YES(O(1),O(1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict Trs: { fst(0(), Z) -> nil() , fst(s(), cons(Y)) -> cons(Y) , from(X) -> cons(X) , add(0(), X) -> X , add(s(), Y) -> s() , len(nil()) -> 0() , len(cons(X)) -> s() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We add following weak dependency pairs: Strict DPs: { fst^#(0(), Z) -> c_1() , fst^#(s(), cons(Y)) -> c_2() , from^#(X) -> c_3() , add^#(0(), X) -> c_4() , add^#(s(), Y) -> c_5() , len^#(nil()) -> c_6() , len^#(cons(X)) -> c_7() } 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: { fst^#(0(), Z) -> c_1() , fst^#(s(), cons(Y)) -> c_2() , from^#(X) -> c_3() , add^#(0(), X) -> c_4() , add^#(s(), Y) -> c_5() , len^#(nil()) -> c_6() , len^#(cons(X)) -> c_7() } Strict Trs: { fst(0(), Z) -> nil() , fst(s(), cons(Y)) -> cons(Y) , from(X) -> cons(X) , add(0(), X) -> X , add(s(), Y) -> s() , len(nil()) -> 0() , len(cons(X)) -> s() } 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: { fst^#(0(), Z) -> c_1() , fst^#(s(), cons(Y)) -> c_2() , from^#(X) -> c_3() , add^#(0(), X) -> c_4() , add^#(s(), Y) -> c_5() , len^#(nil()) -> c_6() , len^#(cons(X)) -> c_7() } 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: none TcT has computed following constructor-restricted matrix interpretation. [0] = [2] [nil] = [1] [s] = [1] [cons](x1) = [2] [fst^#](x1, x2) = [1] x1 + [1] x2 + [1] [c_1] = [2] [c_2] = [1] [from^#](x1) = [2] [c_3] = [1] [add^#](x1, x2) = [1] x1 + [2] [c_4] = [1] [c_5] = [2] [len^#](x1) = [1] x1 + [1] [c_6] = [1] [c_7] = [2] 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),O(1)). Weak DPs: { fst^#(0(), Z) -> c_1() , fst^#(s(), cons(Y)) -> c_2() , from^#(X) -> c_3() , add^#(0(), X) -> c_4() , add^#(s(), Y) -> c_5() , len^#(nil()) -> c_6() , len^#(cons(X)) -> c_7() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { fst^#(0(), Z) -> c_1() , fst^#(s(), cons(Y)) -> c_2() , from^#(X) -> c_3() , add^#(0(), X) -> c_4() , add^#(s(), Y) -> c_5() , len^#(nil()) -> c_6() , len^#(cons(X)) -> c_7() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(1))