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