WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons ,false,nil,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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} Following symbols are considered usable: {append,hd,ifappend,is_empty,tl} TcT has computed the following interpretation: p(append) = [4] x1 + [1] x2 + [4] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [10] p(hd) = [1] x1 + [10] p(ifappend) = [1] x2 + [4] x3 + [2] p(is_empty) = [9] x1 + [7] p(nil) = [1] p(tl) = [12] x1 + [6] p(true) = [1] Following rules are strictly oriented: append(l1,l2) = [4] l1 + [1] l2 + [4] > [4] l1 + [1] l2 + [2] = ifappend(l1,l2,l1) hd(cons(x,l)) = [1] l + [1] x + [11] > [1] x + [0] = x ifappend(l1,l2,cons(x,l)) = [4] l + [1] l2 + [4] x + [6] > [4] l + [1] l2 + [1] x + [5] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [1] l2 + [6] > [1] l2 + [0] = l2 is_empty(cons(x,l)) = [9] l + [9] x + [16] > [10] = false() is_empty(nil()) = [16] > [1] = true() tl(cons(x,l)) = [12] l + [12] x + [18] > [1] l + [0] = l Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))