MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , exp(x, 0()) -> s(0()) , exp(x, s(y)) -> times(x, exp(x, y)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , tower(x, y) -> towerIter(0(), x, y, s(0())) , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) , help(true(), c, x, y, z) -> z , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) } 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: 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) '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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { plus^#(0(), x) -> c_1(x) , plus^#(s(x), y) -> c_2(plus^#(x, y)) , times^#(0(), y) -> c_3() , times^#(s(x), y) -> c_4(plus^#(y, times(x, y))) , exp^#(x, 0()) -> c_5() , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y))) , ge^#(x, 0()) -> c_7() , ge^#(0(), s(x)) -> c_8() , ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0()))) , towerIter^#(c, x, y, z) -> c_11(help^#(ge(c, x), c, x, y, z)) , help^#(true(), c, x, y, z) -> c_12(z) , help^#(false(), c, x, y, z) -> c_13(towerIter^#(s(c), x, y, exp(y, z))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(0(), x) -> c_1(x) , plus^#(s(x), y) -> c_2(plus^#(x, y)) , times^#(0(), y) -> c_3() , times^#(s(x), y) -> c_4(plus^#(y, times(x, y))) , exp^#(x, 0()) -> c_5() , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y))) , ge^#(x, 0()) -> c_7() , ge^#(0(), s(x)) -> c_8() , ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0()))) , towerIter^#(c, x, y, z) -> c_11(help^#(ge(c, x), c, x, y, z)) , help^#(true(), c, x, y, z) -> c_12(z) , help^#(false(), c, x, y, z) -> c_13(towerIter^#(s(c), x, y, exp(y, z))) } Strict Trs: { plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , exp(x, 0()) -> s(0()) , exp(x, s(y)) -> times(x, exp(x, y)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , tower(x, y) -> towerIter(0(), x, y, s(0())) , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) , help(true(), c, x, y, z) -> z , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,5,7,8} by applications of Pre({3,5,7,8}) = {1,6,9,12}. Here rules are labeled as follows: DPs: { 1: plus^#(0(), x) -> c_1(x) , 2: plus^#(s(x), y) -> c_2(plus^#(x, y)) , 3: times^#(0(), y) -> c_3() , 4: times^#(s(x), y) -> c_4(plus^#(y, times(x, y))) , 5: exp^#(x, 0()) -> c_5() , 6: exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y))) , 7: ge^#(x, 0()) -> c_7() , 8: ge^#(0(), s(x)) -> c_8() , 9: ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) , 10: tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0()))) , 11: towerIter^#(c, x, y, z) -> c_11(help^#(ge(c, x), c, x, y, z)) , 12: help^#(true(), c, x, y, z) -> c_12(z) , 13: help^#(false(), c, x, y, z) -> c_13(towerIter^#(s(c), x, y, exp(y, z))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(0(), x) -> c_1(x) , plus^#(s(x), y) -> c_2(plus^#(x, y)) , times^#(s(x), y) -> c_4(plus^#(y, times(x, y))) , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y))) , ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0()))) , towerIter^#(c, x, y, z) -> c_11(help^#(ge(c, x), c, x, y, z)) , help^#(true(), c, x, y, z) -> c_12(z) , help^#(false(), c, x, y, z) -> c_13(towerIter^#(s(c), x, y, exp(y, z))) } Strict Trs: { plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , exp(x, 0()) -> s(0()) , exp(x, s(y)) -> times(x, exp(x, y)) , ge(x, 0()) -> true() , ge(0(), s(x)) -> false() , ge(s(x), s(y)) -> ge(x, y) , tower(x, y) -> towerIter(0(), x, y, s(0())) , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) , help(true(), c, x, y, z) -> z , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) } Weak DPs: { times^#(0(), y) -> c_3() , exp^#(x, 0()) -> c_5() , ge^#(x, 0()) -> c_7() , ge^#(0(), s(x)) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..