/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 245 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 8 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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) 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(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) ---------------------------------------- (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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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 app(cons(x, l1), l2) ->^+ cons(x, app(l1, l2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [l1 / cons(x, l1)]. 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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: if(true, x, y) -> x if(false, x, y) -> y eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) app(nil, l) -> l app(cons(x, l1), l2) -> cons(x, app(l1, l2)) app(app(l1, l2), l3) -> app(l1, app(l2, l3)) mem(x, nil) -> false mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l) ifmem(true, x, l) -> true ifmem(false, x, l) -> mem(x, l) inter(x, nil) -> nil inter(nil, x) -> nil inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)) inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)) inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2) inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1) ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2)) ifinter(false, x, l1, l2) -> inter(l1, l2) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_mem(x_1, x_2)) -> mem(encArg(x_1), encArg(x_2)) encArg(cons_ifmem(x_1, x_2, x_3)) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_inter(x_1, x_2)) -> inter(encArg(x_1), encArg(x_2)) encArg(cons_ifinter(x_1, x_2, x_3, x_4)) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mem(x_1, x_2) -> mem(encArg(x_1), encArg(x_2)) encode_ifmem(x_1, x_2, x_3) -> ifmem(encArg(x_1), encArg(x_2), encArg(x_3)) encode_inter(x_1, x_2) -> inter(encArg(x_1), encArg(x_2)) encode_ifinter(x_1, x_2, x_3, x_4) -> ifinter(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST