/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern mergesort(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 25 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 31 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 2 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 [SOUND, 0 ms] (25) QDP (26) MRRProof [EQUIVALENT, 39 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 0 ms] (29) QDP (30) UsableRulesProof [EQUIVALENT, 0 ms] (31) QDP (32) QReductionProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] (35) YES (36) PiDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) PiDP (39) PiDPToQDPProof [SOUND, 0 ms] (40) QDP (41) QDPSizeChangeProof [EQUIVALENT, 0 ms] (42) YES (43) PiDP (44) PiDPToQDPProof [SOUND, 0 ms] (45) QDP (46) QDPOrderProof [EQUIVALENT, 232 ms] (47) QDP (48) DependencyGraphProof [EQUIVALENT, 0 ms] (49) TRUE ---------------------------------------- (0) Obligation: Clauses: mergesort([], []). mergesort(.(X, []), .(X, [])). mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2)))). split(L1, L2, L3) :- split0(L1, L2, L3). split(L1, L2, L3) :- split1(L1, L2, L3). split(L1, L2, L3) :- split2(L1, L2, L3). split0([], [], []). split1(.(X, []), .(X, []), []). split2(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) :- split(L1, L2, L3). merge([], L1, L1). merge(L1, [], L1). merge(.(X, L1), .(Y, L2), .(X, L3)) :- ','(le(X, Y), merge(L1, .(Y, L2), L3)). merge(.(X, L1), .(Y, L2), .(Y, L3)) :- ','(gt(X, Y), merge(.(X, 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: mergesort(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: mergesort_in_2: (b,f) split2_in_3: (b,f,f) split_in_3: (b,f,f) merge_in_3: (b,b,f) le_in_2: (b,b) gt_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) ---------------------------------------- (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: MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4) SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT0_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT1_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> MERGE_IN_GGA(L5, L6, L2) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U14_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x6) SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) SPLIT0_IN_GAA(x1, x2, x3) = SPLIT0_IN_GAA(x1) U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4) SPLIT1_IN_GAA(x1, x2, x3) = SPLIT1_IN_GAA(x1) U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x1, x2, x3, x5, x6) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U14_GG(x1, x2, x3) = U14_GG(x1, x2, x3) U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> SPLIT2_IN_GAA(.(X, .(Y, L1)), L3, L4) SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_GAA(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U5_GAA(L1, L2, L3, split0_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT0_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U6_GAA(L1, L2, L3, split1_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT1_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> U7_GAA(L1, L2, L3, split2_in_gaa(L1, L2, L3)) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_GA(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_GA(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) U3_GA(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> MERGE_IN_GGA(L5, L6, L2) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> LE_IN_GG(X, Y) LE_IN_GG(s(X), s(Y)) -> U14_GG(X, Y, le_in_gg(X, Y)) LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_GGA(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> GT_IN_GG(X, Y) GT_IN_GG(s(X), s(Y)) -> U13_GG(X, Y, gt_in_gg(X, Y)) GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_GGA(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) U8_GAA(x1, x2, x3, x4, x5, x6) = U8_GAA(x1, x2, x3, x6) SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) SPLIT0_IN_GAA(x1, x2, x3) = SPLIT0_IN_GAA(x1) U6_GAA(x1, x2, x3, x4) = U6_GAA(x1, x4) SPLIT1_IN_GAA(x1, x2, x3) = SPLIT1_IN_GAA(x1) U7_GAA(x1, x2, x3, x4) = U7_GAA(x1, x4) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x1, x2, x3, x5, x6) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) U14_GG(x1, x2, x3) = U14_GG(x1, x2, x3) U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) 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: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) GT_IN_GG(x1, x2) = GT_IN_GG(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: 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 ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) 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: 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. ---------------------------------------- (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: *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 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: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) 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: U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) 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: U9_GGA(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2), L3) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(X, L3)) -> U9_GGA(X, L1, Y, L2, L3, le_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_GGA(X, L1, Y, L2, L3, gt_in_gg(X, Y)) U11_GGA(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2, L3) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_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)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U13_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) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) 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: U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y)) U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_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)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The set Q consists of the following terms: le_in_gg(x0, x1) gt_in_gg(x0, x1) U14_gg(x0, x1, x2) U13_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U11_GGA(X, L1, Y, L2, gt_in_gg(X, Y)) Strictly oriented rules of the TRS R: le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) le_in_gg(0, 0) -> le_out_gg(0, 0) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(0) = 1 POL(MERGE_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U11_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 + 2*x_4 + 2*x_5 POL(U13_gg(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U14_gg(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U9_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + x_1 + 2*x_2 + 2*x_3 + 2*x_4 + x_5 POL(gt_in_gg(x_1, x_2)) = x_1 + x_2 POL(gt_out_gg(x_1, x_2)) = x_1 + x_2 POL(le_in_gg(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(le_out_gg(x_1, x_2)) = 2*x_1 + 2*x_2 POL(s(x_1)) = 2*x_1 ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) U11_GGA(X, L1, Y, L2, gt_out_gg(X, Y)) -> MERGE_IN_GGA(.(X, L1), L2) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The set Q consists of the following terms: le_in_gg(x0, x1) gt_in_gg(x0, x1) U14_gg(x0, x1, x2) U13_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) The set Q consists of the following terms: le_in_gg(x0, x1) gt_in_gg(x0, x1) U14_gg(x0, x1, x2) U13_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) The set Q consists of the following terms: le_in_gg(x0, x1) gt_in_gg(x0, x1) U14_gg(x0, x1, x2) U13_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. gt_in_gg(x0, x1) U13_gg(x0, x1, x2) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) The TRS R consists of the following rules: le_in_gg(s(X), s(Y)) -> U14_gg(X, Y, le_in_gg(X, Y)) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) The set Q consists of the following terms: le_in_gg(x0, x1) U14_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (34) 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: *U9_GGA(X, L1, Y, L2, le_out_gg(X, Y)) -> MERGE_IN_GGA(L1, .(Y, L2)) The graph contains the following edges 2 >= 1 *MERGE_IN_GGA(.(X, L1), .(Y, L2)) -> U9_GGA(X, L1, Y, L2, le_in_gg(X, Y)) The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT2_IN_GAA(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> SPLIT_IN_GAA(L1, L2, L3) SPLIT_IN_GAA(L1, L2, L3) -> SPLIT2_IN_GAA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SPLIT2_IN_GAA(x1, x2, x3) = SPLIT2_IN_GAA(x1) SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: SPLIT2_IN_GAA(.(X, .(Y, L1))) -> SPLIT_IN_GAA(L1) SPLIT_IN_GAA(L1) -> SPLIT2_IN_GAA(L1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) 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_GAA(L1) -> SPLIT2_IN_GAA(L1) The graph contains the following edges 1 >= 1 *SPLIT2_IN_GAA(.(X, .(Y, L1))) -> SPLIT_IN_GAA(L1) The graph contains the following edges 1 > 1 ---------------------------------------- (42) YES ---------------------------------------- (43) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_GA(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4, L6) MERGESORT_IN_GA(.(X, .(Y, L1)), L2) -> U1_GA(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) U1_GA(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3, L5) The TRS R consists of the following rules: mergesort_in_ga([], []) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, []), .(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1)), L2) -> U1_ga(X, Y, L1, L2, split2_in_gaa(.(X, .(Y, L1)), L3, L4)) split2_in_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) -> U8_gaa(X, Y, L1, L2, L3, split_in_gaa(L1, L2, L3)) split_in_gaa(L1, L2, L3) -> U5_gaa(L1, L2, L3, split0_in_gaa(L1, L2, L3)) split0_in_gaa([], [], []) -> split0_out_gaa([], [], []) U5_gaa(L1, L2, L3, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U6_gaa(L1, L2, L3, split1_in_gaa(L1, L2, L3)) split1_in_gaa(.(X, []), .(X, []), []) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, L2, L3, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1, L2, L3) -> U7_gaa(L1, L2, L3, split2_in_gaa(L1, L2, L3)) U7_gaa(L1, L2, L3, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, L2, L3, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, L2, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L2, L4, mergesort_in_ga(L3, L5)) U2_ga(X, Y, L1, L2, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L2, L5, mergesort_in_ga(L4, L6)) U3_ga(X, Y, L1, L2, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, L2, merge_in_gga(L5, L6, L2)) merge_in_gga([], L1, L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, [], L1) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2), .(X, L3)) -> U9_gga(X, L1, Y, L2, L3, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, L3, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, L3, merge_in_gga(L1, .(Y, L2), L3)) merge_in_gga(.(X, L1), .(Y, L2), .(Y, L3)) -> U11_gga(X, L1, Y, L2, L3, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, L3, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, L3, merge_in_gga(.(X, L1), L2, L3)) U12_gga(X, L1, Y, L2, L3, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, L3, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, L2, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The argument filtering Pi contains the following mapping: mergesort_in_ga(x1, x2) = mergesort_in_ga(x1) [] = [] mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, x5) split2_in_gaa(x1, x2, x3) = split2_in_gaa(x1) U8_gaa(x1, x2, x3, x4, x5, x6) = U8_gaa(x1, x2, x3, x6) split_in_gaa(x1, x2, x3) = split_in_gaa(x1) U5_gaa(x1, x2, x3, x4) = U5_gaa(x1, x4) split0_in_gaa(x1, x2, x3) = split0_in_gaa(x1) split0_out_gaa(x1, x2, x3) = split0_out_gaa(x1, x2, x3) split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3) U6_gaa(x1, x2, x3, x4) = U6_gaa(x1, x4) split1_in_gaa(x1, x2, x3) = split1_in_gaa(x1) split1_out_gaa(x1, x2, x3) = split1_out_gaa(x1, x2, x3) U7_gaa(x1, x2, x3, x4) = U7_gaa(x1, x4) split2_out_gaa(x1, x2, x3) = split2_out_gaa(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x2, x3, x5, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x1, x2, x3, x5, x6) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5) merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x1, x2, x3, x4, x6) le_in_gg(x1, x2) = le_in_gg(x1, x2) s(x1) = s(x1) U14_gg(x1, x2, x3) = U14_gg(x1, x2, x3) 0 = 0 le_out_gg(x1, x2) = le_out_gg(x1, x2) U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6) U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6) gt_in_gg(x1, x2) = gt_in_gg(x1, x2) U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) gt_out_gg(x1, x2) = gt_out_gg(x1, x2) U12_gga(x1, x2, x3, x4, x5, x6) = U12_gga(x1, x2, x3, x4, x6) MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (44) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L4, mergesort_in_ga(L3)) U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4) MERGESORT_IN_GA(.(X, .(Y, L1))) -> U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3) The TRS R consists of the following rules: mergesort_in_ga([]) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1))) -> U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) split0_in_gaa([]) -> split0_out_gaa([], [], []) U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L4, mergesort_in_ga(L3)) U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L5, mergesort_in_ga(L4)) U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, merge_in_gga(L5, L6)) merge_in_gga([], L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, []) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2)) -> U9_gga(X, L1, Y, L2, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2))) merge_in_gga(.(X, L1), .(Y, L2)) -> U11_gga(X, L1, Y, L2, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2)) U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The set Q consists of the following terms: mergesort_in_ga(x0) split2_in_gaa(x0) split_in_gaa(x0) split0_in_gaa(x0) U5_gaa(x0, x1) split1_in_gaa(x0) U6_gaa(x0, x1) U7_gaa(x0, x1) U8_gaa(x0, x1, x2, x3) U1_ga(x0, x1, x2, x3) U2_ga(x0, x1, x2, x3, x4) U3_ga(x0, x1, x2, x3, x4) merge_in_gga(x0, x1) le_in_gg(x0, x1) U14_gg(x0, x1, x2) U9_gga(x0, x1, x2, x3, x4) gt_in_gg(x0, x1) U13_gg(x0, x1, x2) U11_gga(x0, x1, x2, x3, x4) U12_gga(x0, x1, x2, x3, x4) U10_gga(x0, x1, x2, x3, x4) U4_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MERGESORT_IN_GA(.(X, .(Y, L1))) -> U1_GA(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U2_GA_5(x_1, ..., x_5) ) = 2x_4 POL( mergesort_in_ga_1(x_1) ) = 0 POL( [] ) = 0 POL( mergesort_out_ga_2(x_1, x_2) ) = max{0, x_2 - 2} POL( ._2(x_1, x_2) ) = 2x_2 + 1 POL( U1_ga_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2 POL( split2_in_gaa_1(x_1) ) = 2x_1 + 1 POL( U1_GA_4(x_1, ..., x_4) ) = max{0, x_4 - 2} POL( U8_gaa_4(x_1, ..., x_4) ) = 2x_4 + 2 POL( split_in_gaa_1(x_1) ) = 2x_1 + 2 POL( split2_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U2_ga_5(x_1, ..., x_5) ) = max{0, x_1 + x_2 + x_4 - 2} POL( U3_ga_5(x_1, ..., x_5) ) = 2x_4 + 2 POL( U4_ga_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2} POL( merge_in_gga_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U7_gaa_2(x_1, x_2) ) = x_2 POL( split_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U5_gaa_2(x_1, x_2) ) = x_2 + 2 POL( split0_in_gaa_1(x_1) ) = 0 POL( U6_gaa_2(x_1, x_2) ) = 2x_2 + 2 POL( split1_in_gaa_1(x_1) ) = x_1 POL( split0_out_gaa_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 POL( split1_out_gaa_3(x_1, ..., x_3) ) = x_2 + x_3 POL( merge_out_gga_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U9_gga_5(x_1, ..., x_5) ) = x_2 + x_4 + 2 POL( le_in_gg_2(x_1, x_2) ) = 2x_1 POL( U11_gga_5(x_1, ..., x_5) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( gt_in_gg_2(x_1, x_2) ) = 0 POL( s_1(x_1) ) = 2x_1 POL( U14_gg_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 POL( 0 ) = 0 POL( le_out_gg_2(x_1, x_2) ) = max{0, -2} POL( U10_gga_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + 2 POL( U13_gg_3(x_1, ..., x_3) ) = 0 POL( gt_out_gg_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U12_gga_5(x_1, ..., x_5) ) = max{0, 2x_4 - 2} POL( MERGESORT_IN_GA_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) split0_in_gaa([]) -> split0_out_gaa([], [], []) U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_GA(X, Y, L1, L4, mergesort_in_ga(L3)) U2_GA(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> MERGESORT_IN_GA(L4) U1_GA(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> MERGESORT_IN_GA(L3) The TRS R consists of the following rules: mergesort_in_ga([]) -> mergesort_out_ga([], []) mergesort_in_ga(.(X, [])) -> mergesort_out_ga(.(X, []), .(X, [])) mergesort_in_ga(.(X, .(Y, L1))) -> U1_ga(X, Y, L1, split2_in_gaa(.(X, .(Y, L1)))) split2_in_gaa(.(X, .(Y, L1))) -> U8_gaa(X, Y, L1, split_in_gaa(L1)) split_in_gaa(L1) -> U5_gaa(L1, split0_in_gaa(L1)) split0_in_gaa([]) -> split0_out_gaa([], [], []) U5_gaa(L1, split0_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1) -> U6_gaa(L1, split1_in_gaa(L1)) split1_in_gaa(.(X, [])) -> split1_out_gaa(.(X, []), .(X, []), []) U6_gaa(L1, split1_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) split_in_gaa(L1) -> U7_gaa(L1, split2_in_gaa(L1)) U7_gaa(L1, split2_out_gaa(L1, L2, L3)) -> split_out_gaa(L1, L2, L3) U8_gaa(X, Y, L1, split_out_gaa(L1, L2, L3)) -> split2_out_gaa(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) U1_ga(X, Y, L1, split2_out_gaa(.(X, .(Y, L1)), L3, L4)) -> U2_ga(X, Y, L1, L4, mergesort_in_ga(L3)) U2_ga(X, Y, L1, L4, mergesort_out_ga(L3, L5)) -> U3_ga(X, Y, L1, L5, mergesort_in_ga(L4)) U3_ga(X, Y, L1, L5, mergesort_out_ga(L4, L6)) -> U4_ga(X, Y, L1, merge_in_gga(L5, L6)) merge_in_gga([], L1) -> merge_out_gga([], L1, L1) merge_in_gga(L1, []) -> merge_out_gga(L1, [], L1) merge_in_gga(.(X, L1), .(Y, L2)) -> U9_gga(X, L1, Y, L2, le_in_gg(X, Y)) le_in_gg(s(X), s(Y)) -> U14_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) U14_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) U9_gga(X, L1, Y, L2, le_out_gg(X, Y)) -> U10_gga(X, L1, Y, L2, merge_in_gga(L1, .(Y, L2))) merge_in_gga(.(X, L1), .(Y, L2)) -> U11_gga(X, L1, Y, L2, gt_in_gg(X, Y)) gt_in_gg(s(X), s(Y)) -> U13_gg(X, Y, gt_in_gg(X, Y)) gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) U13_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) U11_gga(X, L1, Y, L2, gt_out_gg(X, Y)) -> U12_gga(X, L1, Y, L2, merge_in_gga(.(X, L1), L2)) U12_gga(X, L1, Y, L2, merge_out_gga(.(X, L1), L2, L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(Y, L3)) U10_gga(X, L1, Y, L2, merge_out_gga(L1, .(Y, L2), L3)) -> merge_out_gga(.(X, L1), .(Y, L2), .(X, L3)) U4_ga(X, Y, L1, merge_out_gga(L5, L6, L2)) -> mergesort_out_ga(.(X, .(Y, L1)), L2) The set Q consists of the following terms: mergesort_in_ga(x0) split2_in_gaa(x0) split_in_gaa(x0) split0_in_gaa(x0) U5_gaa(x0, x1) split1_in_gaa(x0) U6_gaa(x0, x1) U7_gaa(x0, x1) U8_gaa(x0, x1, x2, x3) U1_ga(x0, x1, x2, x3) U2_ga(x0, x1, x2, x3, x4) U3_ga(x0, x1, x2, x3, x4) merge_in_gga(x0, x1) le_in_gg(x0, x1) U14_gg(x0, x1, x2) U9_gga(x0, x1, x2, x3, x4) gt_in_gg(x0, x1) U13_gg(x0, x1, x2) U11_gga(x0, x1, x2, x3, x4) U12_gga(x0, x1, x2, x3, x4) U10_gga(x0, x1, x2, x3, x4) U4_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (48) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (49) TRUE