MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { diff(x, y) -> cond1(equal(x, y), x, y) , cond1(true(), x, y) -> 0() , cond1(false(), x, y) -> cond2(gt(x, y), x, y) , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , cond2(true(), x, y) -> s(diff(x, s(y))) , cond2(false(), x, y) -> s(diff(s(x), y)) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, 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 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: { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y)) , cond1^#(true(), x, y) -> c_2() , cond1^#(false(), x, y) -> c_3(cond2^#(gt(x, y), x, y)) , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y))) , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y)) , equal^#(0(), 0()) -> c_4() , equal^#(0(), s(y)) -> c_5() , equal^#(s(x), 0()) -> c_6() , equal^#(s(x), s(y)) -> c_7(equal^#(x, y)) , gt^#(0(), v) -> c_10() , gt^#(s(u), 0()) -> c_11() , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y)) , cond1^#(true(), x, y) -> c_2() , cond1^#(false(), x, y) -> c_3(cond2^#(gt(x, y), x, y)) , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y))) , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y)) , equal^#(0(), 0()) -> c_4() , equal^#(0(), s(y)) -> c_5() , equal^#(s(x), 0()) -> c_6() , equal^#(s(x), s(y)) -> c_7(equal^#(x, y)) , gt^#(0(), v) -> c_10() , gt^#(s(u), 0()) -> c_11() , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) } Strict Trs: { diff(x, y) -> cond1(equal(x, y), x, y) , cond1(true(), x, y) -> 0() , cond1(false(), x, y) -> cond2(gt(x, y), x, y) , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , cond2(true(), x, y) -> s(diff(x, s(y))) , cond2(false(), x, y) -> s(diff(s(x), y)) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, v) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,6,7,8,10,11} by applications of Pre({2,6,7,8,10,11}) = {1,9,12}. Here rules are labeled as follows: DPs: { 1: diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y)) , 2: cond1^#(true(), x, y) -> c_2() , 3: cond1^#(false(), x, y) -> c_3(cond2^#(gt(x, y), x, y)) , 4: cond2^#(true(), x, y) -> c_8(diff^#(x, s(y))) , 5: cond2^#(false(), x, y) -> c_9(diff^#(s(x), y)) , 6: equal^#(0(), 0()) -> c_4() , 7: equal^#(0(), s(y)) -> c_5() , 8: equal^#(s(x), 0()) -> c_6() , 9: equal^#(s(x), s(y)) -> c_7(equal^#(x, y)) , 10: gt^#(0(), v) -> c_10() , 11: gt^#(s(u), 0()) -> c_11() , 12: gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { diff^#(x, y) -> c_1(cond1^#(equal(x, y), x, y)) , cond1^#(false(), x, y) -> c_3(cond2^#(gt(x, y), x, y)) , cond2^#(true(), x, y) -> c_8(diff^#(x, s(y))) , cond2^#(false(), x, y) -> c_9(diff^#(s(x), y)) , equal^#(s(x), s(y)) -> c_7(equal^#(x, y)) , gt^#(s(u), s(v)) -> c_12(gt^#(u, v)) } Strict Trs: { diff(x, y) -> cond1(equal(x, y), x, y) , cond1(true(), x, y) -> 0() , cond1(false(), x, y) -> cond2(gt(x, y), x, y) , equal(0(), 0()) -> true() , equal(0(), s(y)) -> false() , equal(s(x), 0()) -> false() , equal(s(x), s(y)) -> equal(x, y) , cond2(true(), x, y) -> s(diff(x, s(y))) , cond2(false(), x, y) -> s(diff(s(x), y)) , gt(0(), v) -> false() , gt(s(u), 0()) -> true() , gt(s(u), s(v)) -> gt(u, v) } Weak DPs: { cond1^#(true(), x, y) -> c_2() , equal^#(0(), 0()) -> c_4() , equal^#(0(), s(y)) -> c_5() , equal^#(s(x), 0()) -> c_6() , gt^#(0(), v) -> c_10() , gt^#(s(u), 0()) -> c_11() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..