MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , ++(nil(), ys) -> ys , ++(:(x, xs), ys) -> :(x, ++(xs, ys)) , sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys))))) , sum(:(x, nil())) -> :(x, nil()) , sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs)) , -(x, 0()) -> x , -(0(), s(y)) -> 0() , -(s(x), s(y)) -> -(x, y) , quot(0(), s(y)) -> 0() , quot(s(x), s(y)) -> s(quot(-(x, y), s(y))) , length(nil()) -> 0() , length(:(x, xs)) -> s(length(xs)) , hd(:(x, xs)) -> x , avg(xs) -> quot(hd(sum(xs)), length(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) '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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { +^#(0(), y) -> c_1(y) , +^#(s(x), y) -> c_2(+^#(x, y)) , ++^#(nil(), ys) -> c_3(ys) , ++^#(:(x, xs), ys) -> c_4(x, ++^#(xs, ys)) , sum^#(++(xs, :(x, :(y, ys)))) -> c_5(sum^#(++(xs, sum(:(x, :(y, ys)))))) , sum^#(:(x, nil())) -> c_6(x) , sum^#(:(x, :(y, xs))) -> c_7(sum^#(:(+(x, y), xs))) , -^#(x, 0()) -> c_8(x) , -^#(0(), s(y)) -> c_9() , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , quot^#(0(), s(y)) -> c_11() , quot^#(s(x), s(y)) -> c_12(quot^#(-(x, y), s(y))) , length^#(nil()) -> c_13() , length^#(:(x, xs)) -> c_14(length^#(xs)) , hd^#(:(x, xs)) -> c_15(x) , avg^#(xs) -> c_16(quot^#(hd(sum(xs)), length(xs))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(0(), y) -> c_1(y) , +^#(s(x), y) -> c_2(+^#(x, y)) , ++^#(nil(), ys) -> c_3(ys) , ++^#(:(x, xs), ys) -> c_4(x, ++^#(xs, ys)) , sum^#(++(xs, :(x, :(y, ys)))) -> c_5(sum^#(++(xs, sum(:(x, :(y, ys)))))) , sum^#(:(x, nil())) -> c_6(x) , sum^#(:(x, :(y, xs))) -> c_7(sum^#(:(+(x, y), xs))) , -^#(x, 0()) -> c_8(x) , -^#(0(), s(y)) -> c_9() , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , quot^#(0(), s(y)) -> c_11() , quot^#(s(x), s(y)) -> c_12(quot^#(-(x, y), s(y))) , length^#(nil()) -> c_13() , length^#(:(x, xs)) -> c_14(length^#(xs)) , hd^#(:(x, xs)) -> c_15(x) , avg^#(xs) -> c_16(quot^#(hd(sum(xs)), length(xs))) } Strict Trs: { +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , ++(nil(), ys) -> ys , ++(:(x, xs), ys) -> :(x, ++(xs, ys)) , sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys))))) , sum(:(x, nil())) -> :(x, nil()) , sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs)) , -(x, 0()) -> x , -(0(), s(y)) -> 0() , -(s(x), s(y)) -> -(x, y) , quot(0(), s(y)) -> 0() , quot(s(x), s(y)) -> s(quot(-(x, y), s(y))) , length(nil()) -> 0() , length(:(x, xs)) -> s(length(xs)) , hd(:(x, xs)) -> x , avg(xs) -> quot(hd(sum(xs)), length(xs)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {9,11,13} by applications of Pre({9,11,13}) = {1,3,4,6,8,10,12,14,15,16}. Here rules are labeled as follows: DPs: { 1: +^#(0(), y) -> c_1(y) , 2: +^#(s(x), y) -> c_2(+^#(x, y)) , 3: ++^#(nil(), ys) -> c_3(ys) , 4: ++^#(:(x, xs), ys) -> c_4(x, ++^#(xs, ys)) , 5: sum^#(++(xs, :(x, :(y, ys)))) -> c_5(sum^#(++(xs, sum(:(x, :(y, ys)))))) , 6: sum^#(:(x, nil())) -> c_6(x) , 7: sum^#(:(x, :(y, xs))) -> c_7(sum^#(:(+(x, y), xs))) , 8: -^#(x, 0()) -> c_8(x) , 9: -^#(0(), s(y)) -> c_9() , 10: -^#(s(x), s(y)) -> c_10(-^#(x, y)) , 11: quot^#(0(), s(y)) -> c_11() , 12: quot^#(s(x), s(y)) -> c_12(quot^#(-(x, y), s(y))) , 13: length^#(nil()) -> c_13() , 14: length^#(:(x, xs)) -> c_14(length^#(xs)) , 15: hd^#(:(x, xs)) -> c_15(x) , 16: avg^#(xs) -> c_16(quot^#(hd(sum(xs)), length(xs))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(0(), y) -> c_1(y) , +^#(s(x), y) -> c_2(+^#(x, y)) , ++^#(nil(), ys) -> c_3(ys) , ++^#(:(x, xs), ys) -> c_4(x, ++^#(xs, ys)) , sum^#(++(xs, :(x, :(y, ys)))) -> c_5(sum^#(++(xs, sum(:(x, :(y, ys)))))) , sum^#(:(x, nil())) -> c_6(x) , sum^#(:(x, :(y, xs))) -> c_7(sum^#(:(+(x, y), xs))) , -^#(x, 0()) -> c_8(x) , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , quot^#(s(x), s(y)) -> c_12(quot^#(-(x, y), s(y))) , length^#(:(x, xs)) -> c_14(length^#(xs)) , hd^#(:(x, xs)) -> c_15(x) , avg^#(xs) -> c_16(quot^#(hd(sum(xs)), length(xs))) } Strict Trs: { +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , ++(nil(), ys) -> ys , ++(:(x, xs), ys) -> :(x, ++(xs, ys)) , sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys))))) , sum(:(x, nil())) -> :(x, nil()) , sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs)) , -(x, 0()) -> x , -(0(), s(y)) -> 0() , -(s(x), s(y)) -> -(x, y) , quot(0(), s(y)) -> 0() , quot(s(x), s(y)) -> s(quot(-(x, y), s(y))) , length(nil()) -> 0() , length(:(x, xs)) -> s(length(xs)) , hd(:(x, xs)) -> x , avg(xs) -> quot(hd(sum(xs)), length(xs)) } Weak DPs: { -^#(0(), s(y)) -> c_9() , quot^#(0(), s(y)) -> c_11() , length^#(nil()) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..