MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(m)) -> false() , eq(s(n), 0()) -> false() , eq(s(n), s(m)) -> eq(n, m) , le(0(), m) -> true() , le(s(n), 0()) -> false() , le(s(n), s(m)) -> le(n, m) , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) , min(cons(0(), nil())) -> 0() , min(cons(s(n), nil())) -> s(n) , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x)) , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x)) , replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) , replace(n, m, nil()) -> nil() , if_replace(true(), n, m, cons(k, x)) -> cons(m, x) , if_replace(false(), n, m, cons(k, x)) -> cons(k, replace(n, m, x)) , sort(cons(n, x)) -> cons(min(cons(n, x)), sort(replace(min(cons(n, x)), n, x))) , sort(nil()) -> nil() } 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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(m)) -> c_2() , eq^#(s(n), 0()) -> c_3() , eq^#(s(n), s(m)) -> c_4(eq^#(n, m)) , le^#(0(), m) -> c_5() , le^#(s(n), 0()) -> c_6() , le^#(s(n), s(m)) -> c_7(le^#(n, m)) , min^#(cons(n, cons(m, x))) -> c_8(if_min^#(le(n, m), cons(n, cons(m, x)))) , min^#(cons(0(), nil())) -> c_9() , min^#(cons(s(n), nil())) -> c_10(n) , if_min^#(true(), cons(n, cons(m, x))) -> c_11(min^#(cons(n, x))) , if_min^#(false(), cons(n, cons(m, x))) -> c_12(min^#(cons(m, x))) , replace^#(n, m, cons(k, x)) -> c_13(if_replace^#(eq(n, k), n, m, cons(k, x))) , replace^#(n, m, nil()) -> c_14() , if_replace^#(true(), n, m, cons(k, x)) -> c_15(m, x) , if_replace^#(false(), n, m, cons(k, x)) -> c_16(k, replace^#(n, m, x)) , sort^#(cons(n, x)) -> c_17(min^#(cons(n, x)), sort^#(replace(min(cons(n, x)), n, x))) , sort^#(nil()) -> c_18() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(m)) -> c_2() , eq^#(s(n), 0()) -> c_3() , eq^#(s(n), s(m)) -> c_4(eq^#(n, m)) , le^#(0(), m) -> c_5() , le^#(s(n), 0()) -> c_6() , le^#(s(n), s(m)) -> c_7(le^#(n, m)) , min^#(cons(n, cons(m, x))) -> c_8(if_min^#(le(n, m), cons(n, cons(m, x)))) , min^#(cons(0(), nil())) -> c_9() , min^#(cons(s(n), nil())) -> c_10(n) , if_min^#(true(), cons(n, cons(m, x))) -> c_11(min^#(cons(n, x))) , if_min^#(false(), cons(n, cons(m, x))) -> c_12(min^#(cons(m, x))) , replace^#(n, m, cons(k, x)) -> c_13(if_replace^#(eq(n, k), n, m, cons(k, x))) , replace^#(n, m, nil()) -> c_14() , if_replace^#(true(), n, m, cons(k, x)) -> c_15(m, x) , if_replace^#(false(), n, m, cons(k, x)) -> c_16(k, replace^#(n, m, x)) , sort^#(cons(n, x)) -> c_17(min^#(cons(n, x)), sort^#(replace(min(cons(n, x)), n, x))) , sort^#(nil()) -> c_18() } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(m)) -> false() , eq(s(n), 0()) -> false() , eq(s(n), s(m)) -> eq(n, m) , le(0(), m) -> true() , le(s(n), 0()) -> false() , le(s(n), s(m)) -> le(n, m) , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) , min(cons(0(), nil())) -> 0() , min(cons(s(n), nil())) -> s(n) , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x)) , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x)) , replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) , replace(n, m, nil()) -> nil() , if_replace(true(), n, m, cons(k, x)) -> cons(m, x) , if_replace(false(), n, m, cons(k, x)) -> cons(k, replace(n, m, x)) , sort(cons(n, x)) -> cons(min(cons(n, x)), sort(replace(min(cons(n, x)), n, x))) , sort(nil()) -> nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,5,6,9,14,18} by applications of Pre({1,2,3,5,6,9,14,18}) = {4,7,10,11,12,15,16,17}. Here rules are labeled as follows: DPs: { 1: eq^#(0(), 0()) -> c_1() , 2: eq^#(0(), s(m)) -> c_2() , 3: eq^#(s(n), 0()) -> c_3() , 4: eq^#(s(n), s(m)) -> c_4(eq^#(n, m)) , 5: le^#(0(), m) -> c_5() , 6: le^#(s(n), 0()) -> c_6() , 7: le^#(s(n), s(m)) -> c_7(le^#(n, m)) , 8: min^#(cons(n, cons(m, x))) -> c_8(if_min^#(le(n, m), cons(n, cons(m, x)))) , 9: min^#(cons(0(), nil())) -> c_9() , 10: min^#(cons(s(n), nil())) -> c_10(n) , 11: if_min^#(true(), cons(n, cons(m, x))) -> c_11(min^#(cons(n, x))) , 12: if_min^#(false(), cons(n, cons(m, x))) -> c_12(min^#(cons(m, x))) , 13: replace^#(n, m, cons(k, x)) -> c_13(if_replace^#(eq(n, k), n, m, cons(k, x))) , 14: replace^#(n, m, nil()) -> c_14() , 15: if_replace^#(true(), n, m, cons(k, x)) -> c_15(m, x) , 16: if_replace^#(false(), n, m, cons(k, x)) -> c_16(k, replace^#(n, m, x)) , 17: sort^#(cons(n, x)) -> c_17(min^#(cons(n, x)), sort^#(replace(min(cons(n, x)), n, x))) , 18: sort^#(nil()) -> c_18() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(s(n), s(m)) -> c_4(eq^#(n, m)) , le^#(s(n), s(m)) -> c_7(le^#(n, m)) , min^#(cons(n, cons(m, x))) -> c_8(if_min^#(le(n, m), cons(n, cons(m, x)))) , min^#(cons(s(n), nil())) -> c_10(n) , if_min^#(true(), cons(n, cons(m, x))) -> c_11(min^#(cons(n, x))) , if_min^#(false(), cons(n, cons(m, x))) -> c_12(min^#(cons(m, x))) , replace^#(n, m, cons(k, x)) -> c_13(if_replace^#(eq(n, k), n, m, cons(k, x))) , if_replace^#(true(), n, m, cons(k, x)) -> c_15(m, x) , if_replace^#(false(), n, m, cons(k, x)) -> c_16(k, replace^#(n, m, x)) , sort^#(cons(n, x)) -> c_17(min^#(cons(n, x)), sort^#(replace(min(cons(n, x)), n, x))) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(m)) -> false() , eq(s(n), 0()) -> false() , eq(s(n), s(m)) -> eq(n, m) , le(0(), m) -> true() , le(s(n), 0()) -> false() , le(s(n), s(m)) -> le(n, m) , min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) , min(cons(0(), nil())) -> 0() , min(cons(s(n), nil())) -> s(n) , if_min(true(), cons(n, cons(m, x))) -> min(cons(n, x)) , if_min(false(), cons(n, cons(m, x))) -> min(cons(m, x)) , replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) , replace(n, m, nil()) -> nil() , if_replace(true(), n, m, cons(k, x)) -> cons(m, x) , if_replace(false(), n, m, cons(k, x)) -> cons(k, replace(n, m, x)) , sort(cons(n, x)) -> cons(min(cons(n, x)), sort(replace(min(cons(n, x)), n, x))) , sort(nil()) -> nil() } Weak DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(m)) -> c_2() , eq^#(s(n), 0()) -> c_3() , le^#(0(), m) -> c_5() , le^#(s(n), 0()) -> c_6() , min^#(cons(0(), nil())) -> c_9() , replace^#(n, m, nil()) -> c_14() , sort^#(nil()) -> c_18() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..