WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: concat(cons(U,V),Y) -> cons(U,concat(V,Y)) concat(leaf(),Y) -> Y lessleaves(X,leaf()) -> false() lessleaves(cons(U,V),cons(W,Z)) -> lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) -> true() - Signature: {concat/2,lessleaves/2} / {cons/2,false/0,leaf/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {concat,lessleaves} and constructors {cons,false,leaf ,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(lessleaves) = {1,2} Following symbols are considered usable: {concat,lessleaves} TcT has computed the following interpretation: p(concat) = [1 2] x1 + [1 0] x2 + [0] [0 2] [0 1] [0] p(cons) = [1 3] x1 + [1 0] x2 + [0] [0 0] [0 1] [7] p(false) = [0] [0] p(leaf) = [1] [0] p(lessleaves) = [1 0] x1 + [4 2] x2 + [0] [4 0] [0 0] [1] p(true) = [0] [1] Following rules are strictly oriented: concat(cons(U,V),Y) = [1 3] U + [1 2] V + [1 0] Y + [14] [0 0] [0 2] [0 1] [14] > [1 3] U + [1 2] V + [1 0] Y + [0] [0 0] [0 2] [0 1] [7] = cons(U,concat(V,Y)) concat(leaf(),Y) = [1 0] Y + [1] [0 1] [0] > [1 0] Y + [0] [0 1] [0] = Y lessleaves(X,leaf()) = [1 0] X + [4] [4 0] [1] > [0] [0] = false() lessleaves(cons(U,V),cons(W,Z)) = [1 3] U + [1 0] V + [4 12] W + [4 2] Z + [14] [4 12] [4 0] [0 0] [0 0] [1] > [1 2] U + [1 0] V + [4 12] W + [4 2] Z + [0] [4 8] [4 0] [0 0] [0 0] [1] = lessleaves(concat(U,V),concat(W,Z)) lessleaves(leaf(),cons(W,Z)) = [4 12] W + [4 2] Z + [15] [0 0] [0 0] [5] > [0] [1] = true() Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))