/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 476 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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) 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(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) 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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) 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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) 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 eq(s(X), s(Y)) ->^+ eq(X, Y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [X / s(X), Y / s(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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) 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: eq(0, 0) -> true eq(0, s(Y)) -> false eq(s(X), 0) -> false eq(s(X), s(Y)) -> eq(X, Y) le(0, Y) -> true le(s(X), 0) -> false le(s(X), s(Y)) -> le(X, Y) min(cons(0, nil)) -> 0 min(cons(s(N), nil)) -> s(N) min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L))) ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L)) ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L)) replace(N, M, nil) -> nil replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L)) ifrepl(true, N, M, cons(K, L)) -> cons(M, L) ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L)) selsort(nil) -> nil selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L)) ifselsort(true, cons(N, L)) -> cons(N, selsort(L)) ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_ifmin(x_1, x_2)) -> ifmin(encArg(x_1), encArg(x_2)) encArg(cons_replace(x_1, x_2, x_3)) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifrepl(x_1, x_2, x_3, x_4)) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_selsort(x_1)) -> selsort(encArg(x_1)) encArg(cons_ifselsort(x_1, x_2)) -> ifselsort(encArg(x_1), encArg(x_2)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_ifmin(x_1, x_2) -> ifmin(encArg(x_1), encArg(x_2)) encode_replace(x_1, x_2, x_3) -> replace(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ifrepl(x_1, x_2, x_3, x_4) -> ifrepl(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_selsort(x_1) -> selsort(encArg(x_1)) encode_ifselsort(x_1, x_2) -> ifselsort(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST