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(x, x) -> 0() , minus(x, 0()) -> x , minus(0(), x) -> 0() , minus(s(x), s(y)) -> minus(x, y) , isZero(0()) -> true() , isZero(s(x)) -> false() , mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y)) , if_mod(true(), b, x, y, z) -> divByZeroError() , if_mod(false(), true(), x, y, z) -> mod(z, y) , if_mod(false(), false(), x, y, z) -> x } 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: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , minus^#(x, x) -> c_4() , minus^#(x, 0()) -> c_5(x) , minus^#(0(), x) -> c_6() , minus^#(s(x), s(y)) -> c_7(minus^#(x, y)) , isZero^#(0()) -> c_8() , isZero^#(s(x)) -> c_9() , mod^#(x, y) -> c_10(if_mod^#(isZero(y), le(y, x), x, y, minus(x, y))) , if_mod^#(true(), b, x, y, z) -> c_11() , if_mod^#(false(), true(), x, y, z) -> c_12(mod^#(z, y)) , if_mod^#(false(), false(), x, y, z) -> c_13(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^#(x, x) -> c_4() , minus^#(x, 0()) -> c_5(x) , minus^#(0(), x) -> c_6() , minus^#(s(x), s(y)) -> c_7(minus^#(x, y)) , isZero^#(0()) -> c_8() , isZero^#(s(x)) -> c_9() , mod^#(x, y) -> c_10(if_mod^#(isZero(y), le(y, x), x, y, minus(x, y))) , if_mod^#(true(), b, x, y, z) -> c_11() , if_mod^#(false(), true(), x, y, z) -> c_12(mod^#(z, y)) , if_mod^#(false(), false(), x, y, z) -> c_13(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, x) -> 0() , minus(x, 0()) -> x , minus(0(), x) -> 0() , minus(s(x), s(y)) -> minus(x, y) , isZero(0()) -> true() , isZero(s(x)) -> false() , mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y)) , if_mod(true(), b, x, y, z) -> divByZeroError() , if_mod(false(), true(), x, y, z) -> mod(z, y) , if_mod(false(), false(), x, y, z) -> x } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,6,8,9,11} by applications of Pre({1,2,4,6,8,9,11}) = {3,5,7,10,13}. 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^#(x, x) -> c_4() , 5: minus^#(x, 0()) -> c_5(x) , 6: minus^#(0(), x) -> c_6() , 7: minus^#(s(x), s(y)) -> c_7(minus^#(x, y)) , 8: isZero^#(0()) -> c_8() , 9: isZero^#(s(x)) -> c_9() , 10: mod^#(x, y) -> c_10(if_mod^#(isZero(y), le(y, x), x, y, minus(x, y))) , 11: if_mod^#(true(), b, x, y, z) -> c_11() , 12: if_mod^#(false(), true(), x, y, z) -> c_12(mod^#(z, y)) , 13: if_mod^#(false(), false(), x, y, z) -> c_13(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^#(x, 0()) -> c_5(x) , minus^#(s(x), s(y)) -> c_7(minus^#(x, y)) , mod^#(x, y) -> c_10(if_mod^#(isZero(y), le(y, x), x, y, minus(x, y))) , if_mod^#(false(), true(), x, y, z) -> c_12(mod^#(z, y)) , if_mod^#(false(), false(), x, y, z) -> c_13(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, x) -> 0() , minus(x, 0()) -> x , minus(0(), x) -> 0() , minus(s(x), s(y)) -> minus(x, y) , isZero(0()) -> true() , isZero(s(x)) -> false() , mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y)) , if_mod(true(), b, x, y, z) -> divByZeroError() , if_mod(false(), true(), x, y, z) -> mod(z, y) , if_mod(false(), false(), x, y, z) -> x } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , minus^#(x, x) -> c_4() , minus^#(0(), x) -> c_6() , isZero^#(0()) -> c_8() , isZero^#(s(x)) -> c_9() , if_mod^#(true(), b, x, y, z) -> c_11() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..