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