MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { qsort(xs) -> qs(half(length(xs)), xs) , qs(n, nil()) -> nil() , qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , length(nil()) -> 0() , length(cons(x, xs)) -> s(length(xs)) , append(nil(), ys()) -> ys() , append(cons(x, xs), ys()) -> cons(x, append(xs, ys())) , filterlow(n, nil()) -> nil() , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) , get(n, nil()) -> 0() , get(n, cons(x, nil())) -> x , get(0(), cons(x, cons(y, xs))) -> x , get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) , filterhigh(n, nil()) -> nil() , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) , if1(true(), n, x, xs) -> filterlow(n, xs) , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , if2(true(), n, x, xs) -> filterhigh(n, xs) , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , qs^#(n, nil()) -> c_2() , qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , append^#(nil(), ys()) -> c_9() , append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , half^#(0()) -> c_4() , half^#(s(0())) -> c_5() , half^#(s(s(x))) -> c_6(half^#(x)) , length^#(nil()) -> c_7() , length^#(cons(x, xs)) -> c_8(length^#(xs)) , filterlow^#(n, nil()) -> c_11() , filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , get^#(n, nil()) -> c_13() , get^#(n, cons(x, nil())) -> c_14(x) , get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , filterhigh^#(n, nil()) -> c_17() , filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , ge^#(x, 0()) -> c_21() , ge^#(0(), s(x)) -> c_22() , ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , qs^#(n, nil()) -> c_2() , qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , append^#(nil(), ys()) -> c_9() , append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , half^#(0()) -> c_4() , half^#(s(0())) -> c_5() , half^#(s(s(x))) -> c_6(half^#(x)) , length^#(nil()) -> c_7() , length^#(cons(x, xs)) -> c_8(length^#(xs)) , filterlow^#(n, nil()) -> c_11() , filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , get^#(n, nil()) -> c_13() , get^#(n, cons(x, nil())) -> c_14(x) , get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , filterhigh^#(n, nil()) -> c_17() , filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , ge^#(x, 0()) -> c_21() , ge^#(0(), s(x)) -> c_22() , ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) } Strict Trs: { qsort(xs) -> qs(half(length(xs)), xs) , qs(n, nil()) -> nil() , qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , length(nil()) -> 0() , length(cons(x, xs)) -> s(length(xs)) , append(nil(), ys()) -> ys() , append(cons(x, xs), ys()) -> cons(x, append(xs, ys())) , filterlow(n, nil()) -> nil() , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) , get(n, nil()) -> 0() , get(n, cons(x, nil())) -> x , get(0(), cons(x, cons(y, xs))) -> x , get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) , filterhigh(n, nil()) -> nil() , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) , if1(true(), n, x, xs) -> filterlow(n, xs) , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , if2(true(), n, x, xs) -> filterhigh(n, xs) , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,3,4,6,7,9,11,15,19,23,24} by applications of Pre({2,3,4,6,7,9,11,15,19,23,24}) = {1,5,8,10,13,14,16,17,21,22,25}. Here rules are labeled as follows: DPs: { 1: qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , 2: qs^#(n, nil()) -> c_2() , 3: qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , 4: append^#(nil(), ys()) -> c_9() , 5: append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , 6: half^#(0()) -> c_4() , 7: half^#(s(0())) -> c_5() , 8: half^#(s(s(x))) -> c_6(half^#(x)) , 9: length^#(nil()) -> c_7() , 10: length^#(cons(x, xs)) -> c_8(length^#(xs)) , 11: filterlow^#(n, nil()) -> c_11() , 12: filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , 13: if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , 14: if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , 15: get^#(n, nil()) -> c_13() , 16: get^#(n, cons(x, nil())) -> c_14(x) , 17: get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , 18: get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , 19: filterhigh^#(n, nil()) -> c_17() , 20: filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , 21: if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , 22: if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , 23: ge^#(x, 0()) -> c_21() , 24: ge^#(0(), s(x)) -> c_22() , 25: ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , half^#(s(s(x))) -> c_6(half^#(x)) , length^#(cons(x, xs)) -> c_8(length^#(xs)) , filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , get^#(n, cons(x, nil())) -> c_14(x) , get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) } Strict Trs: { qsort(xs) -> qs(half(length(xs)), xs) , qs(n, nil()) -> nil() , qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , length(nil()) -> 0() , length(cons(x, xs)) -> s(length(xs)) , append(nil(), ys()) -> ys() , append(cons(x, xs), ys()) -> cons(x, append(xs, ys())) , filterlow(n, nil()) -> nil() , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) , get(n, nil()) -> 0() , get(n, cons(x, nil())) -> x , get(0(), cons(x, cons(y, xs))) -> x , get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) , filterhigh(n, nil()) -> nil() , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) , if1(true(), n, x, xs) -> filterlow(n, xs) , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , if2(true(), n, x, xs) -> filterhigh(n, xs) , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) } Weak DPs: { qs^#(n, nil()) -> c_2() , qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , append^#(nil(), ys()) -> c_9() , half^#(0()) -> c_4() , half^#(s(0())) -> c_5() , length^#(nil()) -> c_7() , filterlow^#(n, nil()) -> c_11() , get^#(n, nil()) -> c_13() , filterhigh^#(n, nil()) -> c_17() , ge^#(x, 0()) -> c_21() , ge^#(0(), s(x)) -> c_22() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1} by applications of Pre({1}) = {2,7,8,9,13}. Here rules are labeled as follows: DPs: { 1: qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , 2: append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , 3: half^#(s(s(x))) -> c_6(half^#(x)) , 4: length^#(cons(x, xs)) -> c_8(length^#(xs)) , 5: filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , 6: if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , 7: if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , 8: get^#(n, cons(x, nil())) -> c_14(x) , 9: get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , 10: get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , 11: filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , 12: if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , 13: if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , 14: ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) , 15: qs^#(n, nil()) -> c_2() , 16: qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , 17: append^#(nil(), ys()) -> c_9() , 18: half^#(0()) -> c_4() , 19: half^#(s(0())) -> c_5() , 20: length^#(nil()) -> c_7() , 21: filterlow^#(n, nil()) -> c_11() , 22: get^#(n, nil()) -> c_13() , 23: filterhigh^#(n, nil()) -> c_17() , 24: ge^#(x, 0()) -> c_21() , 25: ge^#(0(), s(x)) -> c_22() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { append^#(cons(x, xs), ys()) -> c_10(x, append^#(xs, ys())) , half^#(s(s(x))) -> c_6(half^#(x)) , length^#(cons(x, xs)) -> c_8(length^#(xs)) , filterlow^#(n, cons(x, xs)) -> c_12(if1^#(ge(n, x), n, x, xs)) , if1^#(true(), n, x, xs) -> c_19(filterlow^#(n, xs)) , if1^#(false(), n, x, xs) -> c_20(x, filterlow^#(n, xs)) , get^#(n, cons(x, nil())) -> c_14(x) , get^#(0(), cons(x, cons(y, xs))) -> c_15(x) , get^#(s(n), cons(x, cons(y, xs))) -> c_16(get^#(n, cons(y, xs))) , filterhigh^#(n, cons(x, xs)) -> c_18(if2^#(ge(x, n), n, x, xs)) , if2^#(true(), n, x, xs) -> c_24(filterhigh^#(n, xs)) , if2^#(false(), n, x, xs) -> c_25(x, filterhigh^#(n, xs)) , ge^#(s(x), s(y)) -> c_23(ge^#(x, y)) } Strict Trs: { qsort(xs) -> qs(half(length(xs)), xs) , qs(n, nil()) -> nil() , qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , length(nil()) -> 0() , length(cons(x, xs)) -> s(length(xs)) , append(nil(), ys()) -> ys() , append(cons(x, xs), ys()) -> cons(x, append(xs, ys())) , filterlow(n, nil()) -> nil() , filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) , get(n, nil()) -> 0() , get(n, cons(x, nil())) -> x , get(0(), cons(x, cons(y, xs))) -> x , get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) , filterhigh(n, nil()) -> nil() , filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) , if1(true(), n, x, xs) -> filterlow(n, xs) , if1(false(), n, x, xs) -> cons(x, filterlow(n, xs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , if2(true(), n, x, xs) -> filterhigh(n, xs) , if2(false(), n, x, xs) -> cons(x, filterhigh(n, xs)) } Weak DPs: { qsort^#(xs) -> c_1(qs^#(half(length(xs)), xs)) , qs^#(n, nil()) -> c_2() , qs^#(n, cons(x, xs)) -> c_3(append^#(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))) , append^#(nil(), ys()) -> c_9() , half^#(0()) -> c_4() , half^#(s(0())) -> c_5() , length^#(nil()) -> c_7() , filterlow^#(n, nil()) -> c_11() , get^#(n, nil()) -> c_13() , filterhigh^#(n, nil()) -> c_17() , ge^#(x, 0()) -> c_21() , ge^#(0(), s(x)) -> c_22() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..