MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(0(), y) -> 0() , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) , if_minus(true(), s(x), y) -> 0() , if_minus(false(), s(x), y) -> s(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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) '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. 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , minus^#(0(), y) -> c_4() , minus^#(s(x), y) -> c_5(if_minus^#(le(s(x), y), s(x), y)) , if_minus^#(true(), s(x), y) -> c_6() , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y)) , mod^#(0(), y) -> c_8() , mod^#(s(x), 0()) -> c_9() , mod^#(s(x), s(y)) -> c_10(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_11(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_12(x) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , minus^#(0(), y) -> c_4() , minus^#(s(x), y) -> c_5(if_minus^#(le(s(x), y), s(x), y)) , if_minus^#(true(), s(x), y) -> c_6() , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y)) , mod^#(0(), y) -> c_8() , mod^#(s(x), 0()) -> c_9() , mod^#(s(x), s(y)) -> c_10(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_11(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_12(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(0(), y) -> 0() , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) , if_minus(true(), s(x), y) -> 0() , if_minus(false(), s(x), y) -> s(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,6,8,9} by applications of Pre({1,2,4,6,8,9}) = {3,5,7,11,12}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y) -> c_1() , 2: le^#(s(x), 0()) -> c_2() , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y)) , 4: minus^#(0(), y) -> c_4() , 5: minus^#(s(x), y) -> c_5(if_minus^#(le(s(x), y), s(x), y)) , 6: if_minus^#(true(), s(x), y) -> c_6() , 7: if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y)) , 8: mod^#(0(), y) -> c_8() , 9: mod^#(s(x), 0()) -> c_9() , 10: mod^#(s(x), s(y)) -> c_10(if_mod^#(le(y, x), s(x), s(y))) , 11: if_mod^#(true(), s(x), s(y)) -> c_11(mod^#(minus(x, y), s(y))) , 12: if_mod^#(false(), s(x), s(y)) -> c_12(x) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , minus^#(s(x), y) -> c_5(if_minus^#(le(s(x), y), s(x), y)) , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y)) , mod^#(s(x), s(y)) -> c_10(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_11(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_12(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(0(), y) -> 0() , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) , if_minus(true(), s(x), y) -> 0() , if_minus(false(), s(x), y) -> s(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , minus^#(0(), y) -> c_4() , if_minus^#(true(), s(x), y) -> c_6() , mod^#(0(), y) -> c_8() , mod^#(s(x), 0()) -> c_9() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..