MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { division(x, y) -> div(x, y, 0()) , div(x, y, z) -> if(lt(x, y), x, y, inc(z)) , if(true(), x, y, z) -> z , if(false(), x, s(y), z) -> div(minus(x, s(y)), s(y), z) , lt(x, 0()) -> false() , lt(0(), s(y)) -> true() , lt(s(x), s(y)) -> lt(x, y) , inc(0()) -> s(0()) , inc(s(x)) -> s(inc(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, 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: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 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 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: { division^#(x, y) -> c_1(div^#(x, y, 0())) , div^#(x, y, z) -> c_2(if^#(lt(x, y), x, y, inc(z))) , if^#(true(), x, y, z) -> c_3(z) , if^#(false(), x, s(y), z) -> c_4(div^#(minus(x, s(y)), s(y), z)) , lt^#(x, 0()) -> c_5() , lt^#(0(), s(y)) -> c_6() , lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) , inc^#(0()) -> c_8() , inc^#(s(x)) -> c_9(inc^#(x)) , minus^#(x, 0()) -> c_10(x) , minus^#(s(x), s(y)) -> c_11(minus^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { division^#(x, y) -> c_1(div^#(x, y, 0())) , div^#(x, y, z) -> c_2(if^#(lt(x, y), x, y, inc(z))) , if^#(true(), x, y, z) -> c_3(z) , if^#(false(), x, s(y), z) -> c_4(div^#(minus(x, s(y)), s(y), z)) , lt^#(x, 0()) -> c_5() , lt^#(0(), s(y)) -> c_6() , lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) , inc^#(0()) -> c_8() , inc^#(s(x)) -> c_9(inc^#(x)) , minus^#(x, 0()) -> c_10(x) , minus^#(s(x), s(y)) -> c_11(minus^#(x, y)) } Strict Trs: { division(x, y) -> div(x, y, 0()) , div(x, y, z) -> if(lt(x, y), x, y, inc(z)) , if(true(), x, y, z) -> z , if(false(), x, s(y), z) -> div(minus(x, s(y)), s(y), z) , lt(x, 0()) -> false() , lt(0(), s(y)) -> true() , lt(s(x), s(y)) -> lt(x, y) , inc(0()) -> s(0()) , inc(s(x)) -> s(inc(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {5,6,8} by applications of Pre({5,6,8}) = {3,7,9,10}. Here rules are labeled as follows: DPs: { 1: division^#(x, y) -> c_1(div^#(x, y, 0())) , 2: div^#(x, y, z) -> c_2(if^#(lt(x, y), x, y, inc(z))) , 3: if^#(true(), x, y, z) -> c_3(z) , 4: if^#(false(), x, s(y), z) -> c_4(div^#(minus(x, s(y)), s(y), z)) , 5: lt^#(x, 0()) -> c_5() , 6: lt^#(0(), s(y)) -> c_6() , 7: lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) , 8: inc^#(0()) -> c_8() , 9: inc^#(s(x)) -> c_9(inc^#(x)) , 10: minus^#(x, 0()) -> c_10(x) , 11: minus^#(s(x), s(y)) -> c_11(minus^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { division^#(x, y) -> c_1(div^#(x, y, 0())) , div^#(x, y, z) -> c_2(if^#(lt(x, y), x, y, inc(z))) , if^#(true(), x, y, z) -> c_3(z) , if^#(false(), x, s(y), z) -> c_4(div^#(minus(x, s(y)), s(y), z)) , lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) , inc^#(s(x)) -> c_9(inc^#(x)) , minus^#(x, 0()) -> c_10(x) , minus^#(s(x), s(y)) -> c_11(minus^#(x, y)) } Strict Trs: { division(x, y) -> div(x, y, 0()) , div(x, y, z) -> if(lt(x, y), x, y, inc(z)) , if(true(), x, y, z) -> z , if(false(), x, s(y), z) -> div(minus(x, s(y)), s(y), z) , lt(x, 0()) -> false() , lt(0(), s(y)) -> true() , lt(s(x), s(y)) -> lt(x, y) , inc(0()) -> s(0()) , inc(s(x)) -> s(inc(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Weak DPs: { lt^#(x, 0()) -> c_5() , lt^#(0(), s(y)) -> c_6() , inc^#(0()) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..