/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 330 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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) 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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) 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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) 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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) 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 split(N, add(M, Y)) ->^+ f_1(split(N, Y), N, M, Y) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [Y / add(M, Y)]. 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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) 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: lt(0, s(X)) -> true lt(s(X), 0) -> false lt(s(X), s(Y)) -> lt(X, Y) append(nil, Y) -> Y append(add(N, X), Y) -> add(N, append(X, Y)) split(N, nil) -> pair(nil, nil) split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y) f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z) f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z)) f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z) qsort(nil) -> nil qsort(add(N, X)) -> f_3(split(N, X), N, X) f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_split(x_1, x_2)) -> split(encArg(x_1), encArg(x_2)) encArg(cons_f_1(x_1, x_2, x_3, x_4)) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_f_2(x_1, x_2, x_3, x_4, x_5, x_6)) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_f_3(x_1, x_2, x_3)) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_false -> false encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_split(x_1, x_2) -> split(encArg(x_1), encArg(x_2)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_f_1(x_1, x_2, x_3, x_4) -> f_1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_f_2(x_1, x_2, x_3, x_4, x_5, x_6) -> f_2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_f_3(x_1, x_2, x_3) -> f_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST