MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y) , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , length(nil()) -> 0() , length(cons(x, y)) -> s(length(y)) , if(true(), c, l, ys, zs) -> nil() , if(false(), c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs)) , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs) , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys) , helpc(true(), ys, zs) -> ys , helpc(false(), ys, zs) -> zs } 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 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: { app^#(x, y) -> c_1(helpa^#(0(), plus(length(x), length(y)), x, y)) , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs)) , if^#(true(), c, l, ys, zs) -> c_7() , if^#(false(), c, l, ys, zs) -> c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs))) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , length^#(nil()) -> c_5() , length^#(cons(x, y)) -> c_6(length^#(y)) , helpb^#(c, l, cons(y, ys), zs) -> c_12(y, helpa^#(s(c), l, ys, zs)) , ge^#(x, 0()) -> c_9() , ge^#(0(), s(x)) -> c_10() , ge^#(s(x), s(y)) -> c_11(ge^#(x, y)) , greater^#(ys, zs) -> c_13(helpc^#(ge(length(ys), length(zs)), ys, zs)) , helpc^#(true(), ys, zs) -> c_15(ys) , helpc^#(false(), ys, zs) -> c_16(zs) , smaller^#(ys, zs) -> c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(x, y) -> c_1(helpa^#(0(), plus(length(x), length(y)), x, y)) , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs)) , if^#(true(), c, l, ys, zs) -> c_7() , if^#(false(), c, l, ys, zs) -> c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs))) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , length^#(nil()) -> c_5() , length^#(cons(x, y)) -> c_6(length^#(y)) , helpb^#(c, l, cons(y, ys), zs) -> c_12(y, helpa^#(s(c), l, ys, zs)) , ge^#(x, 0()) -> c_9() , ge^#(0(), s(x)) -> c_10() , ge^#(s(x), s(y)) -> c_11(ge^#(x, y)) , greater^#(ys, zs) -> c_13(helpc^#(ge(length(ys), length(zs)), ys, zs)) , helpc^#(true(), ys, zs) -> c_15(ys) , helpc^#(false(), ys, zs) -> c_16(zs) , smaller^#(ys, zs) -> c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) } Strict Trs: { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y) , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , length(nil()) -> 0() , length(cons(x, y)) -> s(length(y)) , if(true(), c, l, ys, zs) -> nil() , if(false(), c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs)) , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs) , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys) , helpc(true(), ys, zs) -> ys , helpc(false(), ys, zs) -> zs } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,7,10,11} by applications of Pre({3,7,10,11}) = {2,5,8,9,12,14,15}. Here rules are labeled as follows: DPs: { 1: app^#(x, y) -> c_1(helpa^#(0(), plus(length(x), length(y)), x, y)) , 2: helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs)) , 3: if^#(true(), c, l, ys, zs) -> c_7() , 4: if^#(false(), c, l, ys, zs) -> c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs))) , 5: plus^#(x, 0()) -> c_3(x) , 6: plus^#(x, s(y)) -> c_4(plus^#(x, y)) , 7: length^#(nil()) -> c_5() , 8: length^#(cons(x, y)) -> c_6(length^#(y)) , 9: helpb^#(c, l, cons(y, ys), zs) -> c_12(y, helpa^#(s(c), l, ys, zs)) , 10: ge^#(x, 0()) -> c_9() , 11: ge^#(0(), s(x)) -> c_10() , 12: ge^#(s(x), s(y)) -> c_11(ge^#(x, y)) , 13: greater^#(ys, zs) -> c_13(helpc^#(ge(length(ys), length(zs)), ys, zs)) , 14: helpc^#(true(), ys, zs) -> c_15(ys) , 15: helpc^#(false(), ys, zs) -> c_16(zs) , 16: smaller^#(ys, zs) -> c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(x, y) -> c_1(helpa^#(0(), plus(length(x), length(y)), x, y)) , helpa^#(c, l, ys, zs) -> c_2(if^#(ge(c, l), c, l, ys, zs)) , if^#(false(), c, l, ys, zs) -> c_8(helpb^#(c, l, greater(ys, zs), smaller(ys, zs))) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , length^#(cons(x, y)) -> c_6(length^#(y)) , helpb^#(c, l, cons(y, ys), zs) -> c_12(y, helpa^#(s(c), l, ys, zs)) , ge^#(s(x), s(y)) -> c_11(ge^#(x, y)) , greater^#(ys, zs) -> c_13(helpc^#(ge(length(ys), length(zs)), ys, zs)) , helpc^#(true(), ys, zs) -> c_15(ys) , helpc^#(false(), ys, zs) -> c_16(zs) , smaller^#(ys, zs) -> c_14(helpc^#(ge(length(ys), length(zs)), zs, ys)) } Strict Trs: { app(x, y) -> helpa(0(), plus(length(x), length(y)), x, y) , helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , length(nil()) -> 0() , length(cons(x, y)) -> s(length(y)) , if(true(), c, l, ys, zs) -> nil() , if(false(), c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs)) , greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs) , smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys) , helpc(true(), ys, zs) -> ys , helpc(false(), ys, zs) -> zs } Weak DPs: { if^#(true(), c, l, ys, zs) -> c_7() , length^#(nil()) -> c_5() , ge^#(x, 0()) -> c_9() , ge^#(0(), s(x)) -> c_10() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..