/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 492 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 2 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c 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(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c ---------------------------------------- (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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c 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 le(s(x), s(y)) ->^+ le(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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c 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: plus(x, y) -> plusIter(x, y, 0) plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) ifPlus(true, x, y, z) -> y ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) sum(xs) -> sumIter(xs, 0) sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) ifSum(true, xs, x, y) -> x ifSum(false, xs, x, y) -> sumIter(tail(xs), y) isempty(nil) -> true isempty(cons(x, xs)) -> false head(nil) -> error head(cons(x, xs)) -> x tail(nil) -> nil tail(cons(x, xs)) -> xs a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(false) -> false 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(error) -> error encArg(b) -> b encArg(c) -> c encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_plusIter(x_1, x_2, x_3)) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_ifPlus(x_1, x_2, x_3, x_4)) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_sum(x_1)) -> sum(encArg(x_1)) encArg(cons_sumIter(x_1, x_2)) -> sumIter(encArg(x_1), encArg(x_2)) encArg(cons_ifSum(x_1, x_2, x_3, x_4)) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_isempty(x_1)) -> isempty(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_a) -> a encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_plusIter(x_1, x_2, x_3) -> plusIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_ifPlus(x_1, x_2, x_3, x_4) -> ifPlus(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_sum(x_1) -> sum(encArg(x_1)) encode_sumIter(x_1, x_2) -> sumIter(encArg(x_1), encArg(x_2)) encode_ifSum(x_1, x_2, x_3, x_4) -> ifSum(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_isempty(x_1) -> isempty(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_error -> error encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST