/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 297 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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes ---------------------------------------- (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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes 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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes 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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes 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 mark(nats(X)) ->^+ a__nats(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / nats(X)]. 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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes 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: a__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(nats(x_1)) -> nats(encArg(x_1)) encArg(zprimes) -> zprimes encArg(cons_a__filter(x_1, x_2, x_3)) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_a__nats(x_1)) -> a__nats(encArg(x_1)) encArg(cons_a__zprimes) -> a__zprimes encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__filter(x_1, x_2, x_3) -> a__filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_a__nats(x_1) -> a__nats(encArg(x_1)) encode_nats(x_1) -> nats(encArg(x_1)) encode_a__zprimes -> a__zprimes encode_zprimes -> zprimes Rewrite Strategy: INNERMOST