/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 123 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 31 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 3 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) MNOCProof [EQUIVALENT, 0 ms] (11) QDP (12) MRRProof [EQUIVALENT, 17 ms] (13) QDP (14) MRRProof [EQUIVALENT, 8 ms] (15) QDP (16) PisEmptyProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) MNOCProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 11 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 6 ms] (26) QDP (27) PisEmptyProof [EQUIVALENT, 0 ms] (28) YES (29) QDP (30) UsableRulesProof [EQUIVALENT, 0 ms] (31) QDP (32) MRRProof [EQUIVALENT, 49 ms] (33) QDP (34) QDPOrderProof [EQUIVALENT, 8 ms] (35) QDP (36) PisEmptyProof [EQUIVALENT, 0 ms] (37) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a12(a12(a12(a12(x1)))) -> x1 a13(a13(a13(a13(x1)))) -> x1 a14(a14(a14(a14(x1)))) -> x1 a15(a15(a15(a15(x1)))) -> x1 a16(a16(a16(a16(x1)))) -> x1 a23(a23(a23(a23(x1)))) -> x1 a24(a24(a24(a24(x1)))) -> x1 a25(a25(a25(a25(x1)))) -> x1 a26(a26(a26(a26(x1)))) -> x1 a34(a34(a34(a34(x1)))) -> x1 a35(a35(a35(a35(x1)))) -> x1 a36(a36(a36(a36(x1)))) -> x1 a45(a45(a45(a45(x1)))) -> x1 a46(a46(a46(a46(x1)))) -> x1 a56(a56(a56(a56(x1)))) -> x1 a13(a13(x1)) -> a12(a12(a23(a23(a12(a12(x1)))))) a14(a14(x1)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1)))))))))) a15(a15(x1)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))) a16(a16(x1)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))))) a24(a24(x1)) -> a23(a23(a34(a34(a23(a23(x1)))))) a25(a25(x1)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1)))))))))) a26(a26(x1)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))))) a35(a35(x1)) -> a34(a34(a45(a45(a34(a34(x1)))))) a36(a36(x1)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1)))))))))) a46(a46(x1)) -> a45(a45(a56(a56(a45(a45(x1)))))) a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) -> x1 a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) -> x1 a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) -> x1 a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) -> x1 a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a12(x_1)) = 1 + x_1 POL(a13(x_1)) = 4 + x_1 POL(a14(x_1)) = 6 + x_1 POL(a15(x_1)) = 8 + x_1 POL(a16(x_1)) = 10 + x_1 POL(a23(x_1)) = 1 + x_1 POL(a24(x_1)) = 4 + x_1 POL(a25(x_1)) = 6 + x_1 POL(a26(x_1)) = 8 + x_1 POL(a34(x_1)) = 1 + x_1 POL(a35(x_1)) = 4 + x_1 POL(a36(x_1)) = 6 + x_1 POL(a45(x_1)) = 1 + x_1 POL(a46(x_1)) = 4 + x_1 POL(a56(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a12(a12(a12(a12(x1)))) -> x1 a13(a13(a13(a13(x1)))) -> x1 a14(a14(a14(a14(x1)))) -> x1 a15(a15(a15(a15(x1)))) -> x1 a16(a16(a16(a16(x1)))) -> x1 a23(a23(a23(a23(x1)))) -> x1 a24(a24(a24(a24(x1)))) -> x1 a25(a25(a25(a25(x1)))) -> x1 a26(a26(a26(a26(x1)))) -> x1 a34(a34(a34(a34(x1)))) -> x1 a35(a35(a35(a35(x1)))) -> x1 a36(a36(a36(a36(x1)))) -> x1 a45(a45(a45(a45(x1)))) -> x1 a46(a46(a46(a46(x1)))) -> x1 a56(a56(a56(a56(x1)))) -> x1 a13(a13(x1)) -> a12(a12(a23(a23(a12(a12(x1)))))) a14(a14(x1)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1)))))))))) a15(a15(x1)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))) a16(a16(x1)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))))) a24(a24(x1)) -> a23(a23(a34(a34(a23(a23(x1)))))) a25(a25(x1)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1)))))))))) a26(a26(x1)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))))) a35(a35(x1)) -> a34(a34(a45(a45(a34(a34(x1)))))) a36(a36(x1)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1)))))))))) a46(a46(x1)) -> a45(a45(a56(a56(a45(a45(x1)))))) a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) -> x1 a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) -> x1 a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) -> x1 a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) -> x1 ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: A12(a12(a34(a34(x1)))) -> A34(a34(a12(a12(x1)))) A12(a12(a34(a34(x1)))) -> A34(a12(a12(x1))) A12(a12(a34(a34(x1)))) -> A12(a12(x1)) A12(a12(a34(a34(x1)))) -> A12(x1) A12(a12(a45(a45(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(x1) A12(a12(a56(a56(x1)))) -> A12(a12(x1)) A12(a12(a56(a56(x1)))) -> A12(x1) A23(a23(a45(a45(x1)))) -> A23(a23(x1)) A23(a23(a45(a45(x1)))) -> A23(x1) A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) A34(a34(a56(a56(x1)))) -> A34(a34(x1)) A34(a34(a56(a56(x1)))) -> A34(x1) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: A34(a34(a56(a56(x1)))) -> A34(x1) A34(a34(a56(a56(x1)))) -> A34(a34(x1)) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: A34(a34(a56(a56(x1)))) -> A34(x1) A34(a34(a56(a56(x1)))) -> A34(a34(x1)) The TRS R consists of the following rules: a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: A34(a34(a56(a56(x1)))) -> A34(x1) A34(a34(a56(a56(x1)))) -> A34(a34(x1)) The TRS R consists of the following rules: a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) The set Q consists of the following terms: a34(a34(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) 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: A34(a34(a56(a56(x1)))) -> A34(x1) Used ordering: Polynomial interpretation [POLO]: POL(A34(x_1)) = 2*x_1 POL(a34(x_1)) = 1 + 2*x_1 POL(a56(x_1)) = x_1 ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A34(a34(a56(a56(x1)))) -> A34(a34(x1)) The TRS R consists of the following rules: a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) The set Q consists of the following terms: a34(a34(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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: A34(a34(a56(a56(x1)))) -> A34(a34(x1)) Strictly oriented rules of the TRS R: a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Used ordering: Polynomial interpretation [POLO]: POL(A34(x_1)) = 2*x_1 POL(a34(x_1)) = 3*x_1 POL(a56(x_1)) = 3 + 3*x_1 ---------------------------------------- (15) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: a34(a34(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: A23(a23(a45(a45(x1)))) -> A23(x1) A23(a23(a45(a45(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: A23(a23(a45(a45(x1)))) -> A23(x1) A23(a23(a45(a45(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) The TRS R consists of the following rules: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A23(a23(a45(a45(x1)))) -> A23(x1) A23(a23(a45(a45(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) The TRS R consists of the following rules: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) The set Q consists of the following terms: a23(a23(a45(a45(x0)))) a23(a23(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A23(a23(a45(a45(x1)))) -> A23(x1) A23(a23(a45(a45(x1)))) -> A23(a23(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A23(x_1)) = x_1 POL(a23(x_1)) = x_1 POL(a45(x_1)) = 1 + x_1 POL(a56(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) The TRS R consists of the following rules: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) The set Q consists of the following terms: a23(a23(a45(a45(x0)))) a23(a23(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A23(a23(a56(a56(x1)))) -> A23(a23(x1)) A23(a23(a56(a56(x1)))) -> A23(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A23(x_1)) = x_1 POL(a23(x_1)) = x_1 POL(a45(x_1)) = 0 POL(a56(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) ---------------------------------------- (26) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) The set Q consists of the following terms: a23(a23(a45(a45(x0)))) a23(a23(a56(a56(x0)))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (28) YES ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: A12(a12(a34(a34(x1)))) -> A12(x1) A12(a12(a34(a34(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(x1) A12(a12(a56(a56(x1)))) -> A12(a12(x1)) A12(a12(a56(a56(x1)))) -> A12(x1) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a23(a23(a45(a45(x1)))) -> a45(a45(a23(a23(x1)))) a23(a23(a56(a56(x1)))) -> a56(a56(a23(a23(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: A12(a12(a34(a34(x1)))) -> A12(x1) A12(a12(a34(a34(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(x1) A12(a12(a56(a56(x1)))) -> A12(a12(x1)) A12(a12(a56(a56(x1)))) -> A12(x1) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) 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: A12(a12(a34(a34(x1)))) -> A12(x1) A12(a12(a34(a34(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(a12(x1)) A12(a12(a45(a45(x1)))) -> A12(x1) Used ordering: Polynomial interpretation [POLO]: POL(A12(x_1)) = 2*x_1 POL(a12(x_1)) = x_1 POL(a34(x_1)) = 1 + x_1 POL(a45(x_1)) = 1 + 3*x_1 POL(a56(x_1)) = x_1 ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: A12(a12(a56(a56(x1)))) -> A12(a12(x1)) A12(a12(a56(a56(x1)))) -> A12(x1) The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A12(a12(a56(a56(x1)))) -> A12(a12(x1)) A12(a12(a56(a56(x1)))) -> A12(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A12(x_1)) = x_1 POL(a12(x_1)) = x_1 POL(a34(x_1)) = x_1 POL(a45(x_1)) = 0 POL(a56(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) ---------------------------------------- (35) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a12(a12(a34(a34(x1)))) -> a34(a34(a12(a12(x1)))) a12(a12(a45(a45(x1)))) -> a45(a45(a12(a12(x1)))) a12(a12(a56(a56(x1)))) -> a56(a56(a12(a12(x1)))) a34(a34(a56(a56(x1)))) -> a56(a56(a34(a34(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (37) YES