/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 6 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 3 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 7 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) QDPOrderProof [EQUIVALENT, 10 ms] (15) QDP (16) UsableRulesProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPOrderProof [EQUIVALENT, 0 ms] (19) QDP (20) PisEmptyProof [EQUIVALENT, 0 ms] (21) YES (22) QDP (23) UsableRulesProof [EQUIVALENT, 0 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 10 ms] (26) QDP (27) PisEmptyProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(a(a(x1)))) -> a(b(b(b(x1)))) b(b(a(a(x1)))) -> a(b(a(b(x1)))) b(a(a(b(x1)))) -> b(a(b(a(x1)))) Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(a(a(x1)))) -> a(b(b(b(x1)))) b(a(a(b(x1)))) -> b(a(b(a(x1)))) a(b(b(a(a(x1))))) -> a(a(b(a(b(x1))))) b(b(b(a(a(x1))))) -> b(a(b(a(b(x1))))) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(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: b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(B_{A_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 0 POL(b_{b_1}(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( B_{A_1}_1(x_1) ) = max{0, 2x_1 - 2} POL( a_{a_1}_1(x_1) ) = x_1 POL( a_{b_1}_1(x_1) ) = 2x_1 POL( b_{a_1}_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (19) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (21) YES ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) 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. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) R is empty. Q is empty. 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. A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (26) Obligation: Q DP problem: P is empty. R is empty. Q is empty. 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