MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , if(true(), x, y) -> x , if(true(), x, y) -> true() , if(false(), x, y) -> y , if(false(), x, y) -> false() , odd(0()) -> false() , odd(s(0())) -> true() , odd(s(s(x))) -> odd(x) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , pow(x, y) -> f(x, y, s(0())) , f(x, 0(), z) -> z , f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(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: 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: { -^#(x, 0()) -> c_1(x) , -^#(s(x), s(y)) -> c_2(-^#(x, y)) , *^#(x, 0()) -> c_3() , *^#(x, s(y)) -> c_4(*^#(x, y), x) , if^#(true(), x, y) -> c_5(x) , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7(y) , if^#(false(), x, y) -> c_8() , odd^#(0()) -> c_9() , odd^#(s(0())) -> c_10() , odd^#(s(s(x))) -> c_11(odd^#(x)) , half^#(0()) -> c_12() , half^#(s(0())) -> c_13() , half^#(s(s(x))) -> c_14(half^#(x)) , pow^#(x, y) -> c_15(f^#(x, y, s(0()))) , f^#(x, 0(), z) -> c_16(z) , f^#(x, s(y), z) -> c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(s(x), s(y)) -> c_2(-^#(x, y)) , *^#(x, 0()) -> c_3() , *^#(x, s(y)) -> c_4(*^#(x, y), x) , if^#(true(), x, y) -> c_5(x) , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7(y) , if^#(false(), x, y) -> c_8() , odd^#(0()) -> c_9() , odd^#(s(0())) -> c_10() , odd^#(s(s(x))) -> c_11(odd^#(x)) , half^#(0()) -> c_12() , half^#(s(0())) -> c_13() , half^#(s(s(x))) -> c_14(half^#(x)) , pow^#(x, y) -> c_15(f^#(x, y, s(0()))) , f^#(x, 0(), z) -> c_16(z) , f^#(x, s(y), z) -> c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))) } Strict Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , if(true(), x, y) -> x , if(true(), x, y) -> true() , if(false(), x, y) -> y , if(false(), x, y) -> false() , odd(0()) -> false() , odd(s(0())) -> true() , odd(s(s(x))) -> odd(x) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , pow(x, y) -> f(x, y, s(0())) , f(x, 0(), z) -> z , f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,6,8,9,10,12,13} by applications of Pre({3,6,8,9,10,12,13}) = {1,4,5,7,11,14,16,17}. Here rules are labeled as follows: DPs: { 1: -^#(x, 0()) -> c_1(x) , 2: -^#(s(x), s(y)) -> c_2(-^#(x, y)) , 3: *^#(x, 0()) -> c_3() , 4: *^#(x, s(y)) -> c_4(*^#(x, y), x) , 5: if^#(true(), x, y) -> c_5(x) , 6: if^#(true(), x, y) -> c_6() , 7: if^#(false(), x, y) -> c_7(y) , 8: if^#(false(), x, y) -> c_8() , 9: odd^#(0()) -> c_9() , 10: odd^#(s(0())) -> c_10() , 11: odd^#(s(s(x))) -> c_11(odd^#(x)) , 12: half^#(0()) -> c_12() , 13: half^#(s(0())) -> c_13() , 14: half^#(s(s(x))) -> c_14(half^#(x)) , 15: pow^#(x, y) -> c_15(f^#(x, y, s(0()))) , 16: f^#(x, 0(), z) -> c_16(z) , 17: f^#(x, s(y), z) -> c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(s(x), s(y)) -> c_2(-^#(x, y)) , *^#(x, s(y)) -> c_4(*^#(x, y), x) , if^#(true(), x, y) -> c_5(x) , if^#(false(), x, y) -> c_7(y) , odd^#(s(s(x))) -> c_11(odd^#(x)) , half^#(s(s(x))) -> c_14(half^#(x)) , pow^#(x, y) -> c_15(f^#(x, y, s(0()))) , f^#(x, 0(), z) -> c_16(z) , f^#(x, s(y), z) -> c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))) } Strict Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , if(true(), x, y) -> x , if(true(), x, y) -> true() , if(false(), x, y) -> y , if(false(), x, y) -> false() , odd(0()) -> false() , odd(s(0())) -> true() , odd(s(s(x))) -> odd(x) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) , pow(x, y) -> f(x, y, s(0())) , f(x, 0(), z) -> z , f(x, s(y), z) -> if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) } Weak DPs: { *^#(x, 0()) -> c_3() , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_8() , odd^#(0()) -> c_9() , odd^#(s(0())) -> c_10() , half^#(0()) -> c_12() , half^#(s(0())) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..