MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { sort(l) -> st(0(), l) , st(n, l) -> cond1(member(n, l), n, l) , cond1(true(), n, l) -> cons(n, st(s(n), l)) , cond1(false(), n, l) -> cond2(gt(n, max(l)), n, l) , member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) , member(n, nil()) -> false() , cond2(true(), n, l) -> nil() , cond2(false(), n, l) -> st(s(n), l) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, v) , max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) , max(nil()) -> 0() , or(x, true()) -> true() , or(x, false()) -> x , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , if(true(), u, v) -> u , if(false(), u, v) -> v } 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) '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) '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: { sort^#(l) -> c_1(st^#(0(), l)) , st^#(n, l) -> c_2(cond1^#(member(n, l), n, l)) , cond1^#(true(), n, l) -> c_3(n, st^#(s(n), l)) , cond1^#(false(), n, l) -> c_4(cond2^#(gt(n, max(l)), n, l)) , cond2^#(true(), n, l) -> c_7() , cond2^#(false(), n, l) -> c_8(st^#(s(n), l)) , member^#(n, cons(m, l)) -> c_5(or^#(equal(n, m), member(n, l))) , member^#(n, nil()) -> c_6() , or^#(x, true()) -> c_14() , or^#(x, false()) -> c_15(x) , gt^#(0(), v) -> c_9() , gt^#(s(u), 0()) -> c_10() , gt^#(s(u), s(v)) -> c_11(gt^#(u, v)) , max^#(cons(u, l)) -> c_12(if^#(gt(u, max(l)), u, max(l))) , max^#(nil()) -> c_13() , if^#(true(), u, v) -> c_20(u) , if^#(false(), u, v) -> c_21(v) , equal^#(0(), 0()) -> c_16() , equal^#(0(), s(y)) -> c_17() , equal^#(s(x), 0()) -> c_18() , equal^#(s(x), s(y)) -> c_19(equal^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sort^#(l) -> c_1(st^#(0(), l)) , st^#(n, l) -> c_2(cond1^#(member(n, l), n, l)) , cond1^#(true(), n, l) -> c_3(n, st^#(s(n), l)) , cond1^#(false(), n, l) -> c_4(cond2^#(gt(n, max(l)), n, l)) , cond2^#(true(), n, l) -> c_7() , cond2^#(false(), n, l) -> c_8(st^#(s(n), l)) , member^#(n, cons(m, l)) -> c_5(or^#(equal(n, m), member(n, l))) , member^#(n, nil()) -> c_6() , or^#(x, true()) -> c_14() , or^#(x, false()) -> c_15(x) , gt^#(0(), v) -> c_9() , gt^#(s(u), 0()) -> c_10() , gt^#(s(u), s(v)) -> c_11(gt^#(u, v)) , max^#(cons(u, l)) -> c_12(if^#(gt(u, max(l)), u, max(l))) , max^#(nil()) -> c_13() , if^#(true(), u, v) -> c_20(u) , if^#(false(), u, v) -> c_21(v) , equal^#(0(), 0()) -> c_16() , equal^#(0(), s(y)) -> c_17() , equal^#(s(x), 0()) -> c_18() , equal^#(s(x), s(y)) -> c_19(equal^#(x, y)) } Strict Trs: { sort(l) -> st(0(), l) , st(n, l) -> cond1(member(n, l), n, l) , cond1(true(), n, l) -> cons(n, st(s(n), l)) , cond1(false(), n, l) -> cond2(gt(n, max(l)), n, l) , member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) , member(n, nil()) -> false() , cond2(true(), n, l) -> nil() , cond2(false(), n, l) -> st(s(n), l) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, v) , max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) , max(nil()) -> 0() , or(x, true()) -> true() , or(x, false()) -> x , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , if(true(), u, v) -> u , if(false(), u, v) -> v } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {5,8,9,11,12,15,18,19,20} by applications of Pre({5,8,9,11,12,15,18,19,20}) = {3,4,7,10,13,16,17,21}. Here rules are labeled as follows: DPs: { 1: sort^#(l) -> c_1(st^#(0(), l)) , 2: st^#(n, l) -> c_2(cond1^#(member(n, l), n, l)) , 3: cond1^#(true(), n, l) -> c_3(n, st^#(s(n), l)) , 4: cond1^#(false(), n, l) -> c_4(cond2^#(gt(n, max(l)), n, l)) , 5: cond2^#(true(), n, l) -> c_7() , 6: cond2^#(false(), n, l) -> c_8(st^#(s(n), l)) , 7: member^#(n, cons(m, l)) -> c_5(or^#(equal(n, m), member(n, l))) , 8: member^#(n, nil()) -> c_6() , 9: or^#(x, true()) -> c_14() , 10: or^#(x, false()) -> c_15(x) , 11: gt^#(0(), v) -> c_9() , 12: gt^#(s(u), 0()) -> c_10() , 13: gt^#(s(u), s(v)) -> c_11(gt^#(u, v)) , 14: max^#(cons(u, l)) -> c_12(if^#(gt(u, max(l)), u, max(l))) , 15: max^#(nil()) -> c_13() , 16: if^#(true(), u, v) -> c_20(u) , 17: if^#(false(), u, v) -> c_21(v) , 18: equal^#(0(), 0()) -> c_16() , 19: equal^#(0(), s(y)) -> c_17() , 20: equal^#(s(x), 0()) -> c_18() , 21: equal^#(s(x), s(y)) -> c_19(equal^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sort^#(l) -> c_1(st^#(0(), l)) , st^#(n, l) -> c_2(cond1^#(member(n, l), n, l)) , cond1^#(true(), n, l) -> c_3(n, st^#(s(n), l)) , cond1^#(false(), n, l) -> c_4(cond2^#(gt(n, max(l)), n, l)) , cond2^#(false(), n, l) -> c_8(st^#(s(n), l)) , member^#(n, cons(m, l)) -> c_5(or^#(equal(n, m), member(n, l))) , or^#(x, false()) -> c_15(x) , gt^#(s(u), s(v)) -> c_11(gt^#(u, v)) , max^#(cons(u, l)) -> c_12(if^#(gt(u, max(l)), u, max(l))) , if^#(true(), u, v) -> c_20(u) , if^#(false(), u, v) -> c_21(v) , equal^#(s(x), s(y)) -> c_19(equal^#(x, y)) } Strict Trs: { sort(l) -> st(0(), l) , st(n, l) -> cond1(member(n, l), n, l) , cond1(true(), n, l) -> cons(n, st(s(n), l)) , cond1(false(), n, l) -> cond2(gt(n, max(l)), n, l) , member(n, cons(m, l)) -> or(equal(n, m), member(n, l)) , member(n, nil()) -> false() , cond2(true(), n, l) -> nil() , cond2(false(), n, l) -> st(s(n), l) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, v) , max(cons(u, l)) -> if(gt(u, max(l)), u, max(l)) , max(nil()) -> 0() , or(x, true()) -> true() , or(x, false()) -> x , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , if(true(), u, v) -> u , if(false(), u, v) -> v } Weak DPs: { cond2^#(true(), n, l) -> c_7() , member^#(n, nil()) -> c_6() , or^#(x, true()) -> c_14() , gt^#(0(), v) -> c_9() , gt^#(s(u), 0()) -> c_10() , max^#(nil()) -> c_13() , equal^#(0(), 0()) -> c_16() , equal^#(0(), s(y)) -> c_17() , equal^#(s(x), 0()) -> c_18() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..