/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 698 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) The (relative) TRS S consists of the following rules: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) The (relative) TRS S consists of the following rules: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) The (relative) TRS S consists of the following rules: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence append(cons(y, ys), x) ->^+ cons(y, append(ys, x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [ys / cons(y, ys)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) The (relative) TRS S consists of the following rules: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: isEmpty(empty) -> true isEmpty(node(l, x, r)) -> false left(empty) -> empty left(node(l, x, r)) -> l right(empty) -> empty right(node(l, x, r)) -> r elem(node(l, x, r)) -> x append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) listify(n, xs) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), elem(left(n)), node(right(left(n)), elem(n), right(n))), xs, append(xs, n)) if(true, b, n, m, xs, ys) -> xs if(false, false, n, m, xs, ys) -> listify(m, xs) if(false, true, n, m, xs, ys) -> listify(n, ys) toList(n) -> listify(n, nil) The (relative) TRS S consists of the following rules: encArg(empty) -> empty encArg(true) -> true encArg(node(x_1, x_2, x_3)) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(false) -> false encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(y) -> y encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_left(x_1)) -> left(encArg(x_1)) encArg(cons_right(x_1)) -> right(encArg(x_1)) encArg(cons_elem(x_1)) -> elem(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_listify(x_1, x_2)) -> listify(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_toList(x_1)) -> toList(encArg(x_1)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_empty -> empty encode_true -> true encode_node(x_1, x_2, x_3) -> node(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_left(x_1) -> left(encArg(x_1)) encode_right(x_1) -> right(encArg(x_1)) encode_elem(x_1) -> elem(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_y -> y encode_listify(x_1, x_2) -> listify(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_toList(x_1) -> toList(encArg(x_1)) Rewrite Strategy: INNERMOST