/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 482 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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(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 length(cons(x, y)) ->^+ s(length(y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [y / cons(x, 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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(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: ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x)) if(x, true, z, c, l) -> z if(x, false, z, c, l) -> help(s(c), l, x, z) help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l) append(nil, y) -> y append(cons(x, y), z) -> cons(x, append(y, z)) length(nil) -> 0 length(cons(x, y)) -> s(length(y)) 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(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_rev(x_1)) -> rev(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_help(x_1, x_2, x_3, x_4)) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_rev(x_1) -> rev(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_nil -> nil encode_help(x_1, x_2, x_3, x_4) -> help(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST