WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: divp(x,y) -> =(rem(x,y),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(x))) -> prime1(s(s(x)),s(x)) prime1(x,0()) -> false() prime1(x,s(0())) -> true() prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y))) - Signature: {divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not ,rem,s,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(and) = {1,2}, uargs(not) = {1} Following symbols are considered usable: {divp,prime,prime1} TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [0] p(divp) = [2] p(false) = [0] p(not) = [1] x1 + [0] p(prime) = [2] x1 + [1] p(prime1) = [2] x2 + [2] p(rem) = [0] p(s) = [1] x1 + [2] p(true) = [5] Following rules are strictly oriented: divp(x,y) = [2] > [0] = =(rem(x,y),0()) prime(0()) = [1] > [0] = false() prime(s(0())) = [5] > [0] = false() prime(s(s(x))) = [2] x + [9] > [2] x + [6] = prime1(s(s(x)),s(x)) prime1(x,0()) = [2] > [0] = false() prime1(x,s(0())) = [6] > [5] = true() prime1(x,s(s(y))) = [2] y + [10] > [2] y + [8] = and(not(divp(s(s(y)),x)),prime1(x,s(y))) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))