="solid", color="burlywood", weight=3]; 49[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];302[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];49 -> 302[label="",style="solid", color="burlywood", weight=9]; 302 -> 52[label="",style="solid", color="burlywood", weight=3]; 303[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 303[label="",style="solid", color="burlywood", weight=9]; 303 -> 53[label="",style="solid", color="burlywood", weight=3]; 50[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3]; 51[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3]; 52[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3]; 53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3]; 54 -> 218[label="",style="dashed", color="red", weight=0]; 54[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];54 -> 219[label="",style="dashed", color="magenta", weight=3]; 54 -> 220[label="",style="dashed", color="magenta", weight=3]; 54 -> 221[label="",style="dashed", color="magenta", weight=3]; 54 -> 222[label="",style="dashed", color="magenta", weight=3]; 55[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];55 -> 60[label="",style="solid", color="black", weight=3]; 56[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3]; 57[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];57 -> 62[label="",style="solid", color="black", weight=3]; 219[label="xz31000",fontsize=16,color="green",shape="box"];220[label="xz30000",fontsize=16,color="green",shape="box"];221[label="xz31000",fontsize=16,color="green",shape="box"];222[label="xz30000",fontsize=16,color="green",shape="box"];218[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz22 xz23)",fontsize=16,color="burlywood",shape="triangle"];304[label="xz22/Succ xz220",fontsize=10,color="white",style="solid",shape="box"];218 -> 304[label="",style="solid", color="burlywood", weight=9]; 304 -> 251[label="",style="solid", color="burlywood", weight=3]; 305[label="xz22/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 305[label="",style="solid", color="burlywood", weight=9]; 305 -> 252[label="",style="solid", color="burlywood", weight=3]; 60[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];60 -> 67[label="",style="dashed", color="green", weight=3]; 61[label="Zero",fontsize=16,color="green",shape="box"];62[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 68[label="",style="dashed", color="green", weight=3]; 251[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) xz23)",fontsize=16,color="burlywood",shape="box"];306[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];251 -> 306[label="",style="solid", color="burlywood", weight=9]; 306 -> 253[label="",style="solid", color="burlywood", weight=3]; 307[label="xz23/Zero",fontsize=10,color="white",style="solid",shape="box"];251 -> 307[label="",style="solid", color="burlywood", weight=9]; 307 -> 254[label="",style="solid", color="burlywood", weight=3]; 252[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero xz23)",fontsize=16,color="burlywood",shape="box"];308[label="xz23/Succ xz230",fontsize=10,color="white",style="solid",shape="box"];252 -> 308[label="",style="solid", color="burlywood", weight=9]; 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254[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) Zero)",fontsize=16,color="black",shape="box"];254 -> 258[label="",style="solid", color="black", weight=3]; 255[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero (Succ xz230))",fontsize=16,color="black",shape="box"];255 -> 259[label="",style="solid", color="black", weight=3]; 256[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];256 -> 260[label="",style="solid", color="black", weight=3]; 73[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];73 -> 82[label="",style="solid", color="black", weight=3]; 74[label="Zero",fontsize=16,color="green",shape="box"];75[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];75 -> 83[label="",style="solid", color="black", weight=3]; 76[label="Zero",fontsize=16,color="green",shape="box"];257 -> 218[label="",style="dashed", color="red", weight=0]; 257[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz220 xz230)",fontsize=16,color="magenta"];257 -> 261[label="",style="dashed", color="magenta", weight=3]; 257 -> 262[label="",style="dashed", color="magenta", weight=3]; 258[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="black",shape="triangle"];258 -> 263[label="",style="solid", color="black", weight=3]; 259[label="primDivNatS0 (Succ xz20) (Succ xz21) False",fontsize=16,color="black",shape="box"];259 -> 264[label="",style="solid", color="black", weight=3]; 260 -> 258[label="",style="dashed", color="red", weight=0]; 260[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="magenta"];82[label="Succ xz30000",fontsize=16,color="green",shape="box"];83[label="Zero",fontsize=16,color="green",shape="box"];261[label="xz230",fontsize=16,color="green",shape="box"];262[label="xz220",fontsize=16,color="green",shape="box"];263[label="Succ (primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21)))",fontsize=16,color="green",shape="box"];263 -> 265[label="",style="dashed", color="green", weight=3]; 264[label="Zero",fontsize=16,color="green",shape="box"];265 -> 35[label="",style="dashed", color="red", weight=0]; 265[label="primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21))",fontsize=16,color="magenta"];265 -> 266[label="",style="dashed", color="magenta", weight=3]; 265 -> 267[label="",style="dashed", color="magenta", weight=3]; 266[label="primMinusNatS (Succ xz20) (Succ xz21)",fontsize=16,color="black",shape="box"];266 -> 268[label="",style="solid", color="black", weight=3]; 267[label="Succ xz21",fontsize=16,color="green",shape="box"];268[label="primMinusNatS xz20 xz21",fontsize=16,color="burlywood",shape="triangle"];310[label="xz20/Succ xz200",fontsize=10,color="white",style="solid",shape="box"];268 -> 310[label="",style="solid", color="burlywood", weight=9]; 310 -> 269[label="",style="solid", color="burlywood", weight=3]; 311[label="xz20/Zero",fontsize=10,color="white",style="solid",shape="box"];268 -> 311[label="",style="solid", color="burlywood", weight=9]; 311 -> 270[label="",style="solid", color="burlywood", weight=3]; 269[label="primMinusNatS (Succ xz200) xz21",fontsize=16,color="burlywood",shape="box"];312[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];269 -> 312[label="",style="solid", color="burlywood", weight=9]; 312 -> 271[label="",style="solid", color="burlywood", weight=3]; 313[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];269 -> 313[label="",style="solid", color="burlywood", weight=9]; 313 -> 272[label="",style="solid", color="burlywood", weight=3]; 270[label="primMinusNatS Zero xz21",fontsize=16,color="burlywood",shape="box"];314[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];270 -> 314[label="",style="solid", color="burlywood", weight=9]; 314 -> 273[label="",style="solid", color="burlywood", weight=3]; 315[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];270 -> 315[label="",style="solid", color="burlywood", weight=9]; 315 -> 274[label="",style="solid", color="burlywood", weight=3]; 271[label="primMinusNatS (Succ xz200) (Succ xz210)",fontsize=16,color="black",shape="box"];271 -> 275[label="",style="solid", color="black", weight=3]; 272[label="primMinusNatS (Succ xz200) Zero",fontsize=16,color="black",shape="box"];272 -> 276[label="",style="solid", color="black", weight=3]; 273[label="primMinusNatS Zero (Succ xz210)",fontsize=16,color="black",shape="box"];273 -> 277[label="",style="solid", color="black", weight=3]; 274[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];274 -> 278[label="",style="solid", color="black", weight=3]; 275 -> 268[label="",style="dashed", color="red", weight=0]; 275[label="primMinusNatS xz200 xz210",fontsize=16,color="magenta"];275 -> 279[label="",style="dashed", color="magenta", weight=3]; 275 -> 280[label="",style="dashed", color="magenta", weight=3]; 276[label="Succ xz200",fontsize=16,color="green",shape="box"];277[label="Zero",fontsize=16,color="green",shape="box"];278[label="Zero",fontsize=16,color="green",shape="box"];279[label="xz210",fontsize=16,color="green",shape="box"];280[label="xz200",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(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) 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(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (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(xz210)) -> Zero new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210) 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(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (28) YES