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)) , p(0()) -> 0() , p(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> p(minus(x, y)) , minus(0(), x) -> 0() , isZero(0()) -> true() , isZero(s(x)) -> false() , facIter(x, y) -> if(isZero(x), minus(x, s(0())), y, times(y, x)) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> facIter(x, z) , factorial(x) -> facIter(x, s(0())) } 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: { 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))) , p^#(0()) -> c_5() , p^#(s(x)) -> c_6(x) , minus^#(x, 0()) -> c_7(x) , minus^#(x, s(y)) -> c_8(p^#(minus(x, y))) , minus^#(0(), x) -> c_9() , isZero^#(0()) -> c_10() , isZero^#(s(x)) -> c_11() , facIter^#(x, y) -> c_12(if^#(isZero(x), minus(x, s(0())), y, times(y, x))) , if^#(true(), x, y, z) -> c_13(y) , if^#(false(), x, y, z) -> c_14(facIter^#(x, z)) , factorial^#(x) -> c_15(facIter^#(x, s(0()))) } 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))) , p^#(0()) -> c_5() , p^#(s(x)) -> c_6(x) , minus^#(x, 0()) -> c_7(x) , minus^#(x, s(y)) -> c_8(p^#(minus(x, y))) , minus^#(0(), x) -> c_9() , isZero^#(0()) -> c_10() , isZero^#(s(x)) -> c_11() , facIter^#(x, y) -> c_12(if^#(isZero(x), minus(x, s(0())), y, times(y, x))) , if^#(true(), x, y, z) -> c_13(y) , if^#(false(), x, y, z) -> c_14(facIter^#(x, z)) , factorial^#(x) -> c_15(facIter^#(x, s(0()))) } 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)) , p(0()) -> 0() , p(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> p(minus(x, y)) , minus(0(), x) -> 0() , isZero(0()) -> true() , isZero(s(x)) -> false() , facIter(x, y) -> if(isZero(x), minus(x, s(0())), y, times(y, x)) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> facIter(x, z) , factorial(x) -> facIter(x, s(0())) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,5,9,10,11} by applications of Pre({3,5,9,10,11}) = {1,6,7,8,13}. 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: p^#(0()) -> c_5() , 6: p^#(s(x)) -> c_6(x) , 7: minus^#(x, 0()) -> c_7(x) , 8: minus^#(x, s(y)) -> c_8(p^#(minus(x, y))) , 9: minus^#(0(), x) -> c_9() , 10: isZero^#(0()) -> c_10() , 11: isZero^#(s(x)) -> c_11() , 12: facIter^#(x, y) -> c_12(if^#(isZero(x), minus(x, s(0())), y, times(y, x))) , 13: if^#(true(), x, y, z) -> c_13(y) , 14: if^#(false(), x, y, z) -> c_14(facIter^#(x, z)) , 15: factorial^#(x) -> c_15(facIter^#(x, s(0()))) } 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))) , p^#(s(x)) -> c_6(x) , minus^#(x, 0()) -> c_7(x) , minus^#(x, s(y)) -> c_8(p^#(minus(x, y))) , facIter^#(x, y) -> c_12(if^#(isZero(x), minus(x, s(0())), y, times(y, x))) , if^#(true(), x, y, z) -> c_13(y) , if^#(false(), x, y, z) -> c_14(facIter^#(x, z)) , factorial^#(x) -> c_15(facIter^#(x, s(0()))) } 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)) , p(0()) -> 0() , p(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> p(minus(x, y)) , minus(0(), x) -> 0() , isZero(0()) -> true() , isZero(s(x)) -> false() , facIter(x, y) -> if(isZero(x), minus(x, s(0())), y, times(y, x)) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> facIter(x, z) , factorial(x) -> facIter(x, s(0())) } Weak DPs: { times^#(0(), y) -> c_3() , p^#(0()) -> c_5() , minus^#(0(), x) -> c_9() , isZero^#(0()) -> c_10() , isZero^#(s(x)) -> c_11() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..