/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern qs(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 3 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 14 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) PiDP (36) PiDPToQDPProof [SOUND, 0 ms] (37) QDP (38) QDPOrderProof [EQUIVALENT, 49 ms] (39) QDP (40) DependencyGraphProof [EQUIVALENT, 0 ms] (41) TRUE ---------------------------------------- (0) Obligation: Clauses: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))). qs([], []). part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(gt(X, Y), part(X, Xs, Ls, Bs)). part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(le(X, Y), part(X, Xs, Ls, Bs)). part(X, [], [], []). app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). app([], Ys, Ys). gt(s(X), s(Y)) :- gt(X, Y). gt(s(0), 0). le(s(X), s(Y)) :- le(X, Y). le(0, s(0)). le(0, 0). Query: qs(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: qs_in_2: (b,f) part_in_4: (b,b,f,f) gt_in_2: (b,b) le_in_2: (b,b) app_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) QS_IN_GA(x1, x2) = QS_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x2, x6) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) U4_GA(x1, x2, x3, x4) = U4_GA(x4) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) QS_IN_GA(x1, x2) = QS_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x2, x6) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5) U4_GA(x1, x2, x3, x4) = U4_GA(x4) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs) PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y)) PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y)) U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs) The TRS R consists of the following rules: gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U10_gg(gt_out_gg) -> gt_out_gg U11_gg(le_out_gg) -> le_out_gg The set Q consists of the following terms: gt_in_gg(x0, x1) le_in_gg(x0, x1) U10_gg(x0) U11_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) 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: *PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs) The graph contains the following edges 1 >= 1, 3 >= 2 *U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs) The graph contains the following edges 1 >= 1, 3 >= 2 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs) QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls) The TRS R consists of the following rules: qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs)) part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg(0, s(0)) le_in_gg(0, 0) -> le_out_gg(0, 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs)) part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], []) U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls)) qs_in_ga([], []) -> qs_out_ga([], []) U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs)) U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys)) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys) U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ga(x1, x2) = qs_in_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) s(x1) = s(x1) U10_gg(x1, x2, x3) = U10_gg(x3) 0 = 0 gt_out_gg(x1, x2) = gt_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) U11_gg(x1, x2, x3) = U11_gg(x3) le_out_gg(x1, x2) = le_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6) [] = [] part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5) qs_out_ga(x1, x2) = qs_out_ga(x2) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5) U4_ga(x1, x2, x3, x4) = U4_ga(x4) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) app_out_gga(x1, x2, x3) = app_out_gga(x3) QS_IN_GA(x1, x2) = QS_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) The TRS R consists of the following rules: qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg U10_gg(gt_out_gg) -> gt_out_gg U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U11_gg(le_out_gg) -> le_out_gg U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) part_in_ggaa(X, []) -> part_out_ggaa([], []) U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) qs_in_ga([]) -> qs_out_ga([]) U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys)) app_in_gga([], Ys) -> app_out_gga(Ys) U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) The set Q consists of the following terms: qs_in_ga(x0) part_in_ggaa(x0, x1) gt_in_gg(x0, x1) U10_gg(x0) U5_ggaa(x0, x1, x2, x3) le_in_gg(x0, x1) U11_gg(x0) U7_ggaa(x0, x1, x2, x3) U8_ggaa(x0, x1) U6_ggaa(x0, x1) U1_ga(x0, x1) U2_ga(x0, x1, x2) U3_ga(x0, x1, x2) app_in_gga(x0, x1) U9_gga(x0, x1) U4_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (38) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles)) U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U2_GA_3(x_1, ..., x_3) ) = x_2 + 1 POL( qs_in_ga_1(x_1) ) = 2 POL( ._2(x_1, x_2) ) = x_2 + 1 POL( U1_ga_2(x_1, x_2) ) = 2x_1 + 2 POL( part_in_ggaa_2(x_1, x_2) ) = x_2 POL( [] ) = 0 POL( qs_out_ga_1(x_1) ) = max{0, x_1 - 2} POL( U1_GA_2(x_1, x_2) ) = x_2 + 2 POL( U5_ggaa_4(x_1, ..., x_4) ) = x_3 + 1 POL( gt_in_gg_2(x_1, x_2) ) = 0 POL( U7_ggaa_4(x_1, ..., x_4) ) = x_3 + 1 POL( le_in_gg_2(x_1, x_2) ) = 0 POL( part_out_ggaa_2(x_1, x_2) ) = x_1 + x_2 POL( U2_ga_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 POL( U3_ga_3(x_1, ..., x_3) ) = max{0, -2} POL( U4_ga_1(x_1) ) = 2 POL( app_in_gga_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( gt_out_gg ) = 0 POL( U6_ggaa_2(x_1, x_2) ) = x_2 + 1 POL( s_1(x_1) ) = 0 POL( U10_gg_1(x_1) ) = 2 POL( 0 ) = 0 POL( U11_gg_1(x_1) ) = 2 POL( le_out_gg ) = 0 POL( U8_ggaa_2(x_1, x_2) ) = x_2 + 1 POL( U9_gga_2(x_1, x_2) ) = x_1 + 2 POL( app_out_gga_1(x_1) ) = max{0, x_1 - 2} POL( QS_IN_GA_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) part_in_ggaa(X, []) -> part_out_ggaa([], []) U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs) QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs)) The TRS R consists of the following rules: qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs)) part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(0), 0) -> gt_out_gg U10_gg(gt_out_gg) -> gt_out_gg U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs)) part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(0)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U11_gg(le_out_gg) -> le_out_gg U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs)) part_in_ggaa(X, []) -> part_out_ggaa([], []) U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs)) U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs) U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles)) qs_in_ga([]) -> qs_out_ga([]) U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs)) U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs))) app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys)) app_in_gga([], Ys) -> app_out_gga(Ys) U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys) The set Q consists of the following terms: qs_in_ga(x0) part_in_ggaa(x0, x1) gt_in_gg(x0, x1) U10_gg(x0) U5_ggaa(x0, x1, x2, x3) le_in_gg(x0, x1) U11_gg(x0) U7_ggaa(x0, x1, x2, x3) U8_ggaa(x0, x1) U6_ggaa(x0, x1) U1_ga(x0, x1) U2_ga(x0, x1, x2) U3_ga(x0, x1, x2) app_in_gga(x0, x1) U9_gga(x0, x1) U4_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (41) TRUE