/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 399 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) 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(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) ---------------------------------------- (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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) app(nil, y) -> y app(add(n, x), y) -> add(n, app(x, y)) low(n, nil) -> nil low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) if_low(true, n, add(m, x)) -> add(m, low(n, x)) if_low(false, n, add(m, x)) -> low(n, x) high(n, nil) -> nil high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) if_high(true, n, add(m, x)) -> high(n, x) if_high(false, n, add(m, x)) -> add(m, high(n, x)) quicksort(nil) -> nil quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(nil) -> nil encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2)) encArg(cons_if_low(x_1, x_2, x_3)) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2)) encArg(cons_if_high(x_1, x_2, x_3)) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2)) encode_if_low(x_1, x_2, x_3) -> if_low(encArg(x_1), encArg(x_2), encArg(x_3)) encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2)) encode_if_high(x_1, x_2, x_3) -> if_high(encArg(x_1), encArg(x_2), encArg(x_3)) encode_quicksort(x_1) -> quicksort(encArg(x_1)) Rewrite Strategy: INNERMOST