MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { empty(nil()) -> true() , empty(cons(x, y)) -> false() , tail(nil()) -> nil() , tail(cons(x, y)) -> y , head(cons(x, y)) -> x , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(0())) -> 0() , p(s(s(x))) -> s(p(s(x))) , intlist(x) -> if_intlist(empty(x), x) , if_intlist(true(), x) -> nil() , if_intlist(false(), x) -> cons(s(head(x)), intlist(tail(x))) , int(x, y) -> if_int(zero(x), zero(y), x, y) , if_int(true(), b, x, y) -> if1(b, x, y) , if_int(false(), b, x, y) -> if2(b, x, y) , if1(true(), x, y) -> cons(0(), nil()) , if1(false(), x, y) -> cons(0(), int(s(0()), y)) , if2(true(), x, y) -> nil() , if2(false(), x, y) -> intlist(int(p(x), p(y))) } 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: { empty^#(nil()) -> c_1() , empty^#(cons(x, y)) -> c_2() , tail^#(nil()) -> c_3() , tail^#(cons(x, y)) -> c_4(y) , head^#(cons(x, y)) -> c_5(x) , zero^#(0()) -> c_6() , zero^#(s(x)) -> c_7() , p^#(0()) -> c_8() , p^#(s(0())) -> c_9() , p^#(s(s(x))) -> c_10(p^#(s(x))) , intlist^#(x) -> c_11(if_intlist^#(empty(x), x)) , if_intlist^#(true(), x) -> c_12() , if_intlist^#(false(), x) -> c_13(head^#(x), intlist^#(tail(x))) , int^#(x, y) -> c_14(if_int^#(zero(x), zero(y), x, y)) , if_int^#(true(), b, x, y) -> c_15(if1^#(b, x, y)) , if_int^#(false(), b, x, y) -> c_16(if2^#(b, x, y)) , if1^#(true(), x, y) -> c_17() , if1^#(false(), x, y) -> c_18(int^#(s(0()), y)) , if2^#(true(), x, y) -> c_19() , if2^#(false(), x, y) -> c_20(intlist^#(int(p(x), p(y)))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { empty^#(nil()) -> c_1() , empty^#(cons(x, y)) -> c_2() , tail^#(nil()) -> c_3() , tail^#(cons(x, y)) -> c_4(y) , head^#(cons(x, y)) -> c_5(x) , zero^#(0()) -> c_6() , zero^#(s(x)) -> c_7() , p^#(0()) -> c_8() , p^#(s(0())) -> c_9() , p^#(s(s(x))) -> c_10(p^#(s(x))) , intlist^#(x) -> c_11(if_intlist^#(empty(x), x)) , if_intlist^#(true(), x) -> c_12() , if_intlist^#(false(), x) -> c_13(head^#(x), intlist^#(tail(x))) , int^#(x, y) -> c_14(if_int^#(zero(x), zero(y), x, y)) , if_int^#(true(), b, x, y) -> c_15(if1^#(b, x, y)) , if_int^#(false(), b, x, y) -> c_16(if2^#(b, x, y)) , if1^#(true(), x, y) -> c_17() , if1^#(false(), x, y) -> c_18(int^#(s(0()), y)) , if2^#(true(), x, y) -> c_19() , if2^#(false(), x, y) -> c_20(intlist^#(int(p(x), p(y)))) } Strict Trs: { empty(nil()) -> true() , empty(cons(x, y)) -> false() , tail(nil()) -> nil() , tail(cons(x, y)) -> y , head(cons(x, y)) -> x , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(0())) -> 0() , p(s(s(x))) -> s(p(s(x))) , intlist(x) -> if_intlist(empty(x), x) , if_intlist(true(), x) -> nil() , if_intlist(false(), x) -> cons(s(head(x)), intlist(tail(x))) , int(x, y) -> if_int(zero(x), zero(y), x, y) , if_int(true(), b, x, y) -> if1(b, x, y) , if_int(false(), b, x, y) -> if2(b, x, y) , if1(true(), x, y) -> cons(0(), nil()) , if1(false(), x, y) -> cons(0(), int(s(0()), y)) , if2(true(), x, y) -> nil() , if2(false(), x, y) -> intlist(int(p(x), p(y))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,6,7,8,9,12,17,19} by applications of Pre({1,2,3,6,7,8,9,12,17,19}) = {4,5,10,11,15,16}. Here rules are labeled as follows: DPs: { 1: empty^#(nil()) -> c_1() , 2: empty^#(cons(x, y)) -> c_2() , 3: tail^#(nil()) -> c_3() , 4: tail^#(cons(x, y)) -> c_4(y) , 5: head^#(cons(x, y)) -> c_5(x) , 6: zero^#(0()) -> c_6() , 7: zero^#(s(x)) -> c_7() , 8: p^#(0()) -> c_8() , 9: p^#(s(0())) -> c_9() , 10: p^#(s(s(x))) -> c_10(p^#(s(x))) , 11: intlist^#(x) -> c_11(if_intlist^#(empty(x), x)) , 12: if_intlist^#(true(), x) -> c_12() , 13: if_intlist^#(false(), x) -> c_13(head^#(x), intlist^#(tail(x))) , 14: int^#(x, y) -> c_14(if_int^#(zero(x), zero(y), x, y)) , 15: if_int^#(true(), b, x, y) -> c_15(if1^#(b, x, y)) , 16: if_int^#(false(), b, x, y) -> c_16(if2^#(b, x, y)) , 17: if1^#(true(), x, y) -> c_17() , 18: if1^#(false(), x, y) -> c_18(int^#(s(0()), y)) , 19: if2^#(true(), x, y) -> c_19() , 20: if2^#(false(), x, y) -> c_20(intlist^#(int(p(x), p(y)))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { tail^#(cons(x, y)) -> c_4(y) , head^#(cons(x, y)) -> c_5(x) , p^#(s(s(x))) -> c_10(p^#(s(x))) , intlist^#(x) -> c_11(if_intlist^#(empty(x), x)) , if_intlist^#(false(), x) -> c_13(head^#(x), intlist^#(tail(x))) , int^#(x, y) -> c_14(if_int^#(zero(x), zero(y), x, y)) , if_int^#(true(), b, x, y) -> c_15(if1^#(b, x, y)) , if_int^#(false(), b, x, y) -> c_16(if2^#(b, x, y)) , if1^#(false(), x, y) -> c_18(int^#(s(0()), y)) , if2^#(false(), x, y) -> c_20(intlist^#(int(p(x), p(y)))) } Strict Trs: { empty(nil()) -> true() , empty(cons(x, y)) -> false() , tail(nil()) -> nil() , tail(cons(x, y)) -> y , head(cons(x, y)) -> x , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(0())) -> 0() , p(s(s(x))) -> s(p(s(x))) , intlist(x) -> if_intlist(empty(x), x) , if_intlist(true(), x) -> nil() , if_intlist(false(), x) -> cons(s(head(x)), intlist(tail(x))) , int(x, y) -> if_int(zero(x), zero(y), x, y) , if_int(true(), b, x, y) -> if1(b, x, y) , if_int(false(), b, x, y) -> if2(b, x, y) , if1(true(), x, y) -> cons(0(), nil()) , if1(false(), x, y) -> cons(0(), int(s(0()), y)) , if2(true(), x, y) -> nil() , if2(false(), x, y) -> intlist(int(p(x), p(y))) } Weak DPs: { empty^#(nil()) -> c_1() , empty^#(cons(x, y)) -> c_2() , tail^#(nil()) -> c_3() , zero^#(0()) -> c_6() , zero^#(s(x)) -> c_7() , p^#(0()) -> c_8() , p^#(s(0())) -> c_9() , if_intlist^#(true(), x) -> c_12() , if1^#(true(), x, y) -> c_17() , if2^#(true(), x, y) -> c_19() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..