MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { 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)))) } 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) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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 2) '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 3) '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 perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(0(), s(y)) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , low^#(n, nil()) -> c_10() , low^#(n, add(m, x)) -> c_11(if_low^#(le(m, n), n, add(m, x))) , if_low^#(true(), n, add(m, x)) -> c_12(m, low^#(n, x)) , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x)) , high^#(n, nil()) -> c_14() , high^#(n, add(m, x)) -> c_15(if_high^#(le(m, n), n, add(m, x))) , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x)) , if_high^#(false(), n, add(m, x)) -> c_17(m, high^#(n, x)) , quicksort^#(nil()) -> c_18() , quicksort^#(add(n, x)) -> c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x))))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(0(), s(y)) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , low^#(n, nil()) -> c_10() , low^#(n, add(m, x)) -> c_11(if_low^#(le(m, n), n, add(m, x))) , if_low^#(true(), n, add(m, x)) -> c_12(m, low^#(n, x)) , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x)) , high^#(n, nil()) -> c_14() , high^#(n, add(m, x)) -> c_15(if_high^#(le(m, n), n, add(m, x))) , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x)) , if_high^#(false(), n, add(m, x)) -> c_17(m, high^#(n, x)) , quicksort^#(nil()) -> c_18() , quicksort^#(add(n, x)) -> c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x))))) } Strict Trs: { 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)))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,5,6,10,14,18} by applications of Pre({3,5,6,10,14,18}) = {1,4,7,8,9,12,13,16,17}. Here rules are labeled as follows: DPs: { 1: minus^#(x, 0()) -> c_1(x) , 2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , 3: quot^#(0(), s(y)) -> c_3() , 4: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , 5: le^#(0(), y) -> c_5() , 6: le^#(s(x), 0()) -> c_6() , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y)) , 8: app^#(nil(), y) -> c_8(y) , 9: app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , 10: low^#(n, nil()) -> c_10() , 11: low^#(n, add(m, x)) -> c_11(if_low^#(le(m, n), n, add(m, x))) , 12: if_low^#(true(), n, add(m, x)) -> c_12(m, low^#(n, x)) , 13: if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x)) , 14: high^#(n, nil()) -> c_14() , 15: high^#(n, add(m, x)) -> c_15(if_high^#(le(m, n), n, add(m, x))) , 16: if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x)) , 17: if_high^#(false(), n, add(m, x)) -> c_17(m, high^#(n, x)) , 18: quicksort^#(nil()) -> c_18() , 19: quicksort^#(add(n, x)) -> c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x))))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , low^#(n, add(m, x)) -> c_11(if_low^#(le(m, n), n, add(m, x))) , if_low^#(true(), n, add(m, x)) -> c_12(m, low^#(n, x)) , if_low^#(false(), n, add(m, x)) -> c_13(low^#(n, x)) , high^#(n, add(m, x)) -> c_15(if_high^#(le(m, n), n, add(m, x))) , if_high^#(true(), n, add(m, x)) -> c_16(high^#(n, x)) , if_high^#(false(), n, add(m, x)) -> c_17(m, high^#(n, x)) , quicksort^#(add(n, x)) -> c_19(app^#(quicksort(low(n, x)), add(n, quicksort(high(n, x))))) } Strict Trs: { 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)))) } Weak DPs: { quot^#(0(), s(y)) -> c_3() , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , low^#(n, nil()) -> c_10() , high^#(n, nil()) -> c_14() , quicksort^#(nil()) -> c_18() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..