/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 3 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 ---------------------------------------- (0) Obligation: Clauses: 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). goal(X) :- ','(s2t(X, T1), tappend(T1, T2, T3)). Query: goal(g) ---------------------------------------- (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_1: (b) s2t_in_2: (b,f) tappend_in_3: (b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g 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_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g ---------------------------------------- (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_G(X) -> U6_G(X, s2t_in_ga(X, T1)) GOAL_IN_G(X) -> S2T_IN_GA(X, T1) S2T_IN_GA(s(X), node(T, Y, T)) -> U3_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)) -> U4_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)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U6_G(X, s2t_out_ga(X, T1)) -> U7_G(X, tappend_in_gaa(T1, T2, T3)) U6_G(X, s2t_out_ga(X, T1)) -> TAPPEND_IN_GAA(T1, T2, T3) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> U1_GAA(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GAA(T1, T3, U) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> U2_GAA(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GAA(T2, T3, U) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U6_G(x1, x2) = U6_G(x2) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4) = U4_GA(x4) U5_GA(x1, x2, x3, x4) = U5_GA(x4) U7_G(x1, x2) = U7_G(x2) TAPPEND_IN_GAA(x1, x2, x3) = TAPPEND_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_G(X) -> U6_G(X, s2t_in_ga(X, T1)) GOAL_IN_G(X) -> S2T_IN_GA(X, T1) S2T_IN_GA(s(X), node(T, Y, T)) -> U3_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)) -> U4_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)) -> U5_GA(X, T, Y, s2t_in_ga(X, T)) S2T_IN_GA(s(X), node(T, Y, nil)) -> S2T_IN_GA(X, T) U6_G(X, s2t_out_ga(X, T1)) -> U7_G(X, tappend_in_gaa(T1, T2, T3)) U6_G(X, s2t_out_ga(X, T1)) -> TAPPEND_IN_GAA(T1, T2, T3) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> U1_GAA(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GAA(T1, T3, U) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> U2_GAA(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GAA(T2, T3, U) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g GOAL_IN_G(x1) = GOAL_IN_G(x1) U6_G(x1, x2) = U6_G(x2) S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U4_GA(x1, x2, x3, x4) = U4_GA(x4) U5_GA(x1, x2, x3, x4) = U5_GA(x4) U7_G(x1, x2) = U7_G(x2) TAPPEND_IN_GAA(x1, x2, x3) = TAPPEND_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GAA(T2, T3, U) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GAA(T1, T3, U) The TRS R consists of the following rules: goal_in_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g TAPPEND_IN_GAA(x1, x2, x3) = TAPPEND_IN_GAA(x1) 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: TAPPEND_IN_GAA(node(T1, X, T2), T3, node(T1, X, U)) -> TAPPEND_IN_GAA(T2, T3, U) TAPPEND_IN_GAA(node(T1, X, T2), T3, node(U, X, T2)) -> TAPPEND_IN_GAA(T1, T3, U) R is empty. The argument filtering Pi contains the following mapping: node(x1, x2, x3) = node(x1, x3) TAPPEND_IN_GAA(x1, x2, x3) = TAPPEND_IN_GAA(x1) 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: TAPPEND_IN_GAA(node(T1, T2)) -> TAPPEND_IN_GAA(T2) TAPPEND_IN_GAA(node(T1, T2)) -> TAPPEND_IN_GAA(T1) 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: *TAPPEND_IN_GAA(node(T1, T2)) -> TAPPEND_IN_GAA(T2) The graph contains the following edges 1 > 1 *TAPPEND_IN_GAA(node(T1, T2)) -> TAPPEND_IN_GAA(T1) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) 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_g(X) -> U6_g(X, s2t_in_ga(X, T1)) s2t_in_ga(s(X), node(T, Y, T)) -> U3_ga(X, T, Y, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(nil, Y, T)) -> U4_ga(X, Y, T, s2t_in_ga(X, T)) s2t_in_ga(s(X), node(T, Y, nil)) -> U5_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) U5_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, nil)) U4_ga(X, Y, T, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(nil, Y, T)) U3_ga(X, T, Y, s2t_out_ga(X, T)) -> s2t_out_ga(s(X), node(T, Y, T)) U6_g(X, s2t_out_ga(X, T1)) -> U7_g(X, tappend_in_gaa(T1, T2, T3)) tappend_in_gaa(nil, T, T) -> tappend_out_gaa(nil, T, T) tappend_in_gaa(node(nil, X, T2), T1, node(T1, X, T2)) -> tappend_out_gaa(node(nil, X, T2), T1, node(T1, X, T2)) tappend_in_gaa(node(T1, X, nil), T2, node(T1, X, T2)) -> tappend_out_gaa(node(T1, X, nil), T2, node(T1, X, T2)) tappend_in_gaa(node(T1, X, T2), T3, node(U, X, T2)) -> U1_gaa(T1, X, T2, T3, U, tappend_in_gaa(T1, T3, U)) tappend_in_gaa(node(T1, X, T2), T3, node(T1, X, U)) -> U2_gaa(T1, X, T2, T3, U, tappend_in_gaa(T2, T3, U)) U2_gaa(T1, X, T2, T3, U, tappend_out_gaa(T2, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(T1, X, U)) U1_gaa(T1, X, T2, T3, U, tappend_out_gaa(T1, T3, U)) -> tappend_out_gaa(node(T1, X, T2), T3, node(U, X, T2)) U7_g(X, tappend_out_gaa(T1, T2, T3)) -> goal_out_g(X) The argument filtering Pi contains the following mapping: goal_in_g(x1) = goal_in_g(x1) U6_g(x1, x2) = U6_g(x2) s2t_in_ga(x1, x2) = s2t_in_ga(x1) s(x1) = s(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x4) U4_ga(x1, x2, x3, x4) = U4_ga(x4) U5_ga(x1, x2, x3, x4) = U5_ga(x4) s2t_out_ga(x1, x2) = s2t_out_ga(x2) node(x1, x2, x3) = node(x1, x3) 0 = 0 U7_g(x1, x2) = U7_g(x2) tappend_in_gaa(x1, x2, x3) = tappend_in_gaa(x1) nil = nil tappend_out_gaa(x1, x2, x3) = tappend_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) goal_out_g(x1) = goal_out_g S2T_IN_GA(x1, x2) = S2T_IN_GA(x1) 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: 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 ---------------------------------------- (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: S2T_IN_GA(s(X)) -> S2T_IN_GA(X) 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: *S2T_IN_GA(s(X)) -> S2T_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES