MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a(l, x, s(y), h()) -> a(l, x, y, s(h())) , a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z)) , a(l, s(x), h(), z) -> a(l, x, z, z) , a(h(), h(), h(), x) -> s(x) , a(s(l), h(), h(), z) -> a(l, z, h(), z) , s(h()) -> 1() , +(x, h()) -> x , +(h(), x) -> x , +(s(x), s(y)) -> s(s(+(x, y))) , +(+(x, y), z) -> +(x, +(y, z)) , app(l, nil()) -> l , app(nil(), k) -> k , app(cons(x, l), k) -> cons(x, app(l, k)) , sum(cons(x, nil())) -> cons(x, nil()) , sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h(), h()), l)) } 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) '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) '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: { a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , a^#(h(), h(), h(), x) -> c_4(s^#(x)) , a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , s^#(h()) -> c_6() , +^#(x, h()) -> c_7(x) , +^#(h(), x) -> c_8(x) , +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) , +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , app^#(l, nil()) -> c_11(l) , app^#(nil(), k) -> c_12(k) , app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , sum^#(cons(x, nil())) -> c_14(x) , sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , a^#(h(), h(), h(), x) -> c_4(s^#(x)) , a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , s^#(h()) -> c_6() , +^#(x, h()) -> c_7(x) , +^#(h(), x) -> c_8(x) , +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) , +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , app^#(l, nil()) -> c_11(l) , app^#(nil(), k) -> c_12(k) , app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , sum^#(cons(x, nil())) -> c_14(x) , sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) } Strict Trs: { a(l, x, s(y), h()) -> a(l, x, y, s(h())) , a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z)) , a(l, s(x), h(), z) -> a(l, x, z, z) , a(h(), h(), h(), x) -> s(x) , a(s(l), h(), h(), z) -> a(l, z, h(), z) , s(h()) -> 1() , +(x, h()) -> x , +(h(), x) -> x , +(s(x), s(y)) -> s(s(+(x, y))) , +(+(x, y), z) -> +(x, +(y, z)) , app(l, nil()) -> l , app(nil(), k) -> k , app(cons(x, l), k) -> cons(x, app(l, k)) , sum(cons(x, nil())) -> cons(x, nil()) , sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h(), h()), l)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {6} by applications of Pre({6}) = {4,7,8,9,11,12,13,14}. Here rules are labeled as follows: DPs: { 1: a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , 2: a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , 3: a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , 4: a^#(h(), h(), h(), x) -> c_4(s^#(x)) , 5: a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , 6: s^#(h()) -> c_6() , 7: +^#(x, h()) -> c_7(x) , 8: +^#(h(), x) -> c_8(x) , 9: +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) , 10: +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , 11: app^#(l, nil()) -> c_11(l) , 12: app^#(nil(), k) -> c_12(k) , 13: app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , 14: sum^#(cons(x, nil())) -> c_14(x) , 15: sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , a^#(h(), h(), h(), x) -> c_4(s^#(x)) , a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , +^#(x, h()) -> c_7(x) , +^#(h(), x) -> c_8(x) , +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) , +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , app^#(l, nil()) -> c_11(l) , app^#(nil(), k) -> c_12(k) , app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , sum^#(cons(x, nil())) -> c_14(x) , sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) } Strict Trs: { a(l, x, s(y), h()) -> a(l, x, y, s(h())) , a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z)) , a(l, s(x), h(), z) -> a(l, x, z, z) , a(h(), h(), h(), x) -> s(x) , a(s(l), h(), h(), z) -> a(l, z, h(), z) , s(h()) -> 1() , +(x, h()) -> x , +(h(), x) -> x , +(s(x), s(y)) -> s(s(+(x, y))) , +(+(x, y), z) -> +(x, +(y, z)) , app(l, nil()) -> l , app(nil(), k) -> k , app(cons(x, l), k) -> cons(x, app(l, k)) , sum(cons(x, nil())) -> cons(x, nil()) , sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h(), h()), l)) } Weak DPs: { s^#(h()) -> c_6() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,8} by applications of Pre({4,8}) = {1,2,3,5,6,7,9,10,11,12,13}. Here rules are labeled as follows: DPs: { 1: a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , 2: a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , 3: a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , 4: a^#(h(), h(), h(), x) -> c_4(s^#(x)) , 5: a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , 6: +^#(x, h()) -> c_7(x) , 7: +^#(h(), x) -> c_8(x) , 8: +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) , 9: +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , 10: app^#(l, nil()) -> c_11(l) , 11: app^#(nil(), k) -> c_12(k) , 12: app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , 13: sum^#(cons(x, nil())) -> c_14(x) , 14: sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) , 15: s^#(h()) -> c_6() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a^#(l, x, s(y), h()) -> c_1(a^#(l, x, y, s(h()))) , a^#(l, x, s(y), s(z)) -> c_2(a^#(l, x, y, a(l, x, s(y), z))) , a^#(l, s(x), h(), z) -> c_3(a^#(l, x, z, z)) , a^#(s(l), h(), h(), z) -> c_5(a^#(l, z, h(), z)) , +^#(x, h()) -> c_7(x) , +^#(h(), x) -> c_8(x) , +^#(+(x, y), z) -> c_10(+^#(x, +(y, z))) , app^#(l, nil()) -> c_11(l) , app^#(nil(), k) -> c_12(k) , app^#(cons(x, l), k) -> c_13(x, app^#(l, k)) , sum^#(cons(x, nil())) -> c_14(x) , sum^#(cons(x, cons(y, l))) -> c_15(sum^#(cons(a(x, y, h(), h()), l))) } Strict Trs: { a(l, x, s(y), h()) -> a(l, x, y, s(h())) , a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z)) , a(l, s(x), h(), z) -> a(l, x, z, z) , a(h(), h(), h(), x) -> s(x) , a(s(l), h(), h(), z) -> a(l, z, h(), z) , s(h()) -> 1() , +(x, h()) -> x , +(h(), x) -> x , +(s(x), s(y)) -> s(s(+(x, y))) , +(+(x, y), z) -> +(x, +(y, z)) , app(l, nil()) -> l , app(nil(), k) -> k , app(cons(x, l), k) -> cons(x, app(l, k)) , sum(cons(x, nil())) -> cons(x, nil()) , sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h(), h()), l)) } Weak DPs: { a^#(h(), h(), h(), x) -> c_4(s^#(x)) , s^#(h()) -> c_6() , +^#(s(x), s(y)) -> c_9(s^#(s(+(x, y)))) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..