/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 6 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 0 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 14 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) QDPOrderProof [EQUIVALENT, 31 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 0 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: C(x1) -> c(x1) c(c(x1)) -> x1 b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) b(B(x1)) -> x1 B(b(x1)) -> x1 c(C(x1)) -> x1 C(c(x1)) -> x1 Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: C(x1) -> c(x1) c(c(x1)) -> x1 b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 C(c(x1)) -> x1 c(C(x1)) -> x1 Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(B(x_1)) = x_1 POL(C(x_1)) = 1 + x_1 POL(b(x_1)) = x_1 POL(c(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: C(x1) -> c(x1) C(c(x1)) -> x1 c(C(x1)) -> x1 ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(c(x1)) -> x1 b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(B(x_1)) = x_1 POL(b(x_1)) = x_1 POL(c(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: c(c(x1)) -> x1 ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: B^1(b(x1)) -> B^2(x1) B^2(B(x1)) -> B^1(x1) C(b(c(B(c(x1))))) -> B^1(c(B(c(b(c(B(x1))))))) C(b(c(B(c(x1))))) -> C(B(c(b(c(B(x1)))))) C(b(c(B(c(x1))))) -> B^2(c(b(c(B(x1))))) C(b(c(B(c(x1))))) -> C(b(c(B(x1)))) C(b(c(B(c(x1))))) -> B^1(c(B(x1))) C(b(c(B(c(x1))))) -> C(B(x1)) C(b(c(B(c(x1))))) -> B^2(x1) The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: B^2(B(x1)) -> B^1(x1) B^1(b(x1)) -> B^2(x1) The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) 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. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: B^2(B(x1)) -> B^1(x1) B^1(b(x1)) -> B^2(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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: *B^1(b(x1)) -> B^2(x1) The graph contains the following edges 1 > 1 *B^2(B(x1)) -> B^1(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: C(b(c(B(c(x1))))) -> C(b(c(B(x1)))) C(b(c(B(c(x1))))) -> C(B(c(b(c(B(x1)))))) C(b(c(B(c(x1))))) -> C(B(x1)) The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C(b(c(B(c(x1))))) -> C(b(c(B(x1)))) C(b(c(B(c(x1))))) -> C(B(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(B(x_1)) = x_1 POL(C(x_1)) = x_1 POL(b(x_1)) = x_1 POL(c(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: B(B(x1)) -> b(x1) b(b(x1)) -> B(x1) B(b(x1)) -> x1 c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) b(B(x1)) -> x1 ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: C(b(c(B(c(x1))))) -> C(B(c(b(c(B(x1)))))) The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C(b(c(B(c(x1))))) -> C(B(c(b(c(B(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(C(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(b(x_1)) = [[1A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [-I, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(B(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: B(B(x1)) -> b(x1) b(b(x1)) -> B(x1) B(b(x1)) -> x1 c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) b(B(x1)) -> x1 ---------------------------------------- (20) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b(b(x1)) -> B(x1) B(B(x1)) -> b(x1) c(b(c(B(c(x1))))) -> b(c(B(c(b(c(B(x1))))))) B(b(x1)) -> x1 b(B(x1)) -> x1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES