WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> a__c(d(X)) a__h(X) -> h(X) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,a__h,mark} and constructors {c,d,f,g,h} + 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(a__f) = {1}, uargs(a__h) = {1} Following symbols are considered usable: {a__c,a__f,a__h,mark} TcT has computed the following interpretation: p(a__c) = [2] p(a__f) = [1] x1 + [9] p(a__h) = [1] x1 + [10] p(c) = [1] p(d) = [1] p(f) = [1] x1 + [4] p(g) = [4] p(h) = [1] x1 + [4] p(mark) = [4] x1 + [0] Following rules are strictly oriented: a__c(X) = [2] > [1] = c(X) a__c(X) = [2] > [1] = d(X) a__f(X) = [1] X + [9] > [1] X + [4] = f(X) a__f(f(X)) = [1] X + [13] > [2] = a__c(f(g(f(X)))) a__h(X) = [1] X + [10] > [2] = a__c(d(X)) a__h(X) = [1] X + [10] > [1] X + [4] = h(X) mark(c(X)) = [4] > [2] = a__c(X) mark(d(X)) = [4] > [1] = d(X) mark(f(X)) = [4] X + [16] > [4] X + [9] = a__f(mark(X)) mark(g(X)) = [16] > [4] = g(X) mark(h(X)) = [4] X + [16] > [4] X + [10] = a__h(mark(X)) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))