MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) , head(cons(X, XS)) -> X , tail(cons(X, XS)) -> XS } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { nats^#() -> c_1(incr^#(nats()), nats^#()) , incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , pairs^#() -> c_3(incr^#(odds()), odds^#()) , odds^#() -> c_4(incr^#(pairs()), pairs^#()) , head^#(cons(X, XS)) -> c_5() , tail^#(cons(X, XS)) -> c_6() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { nats^#() -> c_1(incr^#(nats()), nats^#()) , incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , pairs^#() -> c_3(incr^#(odds()), odds^#()) , odds^#() -> c_4(incr^#(pairs()), pairs^#()) , head^#(cons(X, XS)) -> c_5() , tail^#(cons(X, XS)) -> c_6() } Weak Trs: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) , head(cons(X, XS)) -> X , tail(cons(X, XS)) -> XS } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {5,6} by applications of Pre({5,6}) = {}. Here rules are labeled as follows: DPs: { 1: nats^#() -> c_1(incr^#(nats()), nats^#()) , 2: incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , 3: pairs^#() -> c_3(incr^#(odds()), odds^#()) , 4: odds^#() -> c_4(incr^#(pairs()), pairs^#()) , 5: head^#(cons(X, XS)) -> c_5() , 6: tail^#(cons(X, XS)) -> c_6() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { nats^#() -> c_1(incr^#(nats()), nats^#()) , incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , pairs^#() -> c_3(incr^#(odds()), odds^#()) , odds^#() -> c_4(incr^#(pairs()), pairs^#()) } Weak DPs: { head^#(cons(X, XS)) -> c_5() , tail^#(cons(X, XS)) -> c_6() } Weak Trs: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) , head(cons(X, XS)) -> X , tail(cons(X, XS)) -> XS } 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. { head^#(cons(X, XS)) -> c_5() , tail^#(cons(X, XS)) -> c_6() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { nats^#() -> c_1(incr^#(nats()), nats^#()) , incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , pairs^#() -> c_3(incr^#(odds()), odds^#()) , odds^#() -> c_4(incr^#(pairs()), pairs^#()) } Weak Trs: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) , head(cons(X, XS)) -> X , tail(cons(X, XS)) -> XS } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { nats^#() -> c_1(incr^#(nats()), nats^#()) , incr^#(cons(X, XS)) -> c_2(incr^#(XS)) , pairs^#() -> c_3(incr^#(odds()), odds^#()) , odds^#() -> c_4(incr^#(pairs()), pairs^#()) } Weak Trs: { nats() -> cons(0(), incr(nats())) , incr(cons(X, XS)) -> cons(s(X), incr(XS)) , pairs() -> cons(0(), incr(odds())) , odds() -> incr(pairs()) } 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..