MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) , sum(cons(0(), x), y) -> sum(x, y) , sum(nil(), y) -> y , empty(cons(n, x)) -> false() , empty(nil()) -> true() , tail(cons(n, x)) -> x , tail(nil()) -> nil() , head(cons(n, x)) -> n , weight(x) -> if(empty(x), empty(tail(x)), x) , if(true(), b, x) -> weight_undefined_error() , if(false(), b, x) -> if2(b, x) , if2(true(), x) -> head(x) , if2(false(), x) -> weight(sum(x, cons(0(), tail(tail(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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { sum^#(cons(s(n), x), cons(m, y)) -> c_1(sum^#(cons(n, x), cons(s(m), y))) , sum^#(cons(0(), x), y) -> c_2(sum^#(x, y)) , sum^#(nil(), y) -> c_3(y) , empty^#(cons(n, x)) -> c_4() , empty^#(nil()) -> c_5() , tail^#(cons(n, x)) -> c_6(x) , tail^#(nil()) -> c_7() , head^#(cons(n, x)) -> c_8(n) , weight^#(x) -> c_9(if^#(empty(x), empty(tail(x)), x)) , if^#(true(), b, x) -> c_10() , if^#(false(), b, x) -> c_11(if2^#(b, x)) , if2^#(true(), x) -> c_12(head^#(x)) , if2^#(false(), x) -> c_13(weight^#(sum(x, cons(0(), tail(tail(x)))))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sum^#(cons(s(n), x), cons(m, y)) -> c_1(sum^#(cons(n, x), cons(s(m), y))) , sum^#(cons(0(), x), y) -> c_2(sum^#(x, y)) , sum^#(nil(), y) -> c_3(y) , empty^#(cons(n, x)) -> c_4() , empty^#(nil()) -> c_5() , tail^#(cons(n, x)) -> c_6(x) , tail^#(nil()) -> c_7() , head^#(cons(n, x)) -> c_8(n) , weight^#(x) -> c_9(if^#(empty(x), empty(tail(x)), x)) , if^#(true(), b, x) -> c_10() , if^#(false(), b, x) -> c_11(if2^#(b, x)) , if2^#(true(), x) -> c_12(head^#(x)) , if2^#(false(), x) -> c_13(weight^#(sum(x, cons(0(), tail(tail(x)))))) } Strict Trs: { sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) , sum(cons(0(), x), y) -> sum(x, y) , sum(nil(), y) -> y , empty(cons(n, x)) -> false() , empty(nil()) -> true() , tail(cons(n, x)) -> x , tail(nil()) -> nil() , head(cons(n, x)) -> n , weight(x) -> if(empty(x), empty(tail(x)), x) , if(true(), b, x) -> weight_undefined_error() , if(false(), b, x) -> if2(b, x) , if2(true(), x) -> head(x) , if2(false(), x) -> weight(sum(x, cons(0(), tail(tail(x))))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,5,7,10} by applications of Pre({4,5,7,10}) = {3,6,8,9}. Here rules are labeled as follows: DPs: { 1: sum^#(cons(s(n), x), cons(m, y)) -> c_1(sum^#(cons(n, x), cons(s(m), y))) , 2: sum^#(cons(0(), x), y) -> c_2(sum^#(x, y)) , 3: sum^#(nil(), y) -> c_3(y) , 4: empty^#(cons(n, x)) -> c_4() , 5: empty^#(nil()) -> c_5() , 6: tail^#(cons(n, x)) -> c_6(x) , 7: tail^#(nil()) -> c_7() , 8: head^#(cons(n, x)) -> c_8(n) , 9: weight^#(x) -> c_9(if^#(empty(x), empty(tail(x)), x)) , 10: if^#(true(), b, x) -> c_10() , 11: if^#(false(), b, x) -> c_11(if2^#(b, x)) , 12: if2^#(true(), x) -> c_12(head^#(x)) , 13: if2^#(false(), x) -> c_13(weight^#(sum(x, cons(0(), tail(tail(x)))))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sum^#(cons(s(n), x), cons(m, y)) -> c_1(sum^#(cons(n, x), cons(s(m), y))) , sum^#(cons(0(), x), y) -> c_2(sum^#(x, y)) , sum^#(nil(), y) -> c_3(y) , tail^#(cons(n, x)) -> c_6(x) , head^#(cons(n, x)) -> c_8(n) , weight^#(x) -> c_9(if^#(empty(x), empty(tail(x)), x)) , if^#(false(), b, x) -> c_11(if2^#(b, x)) , if2^#(true(), x) -> c_12(head^#(x)) , if2^#(false(), x) -> c_13(weight^#(sum(x, cons(0(), tail(tail(x)))))) } Strict Trs: { sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y)) , sum(cons(0(), x), y) -> sum(x, y) , sum(nil(), y) -> y , empty(cons(n, x)) -> false() , empty(nil()) -> true() , tail(cons(n, x)) -> x , tail(nil()) -> nil() , head(cons(n, x)) -> n , weight(x) -> if(empty(x), empty(tail(x)), x) , if(true(), b, x) -> weight_undefined_error() , if(false(), b, x) -> if2(b, x) , if2(true(), x) -> head(x) , if2(false(), x) -> weight(sum(x, cons(0(), tail(tail(x))))) } Weak DPs: { empty^#(cons(n, x)) -> c_4() , empty^#(nil()) -> c_5() , tail^#(nil()) -> c_7() , if^#(true(), b, x) -> c_10() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..