MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { pred^#(s(x)) -> c_1() , minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) , minus^#(x, 0()) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) , quot^#(0(), s(y)) -> c_5() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { pred^#(s(x)) -> c_1() , minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) , minus^#(x, 0()) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) , quot^#(0(), s(y)) -> c_5() } Weak Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5} by applications of Pre({1,3,5}) = {2,4}. Here rules are labeled as follows: DPs: { 1: pred^#(s(x)) -> c_1() , 2: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) , 3: minus^#(x, 0()) -> c_3() , 4: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) , 5: quot^#(0(), s(y)) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak DPs: { pred^#(s(x)) -> c_1() , minus^#(x, 0()) -> c_3() , quot^#(0(), s(y)) -> c_5() } Weak Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { pred^#(s(x)) -> c_1() , minus^#(x, 0()) -> c_3() , quot^#(0(), s(y)) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { minus^#(x, s(y)) -> c_2(pred^#(minus(x, y)), minus^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..