YES(O(1),O(1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict Trs: { and(not(not(x)), y, not(z)) -> and(y, band(x, z), x) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We add following weak dependency pairs: Strict DPs: { and^#(not(not(x)), y, not(z)) -> c_1(and^#(y, band(x, z), x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict DPs: { and^#(not(not(x)), y, not(z)) -> c_1(and^#(y, band(x, z), x)) } Strict Trs: { and(not(not(x)), y, not(z)) -> and(y, band(x, z), x) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict DPs: { and^#(not(not(x)), y, not(z)) -> c_1(and^#(y, band(x, z), x)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {1} TcT has computed following constructor-restricted matrix interpretation. [not](x1) = [2] [band](x1, x2) = [0] [and^#](x1, x2, x3) = [1] x1 + [2] x2 + [0] [c_1](x1) = [1] x1 + [1] This order satisfies following ordering constraints: Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak DPs: { and^#(not(not(x)), y, not(z)) -> c_1(and^#(y, band(x, z), x)) } Obligation: innermost runtime complexity Answer: YES(?,O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { and^#(not(not(x)), y, not(z)) -> c_1(and^#(y, band(x, z), x)) } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(?,O(1)) The input was oriented with the instance of 'Small Polynomial Path Order (PS)' as induced by the safe mapping and precedence empty . Following symbols are considered recursive: {} The recursion depth is 0. Further, following argument filtering is employed: empty Usable defined function symbols are a subset of: {} For your convenience, here are the satisfied ordering constraints: Hurray, we answered YES(O(1),O(1))