/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern q(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 4 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) PiDPToQDPProof [SOUND, 17 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 164 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) UsableRulesProof [EQUIVALENT, 0 ms] (14) QDP (15) QReductionProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: Clauses: e(a, b). q(X, Y) :- e(X, Y). q(X, f(f(X))) :- ','(p(X, f(f(X))), q(X, f(X))). q(X, f(f(Y))) :- p(X, f(Y)). p(X, Y) :- e(X, Y). p(X, f(Y)) :- ','(r(X, f(Y)), p(X, Y)). r(X, Y) :- e(X, Y). r(X, f(Y)) :- ','(q(X, Y), r(X, Y)). r(f(X), f(X)) :- t(f(X), f(X)). t(X, Y) :- e(X, Y). t(f(X), f(Y)) :- ','(q(f(X), f(Y)), t(X, Y)). Query: q(g,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: q_in_2: (b,b) p_in_2: (b,b) r_in_2: (b,b) t_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg(a, b) U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y)) U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y)) U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y))) U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y))) U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y)) U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y) t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y)) U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y)) U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X)) U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y)) U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y)) U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y)) U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X))) U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X))) The argument filtering Pi contains the following mapping: q_in_gg(x1, x2) = q_in_gg(x1, x2) U1_gg(x1, x2, x3) = U1_gg(x3) e_in_gg(x1, x2) = e_in_gg(x1, x2) a = a b = b e_out_gg(x1, x2) = e_out_gg q_out_gg(x1, x2) = q_out_gg f(x1) = f(x1) U2_gg(x1, x2) = U2_gg(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) p_out_gg(x1, x2) = p_out_gg U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) r_in_gg(x1, x2) = r_in_gg(x1, x2) U8_gg(x1, x2, x3) = U8_gg(x3) r_out_gg(x1, x2) = r_out_gg U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3) U4_gg(x1, x2, x3) = U4_gg(x3) U10_gg(x1, x2, x3) = U10_gg(x3) U11_gg(x1, x2) = U11_gg(x2) t_in_gg(x1, x2) = t_in_gg(x1, x2) U12_gg(x1, x2, x3) = U12_gg(x3) t_out_gg(x1, x2) = t_out_gg U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) U14_gg(x1, x2, x3) = U14_gg(x3) U7_gg(x1, x2, x3) = U7_gg(x3) U3_gg(x1, x2) = U3_gg(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg(a, b) U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y)) U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y)) U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y))) U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y))) U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y)) U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y) t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y)) U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y)) U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X)) U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y)) U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y)) U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y)) U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X))) U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X))) The argument filtering Pi contains the following mapping: q_in_gg(x1, x2) = q_in_gg(x1, x2) U1_gg(x1, x2, x3) = U1_gg(x3) e_in_gg(x1, x2) = e_in_gg(x1, x2) a = a b = b e_out_gg(x1, x2) = e_out_gg q_out_gg(x1, x2) = q_out_gg f(x1) = f(x1) U2_gg(x1, x2) = U2_gg(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) p_out_gg(x1, x2) = p_out_gg U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) r_in_gg(x1, x2) = r_in_gg(x1, x2) U8_gg(x1, x2, x3) = U8_gg(x3) r_out_gg(x1, x2) = r_out_gg U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3) U4_gg(x1, x2, x3) = U4_gg(x3) U10_gg(x1, x2, x3) = U10_gg(x3) U11_gg(x1, x2) = U11_gg(x2) t_in_gg(x1, x2) = t_in_gg(x1, x2) U12_gg(x1, x2, x3) = U12_gg(x3) t_out_gg(x1, x2) = t_out_gg U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) U14_gg(x1, x2, x3) = U14_gg(x3) U7_gg(x1, x2, x3) = U7_gg(x3) U3_gg(x1, x2) = U3_gg(x2) ---------------------------------------- (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: Q_IN_GG(X, Y) -> U1_GG(X, Y, e_in_gg(X, Y)) Q_IN_GG(X, Y) -> E_IN_GG(X, Y) Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X)))) Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, Y) -> U5_GG(X, Y, e_in_gg(X, Y)) P_IN_GG(X, Y) -> E_IN_GG(X, Y) P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(X, Y) -> U8_GG(X, Y, e_in_gg(X, Y)) R_IN_GG(X, Y) -> E_IN_GG(X, Y) R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y)) R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y) Q_IN_GG(X, f(f(Y))) -> U4_GG(X, Y, p_in_gg(X, f(Y))) Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y)) U9_GG(X, Y, q_out_gg(X, Y)) -> U10_GG(X, Y, r_in_gg(X, Y)) U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y) R_IN_GG(f(X), f(X)) -> U11_GG(X, t_in_gg(f(X), f(X))) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(X, Y) -> U12_GG(X, Y, e_in_gg(X, Y)) T_IN_GG(X, Y) -> E_IN_GG(X, Y) T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y))) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> U14_GG(X, Y, t_in_gg(X, Y)) U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y) U6_GG(X, Y, r_out_gg(X, f(Y))) -> U7_GG(X, Y, p_in_gg(X, Y)) U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y) U2_GG(X, p_out_gg(X, f(f(X)))) -> U3_GG(X, q_in_gg(X, f(X))) U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg(a, b) U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y)) U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y)) U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y))) U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y))) U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y)) U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y) t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y)) U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y)) U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X)) U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y)) U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y)) U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y)) U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X))) U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X))) The argument filtering Pi contains the following mapping: q_in_gg(x1, x2) = q_in_gg(x1, x2) U1_gg(x1, x2, x3) = U1_gg(x3) e_in_gg(x1, x2) = e_in_gg(x1, x2) a = a b = b e_out_gg(x1, x2) = e_out_gg q_out_gg(x1, x2) = q_out_gg f(x1) = f(x1) U2_gg(x1, x2) = U2_gg(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) p_out_gg(x1, x2) = p_out_gg U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) r_in_gg(x1, x2) = r_in_gg(x1, x2) U8_gg(x1, x2, x3) = U8_gg(x3) r_out_gg(x1, x2) = r_out_gg U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3) U4_gg(x1, x2, x3) = U4_gg(x3) U10_gg(x1, x2, x3) = U10_gg(x3) U11_gg(x1, x2) = U11_gg(x2) t_in_gg(x1, x2) = t_in_gg(x1, x2) U12_gg(x1, x2, x3) = U12_gg(x3) t_out_gg(x1, x2) = t_out_gg U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) U14_gg(x1, x2, x3) = U14_gg(x3) U7_gg(x1, x2, x3) = U7_gg(x3) U3_gg(x1, x2) = U3_gg(x2) Q_IN_GG(x1, x2) = Q_IN_GG(x1, x2) U1_GG(x1, x2, x3) = U1_GG(x3) E_IN_GG(x1, x2) = E_IN_GG(x1, x2) U2_GG(x1, x2) = U2_GG(x1, x2) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) R_IN_GG(x1, x2) = R_IN_GG(x1, x2) U8_GG(x1, x2, x3) = U8_GG(x3) U9_GG(x1, x2, x3) = U9_GG(x1, x2, x3) U4_GG(x1, x2, x3) = U4_GG(x3) U10_GG(x1, x2, x3) = U10_GG(x3) U11_GG(x1, x2) = U11_GG(x2) T_IN_GG(x1, x2) = T_IN_GG(x1, x2) U12_GG(x1, x2, x3) = U12_GG(x3) U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) U14_GG(x1, x2, x3) = U14_GG(x3) U7_GG(x1, x2, x3) = U7_GG(x3) U3_GG(x1, x2) = U3_GG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_GG(X, Y) -> U1_GG(X, Y, e_in_gg(X, Y)) Q_IN_GG(X, Y) -> E_IN_GG(X, Y) Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X)))) Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, Y) -> U5_GG(X, Y, e_in_gg(X, Y)) P_IN_GG(X, Y) -> E_IN_GG(X, Y) P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(X, Y) -> U8_GG(X, Y, e_in_gg(X, Y)) R_IN_GG(X, Y) -> E_IN_GG(X, Y) R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y)) R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y) Q_IN_GG(X, f(f(Y))) -> U4_GG(X, Y, p_in_gg(X, f(Y))) Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y)) U9_GG(X, Y, q_out_gg(X, Y)) -> U10_GG(X, Y, r_in_gg(X, Y)) U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y) R_IN_GG(f(X), f(X)) -> U11_GG(X, t_in_gg(f(X), f(X))) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(X, Y) -> U12_GG(X, Y, e_in_gg(X, Y)) T_IN_GG(X, Y) -> E_IN_GG(X, Y) T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y))) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> U14_GG(X, Y, t_in_gg(X, Y)) U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y) U6_GG(X, Y, r_out_gg(X, f(Y))) -> U7_GG(X, Y, p_in_gg(X, Y)) U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y) U2_GG(X, p_out_gg(X, f(f(X)))) -> U3_GG(X, q_in_gg(X, f(X))) U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg(a, b) U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y)) U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y)) U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y))) U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y))) U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y)) U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y) t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y)) U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y)) U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X)) U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y)) U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y)) U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y)) U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X))) U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X))) The argument filtering Pi contains the following mapping: q_in_gg(x1, x2) = q_in_gg(x1, x2) U1_gg(x1, x2, x3) = U1_gg(x3) e_in_gg(x1, x2) = e_in_gg(x1, x2) a = a b = b e_out_gg(x1, x2) = e_out_gg q_out_gg(x1, x2) = q_out_gg f(x1) = f(x1) U2_gg(x1, x2) = U2_gg(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) p_out_gg(x1, x2) = p_out_gg U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) r_in_gg(x1, x2) = r_in_gg(x1, x2) U8_gg(x1, x2, x3) = U8_gg(x3) r_out_gg(x1, x2) = r_out_gg U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3) U4_gg(x1, x2, x3) = U4_gg(x3) U10_gg(x1, x2, x3) = U10_gg(x3) U11_gg(x1, x2) = U11_gg(x2) t_in_gg(x1, x2) = t_in_gg(x1, x2) U12_gg(x1, x2, x3) = U12_gg(x3) t_out_gg(x1, x2) = t_out_gg U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) U14_gg(x1, x2, x3) = U14_gg(x3) U7_gg(x1, x2, x3) = U7_gg(x3) U3_gg(x1, x2) = U3_gg(x2) Q_IN_GG(x1, x2) = Q_IN_GG(x1, x2) U1_GG(x1, x2, x3) = U1_GG(x3) E_IN_GG(x1, x2) = E_IN_GG(x1, x2) U2_GG(x1, x2) = U2_GG(x1, x2) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U5_GG(x1, x2, x3) = U5_GG(x3) U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) R_IN_GG(x1, x2) = R_IN_GG(x1, x2) U8_GG(x1, x2, x3) = U8_GG(x3) U9_GG(x1, x2, x3) = U9_GG(x1, x2, x3) U4_GG(x1, x2, x3) = U4_GG(x3) U10_GG(x1, x2, x3) = U10_GG(x3) U11_GG(x1, x2) = U11_GG(x2) T_IN_GG(x1, x2) = T_IN_GG(x1, x2) U12_GG(x1, x2, x3) = U12_GG(x3) U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) U14_GG(x1, x2, x3) = U14_GG(x3) U7_GG(x1, x2, x3) = U7_GG(x3) U3_GG(x1, x2) = U3_GG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 14 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X)))) U2_GG(X, p_out_gg(X, f(f(X)))) -> Q_IN_GG(X, f(X)) Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y))) U6_GG(X, Y, r_out_gg(X, f(Y))) -> P_IN_GG(X, Y) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y)) U9_GG(X, Y, q_out_gg(X, Y)) -> R_IN_GG(X, Y) R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y) Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y))) U13_GG(X, Y, q_out_gg(f(X), f(Y))) -> T_IN_GG(X, Y) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(X, Y, e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg(a, b) U1_gg(X, Y, e_out_gg(X, Y)) -> q_out_gg(X, Y) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(X, Y, e_in_gg(X, Y)) U5_gg(X, Y, e_out_gg(X, Y)) -> p_out_gg(X, Y) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(X, Y, e_in_gg(X, Y)) U8_gg(X, Y, e_out_gg(X, Y)) -> r_out_gg(X, Y) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(X, Y, p_in_gg(X, f(Y))) U4_gg(X, Y, p_out_gg(X, f(Y))) -> q_out_gg(X, f(f(Y))) U9_gg(X, Y, q_out_gg(X, Y)) -> U10_gg(X, Y, r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(X, t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(X, Y, e_in_gg(X, Y)) U12_gg(X, Y, e_out_gg(X, Y)) -> t_out_gg(X, Y) t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg(f(X), f(Y))) -> U14_gg(X, Y, t_in_gg(X, Y)) U14_gg(X, Y, t_out_gg(X, Y)) -> t_out_gg(f(X), f(Y)) U11_gg(X, t_out_gg(f(X), f(X))) -> r_out_gg(f(X), f(X)) U10_gg(X, Y, r_out_gg(X, Y)) -> r_out_gg(X, f(Y)) U6_gg(X, Y, r_out_gg(X, f(Y))) -> U7_gg(X, Y, p_in_gg(X, Y)) U7_gg(X, Y, p_out_gg(X, Y)) -> p_out_gg(X, f(Y)) U2_gg(X, p_out_gg(X, f(f(X)))) -> U3_gg(X, q_in_gg(X, f(X))) U3_gg(X, q_out_gg(X, f(X))) -> q_out_gg(X, f(f(X))) The argument filtering Pi contains the following mapping: q_in_gg(x1, x2) = q_in_gg(x1, x2) U1_gg(x1, x2, x3) = U1_gg(x3) e_in_gg(x1, x2) = e_in_gg(x1, x2) a = a b = b e_out_gg(x1, x2) = e_out_gg q_out_gg(x1, x2) = q_out_gg f(x1) = f(x1) U2_gg(x1, x2) = U2_gg(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) U5_gg(x1, x2, x3) = U5_gg(x3) p_out_gg(x1, x2) = p_out_gg U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) r_in_gg(x1, x2) = r_in_gg(x1, x2) U8_gg(x1, x2, x3) = U8_gg(x3) r_out_gg(x1, x2) = r_out_gg U9_gg(x1, x2, x3) = U9_gg(x1, x2, x3) U4_gg(x1, x2, x3) = U4_gg(x3) U10_gg(x1, x2, x3) = U10_gg(x3) U11_gg(x1, x2) = U11_gg(x2) t_in_gg(x1, x2) = t_in_gg(x1, x2) U12_gg(x1, x2, x3) = U12_gg(x3) t_out_gg(x1, x2) = t_out_gg U13_gg(x1, x2, x3) = U13_gg(x1, x2, x3) U14_gg(x1, x2, x3) = U14_gg(x3) U7_gg(x1, x2, x3) = U7_gg(x3) U3_gg(x1, x2) = U3_gg(x2) Q_IN_GG(x1, x2) = Q_IN_GG(x1, x2) U2_GG(x1, x2) = U2_GG(x1, x2) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) R_IN_GG(x1, x2) = R_IN_GG(x1, x2) U9_GG(x1, x2, x3) = U9_GG(x1, x2, x3) T_IN_GG(x1, x2) = T_IN_GG(x1, x2) U13_GG(x1, x2, x3) = U13_GG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X)))) U2_GG(X, p_out_gg) -> Q_IN_GG(X, f(X)) Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y))) U6_GG(X, Y, r_out_gg) -> P_IN_GG(X, Y) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y)) U9_GG(X, Y, q_out_gg) -> R_IN_GG(X, Y) R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y) Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y))) U13_GG(X, Y, q_out_gg) -> T_IN_GG(X, Y) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg U1_gg(e_out_gg) -> q_out_gg q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y)) U5_gg(e_out_gg) -> p_out_gg p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y)) U8_gg(e_out_gg) -> r_out_gg r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y))) U4_gg(p_out_gg) -> q_out_gg U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y)) U12_gg(e_out_gg) -> t_out_gg t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y)) U14_gg(t_out_gg) -> t_out_gg U11_gg(t_out_gg) -> r_out_gg U10_gg(r_out_gg) -> r_out_gg U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y)) U7_gg(p_out_gg) -> p_out_gg U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X))) U3_gg(q_out_gg) -> q_out_gg The set Q consists of the following terms: q_in_gg(x0, x1) e_in_gg(x0, x1) U1_gg(x0) p_in_gg(x0, x1) U5_gg(x0) r_in_gg(x0, x1) U8_gg(x0) U4_gg(x0) U9_gg(x0, x1, x2) t_in_gg(x0, x1) U12_gg(x0) U13_gg(x0, x1, x2) U14_gg(x0) U11_gg(x0) U10_gg(x0) U6_gg(x0, x1, x2) U7_gg(x0) U2_gg(x0, x1) U3_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. P_IN_GG(X, f(Y)) -> U6_GG(X, Y, r_in_gg(X, f(Y))) U6_GG(X, Y, r_out_gg) -> P_IN_GG(X, Y) U9_GG(X, Y, q_out_gg) -> R_IN_GG(X, Y) R_IN_GG(X, f(Y)) -> Q_IN_GG(X, Y) Q_IN_GG(X, f(f(Y))) -> P_IN_GG(X, f(Y)) T_IN_GG(f(X), f(Y)) -> U13_GG(X, Y, q_in_gg(f(X), f(Y))) U13_GG(X, Y, q_out_gg) -> T_IN_GG(X, Y) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U2_GG_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U13_GG_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U2_gg_2(x_1, x_2) ) = 2 POL( U4_gg_1(x_1) ) = max{0, 2x_1 - 2} POL( U7_gg_1(x_1) ) = max{0, -2} POL( U6_GG_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U9_GG_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( p_in_gg_2(x_1, x_2) ) = 2 POL( U5_gg_1(x_1) ) = max{0, -2} POL( e_in_gg_2(x_1, x_2) ) = 0 POL( f_1(x_1) ) = x_1 + 2 POL( U6_gg_3(x_1, ..., x_3) ) = 2 POL( r_in_gg_2(x_1, x_2) ) = max{0, -2} POL( U10_gg_1(x_1) ) = max{0, -2} POL( U8_gg_1(x_1) ) = 1 POL( U9_gg_3(x_1, ..., x_3) ) = max{0, -2} POL( q_in_gg_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( U11_gg_1(x_1) ) = max{0, -2} POL( t_in_gg_2(x_1, x_2) ) = max{0, x_2 - 1} POL( U13_gg_3(x_1, ..., x_3) ) = 1 POL( U3_gg_1(x_1) ) = max{0, -2} POL( U1_gg_1(x_1) ) = max{0, -2} POL( p_out_gg ) = 0 POL( q_out_gg ) = 0 POL( r_out_gg ) = 0 POL( U14_gg_1(x_1) ) = max{0, -2} POL( U12_gg_1(x_1) ) = max{0, -2} POL( t_out_gg ) = 0 POL( a ) = 2 POL( b ) = 2 POL( e_out_gg ) = 0 POL( Q_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( P_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( R_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( T_IN_GG_2(x_1, x_2) ) = max{0, 2x_2 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y)) p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X))) q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y))) U4_gg(p_out_gg) -> q_out_gg U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y)) U10_gg(r_out_gg) -> r_out_gg U11_gg(t_out_gg) -> r_out_gg t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y)) U14_gg(t_out_gg) -> t_out_gg U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y)) U7_gg(p_out_gg) -> p_out_gg U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X))) U3_gg(q_out_gg) -> q_out_gg U5_gg(e_out_gg) -> p_out_gg ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> U2_GG(X, p_in_gg(X, f(f(X)))) U2_GG(X, p_out_gg) -> Q_IN_GG(X, f(X)) Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(X, f(Y)) -> U9_GG(X, Y, q_in_gg(X, Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg U1_gg(e_out_gg) -> q_out_gg q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y)) U5_gg(e_out_gg) -> p_out_gg p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y)) U8_gg(e_out_gg) -> r_out_gg r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y))) U4_gg(p_out_gg) -> q_out_gg U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y)) U12_gg(e_out_gg) -> t_out_gg t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y)) U14_gg(t_out_gg) -> t_out_gg U11_gg(t_out_gg) -> r_out_gg U10_gg(r_out_gg) -> r_out_gg U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y)) U7_gg(p_out_gg) -> p_out_gg U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X))) U3_gg(q_out_gg) -> q_out_gg The set Q consists of the following terms: q_in_gg(x0, x1) e_in_gg(x0, x1) U1_gg(x0) p_in_gg(x0, x1) U5_gg(x0) r_in_gg(x0, x1) U8_gg(x0) U4_gg(x0) U9_gg(x0, x1, x2) t_in_gg(x0, x1) U12_gg(x0) U13_gg(x0, x1, x2) U14_gg(x0) U11_gg(x0) U10_gg(x0) U6_gg(x0, x1, x2) U7_gg(x0) U2_gg(x0, x1) U3_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) The TRS R consists of the following rules: q_in_gg(X, Y) -> U1_gg(e_in_gg(X, Y)) e_in_gg(a, b) -> e_out_gg U1_gg(e_out_gg) -> q_out_gg q_in_gg(X, f(f(X))) -> U2_gg(X, p_in_gg(X, f(f(X)))) p_in_gg(X, Y) -> U5_gg(e_in_gg(X, Y)) U5_gg(e_out_gg) -> p_out_gg p_in_gg(X, f(Y)) -> U6_gg(X, Y, r_in_gg(X, f(Y))) r_in_gg(X, Y) -> U8_gg(e_in_gg(X, Y)) U8_gg(e_out_gg) -> r_out_gg r_in_gg(X, f(Y)) -> U9_gg(X, Y, q_in_gg(X, Y)) q_in_gg(X, f(f(Y))) -> U4_gg(p_in_gg(X, f(Y))) U4_gg(p_out_gg) -> q_out_gg U9_gg(X, Y, q_out_gg) -> U10_gg(r_in_gg(X, Y)) r_in_gg(f(X), f(X)) -> U11_gg(t_in_gg(f(X), f(X))) t_in_gg(X, Y) -> U12_gg(e_in_gg(X, Y)) U12_gg(e_out_gg) -> t_out_gg t_in_gg(f(X), f(Y)) -> U13_gg(X, Y, q_in_gg(f(X), f(Y))) U13_gg(X, Y, q_out_gg) -> U14_gg(t_in_gg(X, Y)) U14_gg(t_out_gg) -> t_out_gg U11_gg(t_out_gg) -> r_out_gg U10_gg(r_out_gg) -> r_out_gg U6_gg(X, Y, r_out_gg) -> U7_gg(p_in_gg(X, Y)) U7_gg(p_out_gg) -> p_out_gg U2_gg(X, p_out_gg) -> U3_gg(q_in_gg(X, f(X))) U3_gg(q_out_gg) -> q_out_gg The set Q consists of the following terms: q_in_gg(x0, x1) e_in_gg(x0, x1) U1_gg(x0) p_in_gg(x0, x1) U5_gg(x0) r_in_gg(x0, x1) U8_gg(x0) U4_gg(x0) U9_gg(x0, x1, x2) t_in_gg(x0, x1) U12_gg(x0) U13_gg(x0, x1, x2) U14_gg(x0) U11_gg(x0) U10_gg(x0) U6_gg(x0, x1, x2) U7_gg(x0) U2_gg(x0, x1) U3_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) 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. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) R is empty. The set Q consists of the following terms: q_in_gg(x0, x1) e_in_gg(x0, x1) U1_gg(x0) p_in_gg(x0, x1) U5_gg(x0) r_in_gg(x0, x1) U8_gg(x0) U4_gg(x0) U9_gg(x0, x1, x2) t_in_gg(x0, x1) U12_gg(x0) U13_gg(x0, x1, x2) U14_gg(x0) U11_gg(x0) U10_gg(x0) U6_gg(x0, x1, x2) U7_gg(x0) U2_gg(x0, x1) U3_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (15) 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]. q_in_gg(x0, x1) e_in_gg(x0, x1) U1_gg(x0) p_in_gg(x0, x1) U5_gg(x0) r_in_gg(x0, x1) U8_gg(x0) U4_gg(x0) U9_gg(x0, x1, x2) t_in_gg(x0, x1) U12_gg(x0) U13_gg(x0, x1, x2) U14_gg(x0) U11_gg(x0) U10_gg(x0) U6_gg(x0, x1, x2) U7_gg(x0) U2_gg(x0, x1) U3_gg(x0) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) 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: *P_IN_GG(X, f(Y)) -> R_IN_GG(X, f(Y)) The graph contains the following edges 1 >= 1, 2 >= 2 *T_IN_GG(f(X), f(Y)) -> Q_IN_GG(f(X), f(Y)) The graph contains the following edges 1 >= 1, 2 >= 2 *R_IN_GG(f(X), f(X)) -> T_IN_GG(f(X), f(X)) The graph contains the following edges 1 >= 1, 2 >= 1, 1 >= 2, 2 >= 2 *Q_IN_GG(X, f(f(X))) -> P_IN_GG(X, f(f(X))) The graph contains the following edges 1 >= 1, 2 > 1, 2 >= 2 ---------------------------------------- (18) YES