/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 0 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (9) QDP (10) MNOCProof [EQUIVALENT, 0 ms] (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QReductionProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2)) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(append, app(app(cons, h), t)), l) -> APP(app(cons, h), app(app(append, t), l)) APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l) APP(app(append, app(app(cons, h), t)), l) -> APP(append, t) APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t)) APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h)) APP(app(map, f), app(app(cons, h), t)) -> APP(f, h) APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t) APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l1), app(app(append, l2), l3)) APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3) APP(app(append, app(app(append, l1), l2)), l3) -> APP(append, l2) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(append, app(app(map, f), l1)), app(app(map, f), l2)) APP(app(map, f), app(app(append, l1), l2)) -> APP(append, app(app(map, f), l1)) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2) The TRS R consists of the following rules: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l1), app(app(append, l2), l3)) APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l) APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3) The TRS R consists of the following rules: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) 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. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l1), app(app(append, l2), l3)) APP(app(append, app(app(cons, h), t)), l) -> APP(app(append, t), l) APP(app(append, app(app(append, l1), l2)), l3) -> APP(app(append, l2), l3) The TRS R consists of the following rules: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesReductionPairsProof (EQUIVALENT) First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem. The pairs P are: append1(append(l1, l2), l3) -> append1(l1, append(l2, l3)) append1(cons(h, t), l) -> append1(t, l) append1(append(l1, l2), l3) -> append1(l2, l3) and the Q and R are: Q restricted rewrite system: The TRS R consists of the following rules: append(nil, l) -> l append(cons(h, t), l) -> cons(h, append(t, l)) append(append(l1, l2), l3) -> append(l1, append(l2, l3)) Q is empty. 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. The following dependency pairs can be deleted: append1(append(l1, l2), l3) -> append1(l1, append(l2, l3)) append1(append(l1, l2), l3) -> append1(l2, l3) The following rules are removed from R: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(append(x_1, x_2)) = 2 + x_1 + x_2 POL(append1(x_1, x_2)) = 2*x_1 + x_2 POL(cons(x_1, x_2)) = 2*x_1 + x_2 POL(nil) = 0 ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: append1(cons(h, t), l) -> append1(t, l) The TRS R consists of the following rules: append(cons(h, t), l) -> cons(h, append(t, l)) append(append(l1, l2), l3) -> append(l1, append(l2, l3)) 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: append1(cons(h, t), l) -> append1(t, l) The TRS R consists of the following rules: append(cons(h, t), l) -> cons(h, append(t, l)) append(append(l1, l2), l3) -> append(l1, append(l2, l3)) The set Q consists of the following terms: append(cons(x0, x1), x2) append(append(x0, x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) 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. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: append1(cons(h, t), l) -> append1(t, l) R is empty. The set Q consists of the following terms: append(cons(x0, x1), x2) append(append(x0, x1), x2) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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]. append(cons(x0, x1), x2) append(append(x0, x1), x2) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: append1(cons(h, t), l) -> append1(t, l) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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: *append1(cons(h, t), l) -> append1(t, l) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t) APP(app(map, f), app(app(cons, h), t)) -> APP(f, h) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2) The TRS R consists of the following rules: app(app(append, nil), l) -> l app(app(append, app(app(cons, h), t)), l) -> app(app(cons, h), app(app(append, t), l)) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t)) app(app(append, app(app(append, l1), l2)), l3) -> app(app(append, l1), app(app(append, l2), l3)) app(app(map, f), app(app(append, l1), l2)) -> app(app(append, app(app(map, f), l1)), app(app(map, f), l2)) 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: APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t) APP(app(map, f), app(app(cons, h), t)) -> APP(f, h) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1) APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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: *APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t) The graph contains the following edges 1 >= 1, 2 > 2 *APP(app(map, f), app(app(cons, h), t)) -> APP(f, h) The graph contains the following edges 1 > 1, 2 > 2 *APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l1) The graph contains the following edges 1 >= 1, 2 > 2 *APP(app(map, f), app(app(append, l1), l2)) -> APP(app(map, f), l2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (22) YES