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, 0()) -> x , minus(0(), s(y)) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , mod(x, s(y)) -> help(x, s(y), 0()) , mod(s(x), 0()) -> 0() , help(x, s(y), c) -> if(le(c, x), x, s(y), c) , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y))) , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) } 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) '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) '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) '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, 0()) -> c_4(x) , minus^#(0(), s(y)) -> c_5() , minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) , plus^#(x, 0()) -> c_7(x) , plus^#(x, s(y)) -> c_8(plus^#(x, y)) , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0())) , mod^#(s(x), 0()) -> c_10() , help^#(x, s(y), c) -> c_11(if^#(le(c, x), x, s(y), c)) , if^#(true(), x, s(y), c) -> c_12(help^#(x, s(y), plus(c, s(y)))) , if^#(false(), x, s(y), c) -> c_13(minus^#(x, minus(c, s(y)))) } 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, 0()) -> c_4(x) , minus^#(0(), s(y)) -> c_5() , minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) , plus^#(x, 0()) -> c_7(x) , plus^#(x, s(y)) -> c_8(plus^#(x, y)) , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0())) , mod^#(s(x), 0()) -> c_10() , help^#(x, s(y), c) -> c_11(if^#(le(c, x), x, s(y), c)) , if^#(true(), x, s(y), c) -> c_12(help^#(x, s(y), plus(c, s(y)))) , if^#(false(), x, s(y), c) -> c_13(minus^#(x, minus(c, s(y)))) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(0(), s(y)) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , mod(x, s(y)) -> help(x, s(y), 0()) , mod(s(x), 0()) -> 0() , help(x, s(y), c) -> if(le(c, x), x, s(y), c) , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y))) , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,5,10} by applications of Pre({1,2,5,10}) = {3,4,6,7,13}. 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, 0()) -> c_4(x) , 5: minus^#(0(), s(y)) -> c_5() , 6: minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) , 7: plus^#(x, 0()) -> c_7(x) , 8: plus^#(x, s(y)) -> c_8(plus^#(x, y)) , 9: mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0())) , 10: mod^#(s(x), 0()) -> c_10() , 11: help^#(x, s(y), c) -> c_11(if^#(le(c, x), x, s(y), c)) , 12: if^#(true(), x, s(y), c) -> c_12(help^#(x, s(y), plus(c, s(y)))) , 13: if^#(false(), x, s(y), c) -> c_13(minus^#(x, minus(c, s(y)))) } 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_4(x) , minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) , plus^#(x, 0()) -> c_7(x) , plus^#(x, s(y)) -> c_8(plus^#(x, y)) , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0())) , help^#(x, s(y), c) -> c_11(if^#(le(c, x), x, s(y), c)) , if^#(true(), x, s(y), c) -> c_12(help^#(x, s(y), plus(c, s(y)))) , if^#(false(), x, s(y), c) -> c_13(minus^#(x, minus(c, s(y)))) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(0(), s(y)) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , mod(x, s(y)) -> help(x, s(y), 0()) , mod(s(x), 0()) -> 0() , help(x, s(y), c) -> if(le(c, x), x, s(y), c) , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y))) , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , minus^#(0(), s(y)) -> c_5() , mod^#(s(x), 0()) -> c_10() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..