/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern lessleaves(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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) 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) UsableRulesReductionPairsProof [EQUIVALENT, 18 ms] (20) QDP (21) MRRProof [EQUIVALENT, 0 ms] (22) QDP (23) PisEmptyProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Clauses: append(nil, Y, Y). append(cons(U, V), Y, cons(U, Z)) :- append(V, Y, Z). lessleaves(nil, cons(W, Z)). lessleaves(cons(U, V), cons(W, Z)) :- ','(append(U, V, U1), ','(append(W, Z, W1), lessleaves(U1, W1))). Query: lessleaves(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: lessleaves_in_2: (b,b) append_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(U, V, W, Z, append_in_gga(U, V, U1)) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> APPEND_IN_GGA(U, V, U1) APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> U1_GGA(U, V, Y, Z, append_in_gga(V, Y, Z)) APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> APPEND_IN_GGA(V, Y, Z) U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_GG(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> APPEND_IN_GGA(W, Z, W1) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_GG(U, V, W, Z, lessleaves_in_gg(U1, W1)) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> LESSLEAVES_IN_GG(U1, W1) The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) LESSLEAVES_IN_GG(x1, x2) = LESSLEAVES_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5) U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x5, x6) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(U, V, W, Z, append_in_gga(U, V, U1)) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> APPEND_IN_GGA(U, V, U1) APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> U1_GGA(U, V, Y, Z, append_in_gga(V, Y, Z)) APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> APPEND_IN_GGA(V, Y, Z) U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_GG(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> APPEND_IN_GGA(W, Z, W1) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_GG(U, V, W, Z, lessleaves_in_gg(U1, W1)) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> LESSLEAVES_IN_GG(U1, W1) The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) LESSLEAVES_IN_GG(x1, x2) = LESSLEAVES_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5) U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x5, x6) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> APPEND_IN_GGA(V, Y, Z) The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(cons(U, V), Y, cons(U, Z)) -> APPEND_IN_GGA(V, Y, Z) R is empty. The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(cons(U, V), Y) -> APPEND_IN_GGA(V, 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: *APPEND_IN_GGA(cons(U, V), Y) -> APPEND_IN_GGA(V, 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: U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_GG(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> LESSLEAVES_IN_GG(U1, W1) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(U, V, W, Z, append_in_gga(U, V, U1)) The TRS R consists of the following rules: lessleaves_in_gg(nil, cons(W, Z)) -> lessleaves_out_gg(nil, cons(W, Z)) lessleaves_in_gg(cons(U, V), cons(W, Z)) -> U2_gg(U, V, W, Z, append_in_gga(U, V, U1)) append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) U2_gg(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_gg(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_gg(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> U4_gg(U, V, W, Z, lessleaves_in_gg(U1, W1)) U4_gg(U, V, W, Z, lessleaves_out_gg(U1, W1)) -> lessleaves_out_gg(cons(U, V), cons(W, Z)) The argument filtering Pi contains the following mapping: lessleaves_in_gg(x1, x2) = lessleaves_in_gg(x1, x2) nil = nil cons(x1, x2) = cons(x1, x2) lessleaves_out_gg(x1, x2) = lessleaves_out_gg U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) U3_gg(x1, x2, x3, x4, x5, x6) = U3_gg(x5, x6) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) LESSLEAVES_IN_GG(x1, x2) = LESSLEAVES_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x5, x6) 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: U2_GG(U, V, W, Z, append_out_gga(U, V, U1)) -> U3_GG(U, V, W, Z, U1, append_in_gga(W, Z, W1)) U3_GG(U, V, W, Z, U1, append_out_gga(W, Z, W1)) -> LESSLEAVES_IN_GG(U1, W1) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(U, V, W, Z, append_in_gga(U, V, U1)) The TRS R consists of the following rules: append_in_gga(nil, Y, Y) -> append_out_gga(nil, Y, Y) append_in_gga(cons(U, V), Y, cons(U, Z)) -> U1_gga(U, V, Y, Z, append_in_gga(V, Y, Z)) U1_gga(U, V, Y, Z, append_out_gga(V, Y, Z)) -> append_out_gga(cons(U, V), Y, cons(U, Z)) The argument filtering Pi contains the following mapping: nil = nil cons(x1, x2) = cons(x1, x2) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) append_out_gga(x1, x2, x3) = append_out_gga(x3) U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5) LESSLEAVES_IN_GG(x1, x2) = LESSLEAVES_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) U3_GG(x1, x2, x3, x4, x5, x6) = U3_GG(x5, x6) 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: U2_GG(W, Z, append_out_gga(U1)) -> U3_GG(U1, append_in_gga(W, Z)) U3_GG(U1, append_out_gga(W1)) -> LESSLEAVES_IN_GG(U1, W1) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(W, Z, append_in_gga(U, V)) The TRS R consists of the following rules: append_in_gga(nil, Y) -> append_out_gga(Y) append_in_gga(cons(U, V), Y) -> U1_gga(U, append_in_gga(V, Y)) U1_gga(U, append_out_gga(Z)) -> append_out_gga(cons(U, Z)) The set Q consists of the following terms: append_in_gga(x0, x1) U1_gga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: append_in_gga(nil, Y) -> append_out_gga(Y) Used ordering: POLO with Polynomial interpretation [POLO]: POL(LESSLEAVES_IN_GG(x_1, x_2)) = x_1 + x_2 POL(U1_gga(x_1, x_2)) = x_1 + x_2 POL(U2_GG(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U3_GG(x_1, x_2)) = x_1 + x_2 POL(append_in_gga(x_1, x_2)) = x_1 + x_2 POL(append_out_gga(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(nil) = 0 ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GG(W, Z, append_out_gga(U1)) -> U3_GG(U1, append_in_gga(W, Z)) U3_GG(U1, append_out_gga(W1)) -> LESSLEAVES_IN_GG(U1, W1) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(W, Z, append_in_gga(U, V)) The TRS R consists of the following rules: append_in_gga(cons(U, V), Y) -> U1_gga(U, append_in_gga(V, Y)) U1_gga(U, append_out_gga(Z)) -> append_out_gga(cons(U, Z)) The set Q consists of the following terms: append_in_gga(x0, x1) U1_gga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) 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: U2_GG(W, Z, append_out_gga(U1)) -> U3_GG(U1, append_in_gga(W, Z)) U3_GG(U1, append_out_gga(W1)) -> LESSLEAVES_IN_GG(U1, W1) LESSLEAVES_IN_GG(cons(U, V), cons(W, Z)) -> U2_GG(W, Z, append_in_gga(U, V)) Strictly oriented rules of the TRS R: append_in_gga(cons(U, V), Y) -> U1_gga(U, append_in_gga(V, Y)) U1_gga(U, append_out_gga(Z)) -> append_out_gga(cons(U, Z)) Used ordering: Knuth-Bendix order [KBO] with precedence:append_in_gga_2 > cons_2 > U2_GG_3 > U3_GG_2 > LESSLEAVES_IN_GG_2 > U1_gga_2 > append_out_gga_1 and weight map: append_out_gga_1=1 cons_2=0 U1_gga_2=0 U2_GG_3=0 U3_GG_2=1 append_in_gga_2=0 LESSLEAVES_IN_GG_2=2 The variable weight is 1 ---------------------------------------- (22) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: append_in_gga(x0, x1) U1_gga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (24) YES