/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern qsort(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND,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, 4 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 14 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 3 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, 103 ms] (39) QDP (40) DependencyGraphProof [EQUIVALENT, 0 ms] (41) TRUE ---------------------------------------- (0) Obligation: Clauses: qsort([], []). qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S)))). split([], Y, [], []). split(.(X, Xs), Y, .(X, Ls), Bs) :- ','(le(X, Y), split(Xs, Y, Ls, Bs)). split(.(X, Xs), Y, Ls, .(X, Bs)) :- ','(gt(X, Y), split(Xs, Y, Ls, Bs)). append([], L, L). append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3). gt(s(X), s(Y)) :- gt(X, Y). gt(s(X), 0). le(s(X), s(Y)) :- le(X, Y). le(0, s(Y)). le(0, 0). Query: qsort(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: qsort_in_2: (b,f) split_in_4: (b,b,f,f) le_in_2: (b,b) gt_in_2: (b,b) append_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: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) ---------------------------------------- (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: QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B)) QSORT_IN_GA(.(H, L), S) -> SPLIT_IN_GGAA(L, H, A, B) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, 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(s(X), s(Y)) -> LE_IN_GG(X, Y) U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, 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) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1)) U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1) U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> U3_GA(H, L, S, A1, qsort_in_ga(B, B1)) U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1) U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_GA(H, L, S, append_in_gga(A1, .(H, B1), S)) U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> APPEND_IN_GGA(A1, .(H, B1), S) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U9_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3)) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) QSORT_IN_GA(x1, x2) = QSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, 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) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B)) QSORT_IN_GA(.(H, L), S) -> SPLIT_IN_GGAA(L, H, A, B) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, 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(s(X), s(Y)) -> LE_IN_GG(X, Y) U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U5_GGAA(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, 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) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_GGAA(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1)) U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1) U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> U3_GA(H, L, S, A1, qsort_in_ga(B, B1)) U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1) U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_GA(H, L, S, append_in_gga(A1, .(H, B1), S)) U3_GA(H, L, S, A1, qsort_out_ga(B, B1)) -> APPEND_IN_GGA(A1, .(H, B1), S) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U9_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3)) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) QSORT_IN_GA(x1, x2) = QSORT_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_IN_GGAA(x1, x2) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U11_GG(x1, x2, x3) = U11_GG(x3) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x6) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U10_GG(x1, x2, x3) = U10_GG(x3) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, 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) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2) 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: *APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2) 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: GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) GT_IN_GG(x1, x2) = GT_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: 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 ---------------------------------------- (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: 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. ---------------------------------------- (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: *GT_IN_GG(s(X), s(Y)) -> GT_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: LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) LE_IN_GG(x1, x2) = LE_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: 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 ---------------------------------------- (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: 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. ---------------------------------------- (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: *LE_IN_GG(s(X), s(Y)) -> LE_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, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y)) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_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, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) SPLIT_IN_GGAA(.(X, Xs), Y, .(X, Ls), Bs) -> U5_GGAA(X, Xs, Y, Ls, Bs, le_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_GGAA(X, Xs, Y, Ls, Bs, gt_in_gg(X, Y)) U7_GGAA(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> SPLIT_IN_GGAA(Xs, Y, Ls, Bs) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg SPLIT_IN_GGAA(x1, x2, x3, x4) = SPLIT_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, Xs, Y, le_out_gg) -> SPLIT_IN_GGAA(Xs, Y) SPLIT_IN_GGAA(.(X, Xs), Y) -> U5_GGAA(X, Xs, Y, le_in_gg(X, Y)) SPLIT_IN_GGAA(.(X, Xs), Y) -> U7_GGAA(X, Xs, Y, gt_in_gg(X, Y)) U7_GGAA(X, Xs, Y, gt_out_gg) -> SPLIT_IN_GGAA(Xs, Y) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U11_gg(le_out_gg) -> le_out_gg U10_gg(gt_out_gg) -> gt_out_gg The set Q consists of the following terms: le_in_gg(x0, x1) gt_in_gg(x0, x1) U11_gg(x0) U10_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: *SPLIT_IN_GGAA(.(X, Xs), Y) -> U5_GGAA(X, Xs, Y, le_in_gg(X, Y)) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3 *SPLIT_IN_GGAA(.(X, Xs), Y) -> U7_GGAA(X, Xs, Y, gt_in_gg(X, Y)) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3 *U5_GGAA(X, Xs, Y, le_out_gg) -> SPLIT_IN_GGAA(Xs, Y) The graph contains the following edges 2 >= 1, 3 >= 2 *U7_GGAA(X, Xs, Y, gt_out_gg) -> SPLIT_IN_GGAA(Xs, Y) The graph contains the following edges 2 >= 1, 3 >= 2 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_GA(H, L, S, B, qsort_in_ga(A, A1)) U2_GA(H, L, S, B, qsort_out_ga(A, A1)) -> QSORT_IN_GA(B, B1) QSORT_IN_GA(.(H, L), S) -> U1_GA(H, L, S, split_in_ggaa(L, H, A, B)) U1_GA(H, L, S, split_out_ggaa(L, H, A, B)) -> QSORT_IN_GA(A, A1) The TRS R consists of the following rules: qsort_in_ga([], []) -> qsort_out_ga([], []) qsort_in_ga(.(H, L), S) -> U1_ga(H, L, S, split_in_ggaa(L, H, A, B)) split_in_ggaa([], Y, [], []) -> split_out_ggaa([], Y, [], []) split_in_ggaa(.(X, Xs), Y, .(X, Ls), Bs) -> U5_ggaa(X, Xs, Y, 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(Y)) -> le_out_gg(0, s(Y)) 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)) U5_ggaa(X, Xs, Y, Ls, Bs, le_out_gg(X, Y)) -> U6_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) split_in_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) -> U7_ggaa(X, 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), 0) -> gt_out_gg(s(X), 0) U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U7_ggaa(X, Xs, Y, Ls, Bs, gt_out_gg(X, Y)) -> U8_ggaa(X, Xs, Y, Ls, Bs, split_in_ggaa(Xs, Y, Ls, Bs)) U8_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, Ls, .(X, Bs)) U6_ggaa(X, Xs, Y, Ls, Bs, split_out_ggaa(Xs, Y, Ls, Bs)) -> split_out_ggaa(.(X, Xs), Y, .(X, Ls), Bs) U1_ga(H, L, S, split_out_ggaa(L, H, A, B)) -> U2_ga(H, L, S, B, qsort_in_ga(A, A1)) U2_ga(H, L, S, B, qsort_out_ga(A, A1)) -> U3_ga(H, L, S, A1, qsort_in_ga(B, B1)) U3_ga(H, L, S, A1, qsort_out_ga(B, B1)) -> U4_ga(H, L, S, append_in_gga(A1, .(H, B1), S)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U9_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U9_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U4_ga(H, L, S, append_out_gga(A1, .(H, B1), S)) -> qsort_out_ga(.(H, L), S) The argument filtering Pi contains the following mapping: qsort_in_ga(x1, x2) = qsort_in_ga(x1) [] = [] qsort_out_ga(x1, x2) = qsort_out_ga(x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4) split_in_ggaa(x1, x2, x3, x4) = split_in_ggaa(x1, x2) split_out_ggaa(x1, x2, x3, x4) = split_out_ggaa(x3, x4) U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U11_gg(x1, x2, x3) = U11_gg(x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x6) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U10_gg(x1, x2, x3) = U10_gg(x3) gt_out_gg(x1, x2) = gt_out_gg U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, 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) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5) QSORT_IN_GA(x1, x2) = QSORT_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(H, split_out_ggaa(A, B)) -> U2_GA(H, B, qsort_in_ga(A)) U2_GA(H, B, qsort_out_ga(A1)) -> QSORT_IN_GA(B) QSORT_IN_GA(.(H, L)) -> U1_GA(H, split_in_ggaa(L, H)) U1_GA(H, split_out_ggaa(A, B)) -> QSORT_IN_GA(A) The TRS R consists of the following rules: qsort_in_ga([]) -> qsort_out_ga([]) qsort_in_ga(.(H, L)) -> U1_ga(H, split_in_ggaa(L, H)) split_in_ggaa([], Y) -> split_out_ggaa([], []) split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U11_gg(le_out_gg) -> le_out_gg U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y)) split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U10_gg(gt_out_gg) -> gt_out_gg U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y)) U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs)) U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs) U1_ga(H, split_out_ggaa(A, B)) -> U2_ga(H, B, qsort_in_ga(A)) U2_ga(H, B, qsort_out_ga(A1)) -> U3_ga(H, A1, qsort_in_ga(B)) U3_ga(H, A1, qsort_out_ga(B1)) -> U4_ga(append_in_gga(A1, .(H, B1))) append_in_gga([], L) -> append_out_gga(L) append_in_gga(.(H, L1), L2) -> U9_gga(H, append_in_gga(L1, L2)) U9_gga(H, append_out_gga(L3)) -> append_out_gga(.(H, L3)) U4_ga(append_out_gga(S)) -> qsort_out_ga(S) The set Q consists of the following terms: qsort_in_ga(x0) split_in_ggaa(x0, x1) le_in_gg(x0, x1) U11_gg(x0) U5_ggaa(x0, x1, x2, x3) gt_in_gg(x0, x1) U10_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) append_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(H, split_out_ggaa(A, B)) -> U2_GA(H, B, qsort_in_ga(A)) QSORT_IN_GA(.(H, L)) -> U1_GA(H, split_in_ggaa(L, H)) U1_GA(H, split_out_ggaa(A, B)) -> QSORT_IN_GA(A) 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) ) = 2x_2 POL( qsort_in_ga_1(x_1) ) = 2 POL( [] ) = 1 POL( qsort_out_ga_1(x_1) ) = max{0, x_1 - 2} POL( ._2(x_1, x_2) ) = x_2 + 2 POL( U1_ga_2(x_1, x_2) ) = max{0, x_1 - 2} POL( split_in_ggaa_2(x_1, x_2) ) = x_1 + 2 POL( U1_GA_2(x_1, x_2) ) = max{0, 2x_2 - 1} POL( split_out_ggaa_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( U5_ggaa_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( le_in_gg_2(x_1, x_2) ) = 2 POL( U7_ggaa_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( gt_in_gg_2(x_1, x_2) ) = 1 POL( U2_ga_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 POL( U3_ga_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U4_ga_1(x_1) ) = 2 POL( append_in_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( s_1(x_1) ) = 0 POL( U11_gg_1(x_1) ) = x_1 POL( 0 ) = 0 POL( le_out_gg ) = 2 POL( U6_ggaa_2(x_1, x_2) ) = x_2 + 2 POL( U10_gg_1(x_1) ) = x_1 POL( gt_out_gg ) = 1 POL( U8_ggaa_2(x_1, x_2) ) = x_2 + 2 POL( append_out_gga_1(x_1) ) = max{0, 2x_1 - 2} POL( U9_gga_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( QSORT_IN_GA_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: split_in_ggaa([], Y) -> split_out_ggaa([], []) split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y)) split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y)) U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs) gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y)) U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs)) U11_gg(le_out_gg) -> le_out_gg U10_gg(gt_out_gg) -> gt_out_gg ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GA(H, B, qsort_out_ga(A1)) -> QSORT_IN_GA(B) The TRS R consists of the following rules: qsort_in_ga([]) -> qsort_out_ga([]) qsort_in_ga(.(H, L)) -> U1_ga(H, split_in_ggaa(L, H)) split_in_ggaa([], Y) -> split_out_ggaa([], []) split_in_ggaa(.(X, Xs), Y) -> U5_ggaa(X, Xs, Y, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y)) le_in_gg(0, s(Y)) -> le_out_gg le_in_gg(0, 0) -> le_out_gg U11_gg(le_out_gg) -> le_out_gg U5_ggaa(X, Xs, Y, le_out_gg) -> U6_ggaa(X, split_in_ggaa(Xs, Y)) split_in_ggaa(.(X, Xs), Y) -> U7_ggaa(X, Xs, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg U10_gg(gt_out_gg) -> gt_out_gg U7_ggaa(X, Xs, Y, gt_out_gg) -> U8_ggaa(X, split_in_ggaa(Xs, Y)) U8_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(Ls, .(X, Bs)) U6_ggaa(X, split_out_ggaa(Ls, Bs)) -> split_out_ggaa(.(X, Ls), Bs) U1_ga(H, split_out_ggaa(A, B)) -> U2_ga(H, B, qsort_in_ga(A)) U2_ga(H, B, qsort_out_ga(A1)) -> U3_ga(H, A1, qsort_in_ga(B)) U3_ga(H, A1, qsort_out_ga(B1)) -> U4_ga(append_in_gga(A1, .(H, B1))) append_in_gga([], L) -> append_out_gga(L) append_in_gga(.(H, L1), L2) -> U9_gga(H, append_in_gga(L1, L2)) U9_gga(H, append_out_gga(L3)) -> append_out_gga(.(H, L3)) U4_ga(append_out_gga(S)) -> qsort_out_ga(S) The set Q consists of the following terms: qsort_in_ga(x0) split_in_ggaa(x0, x1) le_in_gg(x0, x1) U11_gg(x0) U5_ggaa(x0, x1, x2, x3) gt_in_gg(x0, x1) U10_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) append_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 1 less node. ---------------------------------------- (41) TRUE