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