/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 10 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 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) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES ---------------------------------------- (0) Obligation: Clauses: goal(A, B, C) :- ','(s2t(A, T), tapplast(T, B, C)). tapplast(L, X, Last) :- ','(tappend(L, node(nil, X, nil), LX), tlast(Last, LX)). tlast(X, node(nil, X, nil)). tlast(X, node(L, H, R)) :- tlast(X, L). tlast(X, node(L, H, R)) :- tlast(X, R). tappend(nil, T, T). tappend(node(nil, X, T2), T1, node(T1, X, T2)). tappend(node(T1, X, nil), T2, node(T1, X, T2)). tappend(node(T1, X, T2), T3, node(U, X, T2)) :- tappend(T1, T3, U). tappend(node(T1, X, T2), T3, node(T1, X, U)) :- tappend(T2, T3, U). s2t(s(X), node(T, Y, T)) :- s2t(X, T). s2t(s(X), node(nil, Y, T)) :- s2t(X, T). s2t(s(X), node(T, Y, nil)) :- s2t(X, T). s2t(s(X), node(nil, Y, nil)). s2t(0, nil). Query: goal(g,a,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: goal_in_3: (b,f,f) s2t_in_2: (b,f) tapplast_in_3: (b,f,f) tappend_in_3: (b,b,f) tlast_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) ---------------------------------------- (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: GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2t_in_ga(A, T)) GOAL_IN_GAA(A, B, C) -> S2T_IN_GA(A, T) S2T_IN_GA(s(X), node(T, Y, T)) -> U9_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(nil, Y, T)) -> U10_GA(X, Y, T, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> U11_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U1_GAA(A, B, C, s2t_out_ga(A, T)) -> U2_GAA(A, B, C, tapplast_in_gaa(T, B, C)) U1_GAA(A, B, C, s2t_out_ga(A, T)) -> TAPPLAST_IN_GAA(T, B, C) TAPPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) TAPPLAST_IN_GAA(L, X, Last) -> TAPPEND_IN_GGA(L, node(nil, X, nil), LX) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U) U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_GAA(L, X, Last, tlast_in_ag(Last, LX)) U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> TLAST_IN_AG(Last, LX) TLAST_IN_AG(X, node(L, H, R)) -> U5_AG(X, L, H, R, tlast_in_ag(X, L)) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L) TLAST_IN_AG(X, node(L, H, R)) -> U6_AG(X, L, H, R, tlast_in_ag(X, R)) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) GOAL_IN_GAA(x1, x2, x3) = GOAL_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x1, x4) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) U10_GA(x1, x2, x3, x4) = U10_GA(x1, x4) U11_GA(x1, x2, x3, x4) = U11_GA(x1, x4) U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) TAPPLAST_IN_GAA(x1, x2, x3) = TAPPLAST_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) TAPPEND_IN_GGA(x1, x2, x3) = TAPPEND_IN_GGA(x1, x2) U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x3, x4, x6) U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x3, x4, x6) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) TLAST_IN_AG(x1, x2) = TLAST_IN_AG(x2) U5_AG(x1, x2, x3, x4, x5) = U5_AG(x2, x4, x5) U6_AG(x1, x2, x3, x4, x5) = U6_AG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2t_in_ga(A, T)) GOAL_IN_GAA(A, B, C) -> S2T_IN_GA(A, T) S2T_IN_GA(s(X), node(T, Y, T)) -> U9_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(nil, Y, T)) -> U10_GA(X, Y, T, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> U11_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U1_GAA(A, B, C, s2t_out_ga(A, T)) -> U2_GAA(A, B, C, tapplast_in_gaa(T, B, C)) U1_GAA(A, B, C, s2t_out_ga(A, T)) -> TAPPLAST_IN_GAA(T, B, C) TAPPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) TAPPLAST_IN_GAA(L, X, Last) -> TAPPEND_IN_GGA(L, node(nil, X, nil), LX) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> U7_GGA(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> U8_GGA(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U) U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_GAA(L, X, Last, tlast_in_ag(Last, LX)) U3_GAA(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> TLAST_IN_AG(Last, LX) TLAST_IN_AG(X, node(L, H, R)) -> U5_AG(X, L, H, R, tlast_in_ag(X, L)) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L) TLAST_IN_AG(X, node(L, H, R)) -> U6_AG(X, L, H, R, tlast_in_ag(X, R)) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) GOAL_IN_GAA(x1, x2, x3) = GOAL_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x1, x4) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) U10_GA(x1, x2, x3, x4) = U10_GA(x1, x4) U11_GA(x1, x2, x3, x4) = U11_GA(x1, x4) U2_GAA(x1, x2, x3, x4) = U2_GAA(x1, x4) TAPPLAST_IN_GAA(x1, x2, x3) = TAPPLAST_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) TAPPEND_IN_GGA(x1, x2, x3) = TAPPEND_IN_GGA(x1, x2) U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x3, x4, x6) U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x3, x4, x6) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) TLAST_IN_AG(x1, x2) = TLAST_IN_AG(x2) U5_AG(x1, x2, x3, x4, x5) = U5_AG(x2, x4, x5) U6_AG(x1, x2, x3, x4, x5) = U6_AG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) TLAST_IN_AG(x1, x2) = TLAST_IN_AG(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: TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, R) TLAST_IN_AG(X, node(L, H, R)) -> TLAST_IN_AG(X, L) R is empty. The argument filtering Pi contains the following mapping: node(x1, x2, x3) = node(x1, x3) TLAST_IN_AG(x1, x2) = TLAST_IN_AG(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: TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(R) TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(L) 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: *TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(R) The graph contains the following edges 1 > 1 *TLAST_IN_AG(node(L, R)) -> TLAST_IN_AG(L) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) TAPPEND_IN_GGA(x1, x2, x3) = TAPPEND_IN_GGA(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: TAPPEND_IN_GGA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GGA(T2, T3, U) TAPPEND_IN_GGA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GGA(T1, T3, U) R is empty. The argument filtering Pi contains the following mapping: node(x1, x2, x3) = node(x1, x3) TAPPEND_IN_GGA(x1, x2, x3) = TAPPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) 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: TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T2, T3) TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T1, T3) 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: *TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T2, T3) The graph contains the following edges 1 > 1, 2 >= 2 *TAPPEND_IN_GGA(node(T1, T2), T3) -> TAPPEND_IN_GGA(T1, T3) 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: S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2t_in_ga(A, T)) s2t_in_ga(s(X), node(T, Y, T)) -> U9_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U10_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U11_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, nil)) -> s2t_out_ga(s(X), node(nil, Y, nil)) s2t_in_ga(0, nil) -> s2t_out_ga(0, nil) U11_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U10_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U9_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U1_gaa(A, B, C, s2t_out_ga(A, T)) -> U2_gaa(A, B, C, tapplast_in_gaa(T, B, C)) tapplast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, tappend_in_gga(L, node(nil, X, nil), LX)) tappend_in_gga(nil, T, T) -> tappend_out_gga(nil, T, T) tappend_in_gga(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gga(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gga(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gga(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gga(node(T1, X, T2), T3, node(U, X, T2)) -> U7_gga(T1, X, T2, T3, U, tappend_in_gga(T1, T3, U)) tappend_in_gga(node(T1, X, T2), T3, node(T1, X, U)) -> U8_gga(T1, X, T2, T3, U, tappend_in_gga(T2, T3, U)) U8_gga(T1, X, T2, T3, U, tappend_out_gga(T2, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(T1, X, U)) U7_gga(T1, X, T2, T3, U, tappend_out_gga(T1, T3, U)) -> tappend_out_gga(node(T1, X, T2), T3, node(U, X, T2)) U3_gaa(L, X, Last, tappend_out_gga(L, node(nil, X, nil), LX)) -> U4_gaa(L, X, Last, tlast_in_ag(Last, LX)) tlast_in_ag(X, node(nil, X, nil)) -> tlast_out_ag(X, node(nil, X, nil)) tlast_in_ag(X, node(L, H, R)) -> U5_ag(X, L, H, R, tlast_in_ag(X, L)) tlast_in_ag(X, node(L, H, R)) -> U6_ag(X, L, H, R, tlast_in_ag(X, R)) U6_ag(X, L, H, R, tlast_out_ag(X, R)) -> tlast_out_ag(X, node(L, H, R)) U5_ag(X, L, H, R, tlast_out_ag(X, L)) -> tlast_out_ag(X, node(L, H, R)) U4_gaa(L, X, Last, tlast_out_ag(Last, LX)) -> tapplast_out_gaa(L, X, Last) U2_gaa(A, B, C, tapplast_out_gaa(T, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x1, x4) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3, x4) = U9_ga(x1, x4) U10_ga(x1, x2, x3, x4) = U10_ga(x1, x4) U11_ga(x1, x2, x3, x4) = U11_ga(x1, x4) s2t_out_ga(x1, x2) = s2t_out_ga(x1, x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U2_gaa(x1, x2, x3, x4) = U2_gaa(x1, x4) tapplast_in_gaa(x1, x2, x3) = tapplast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) tappend_in_gga(x1, x2, x3) = tappend_in_gga(x1, x2) nil = nil tappend_out_gga(x1, x2, x3) = tappend_out_gga(x1, x2, x3) U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x3, x4, x6) U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x3, x4, x6) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) tlast_in_ag(x1, x2) = tlast_in_ag(x2) tlast_out_ag(x1, x2) = tlast_out_ag(x2) U5_ag(x1, x2, x3, x4, x5) = U5_ag(x2, x4, x5) U6_ag(x1, x2, x3, x4, x5) = U6_ag(x2, x4, x5) tapplast_out_gaa(x1, x2, x3) = tapplast_out_gaa(x1) goal_out_gaa(x1, x2, x3) = goal_out_gaa(x1) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 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: S2T_IN_GA(s(X), node(nil, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, T)) -> S2T_IN_GA(X, T) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) node(x1, x2, x3) = node(x1, x3) nil = nil S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 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: S2T_IN_GA(s(X)) -> S2T_IN_GA(X) 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: *S2T_IN_GA(s(X)) -> S2T_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES