WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: even(Cons(x,Nil())) -> False() even(Cons(x',Cons(x,xs))) -> even(xs) even(Nil()) -> True() goal(x,y) -> and(lte(x,y),even(x)) lte(Cons(x,xs),Nil()) -> False() lte(Cons(x',xs'),Cons(x,xs)) -> lte(xs',xs) lte(Nil(),y) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() - Signature: {and/2,even/1,goal/2,lte/2,notEmpty/1} / {Cons/2,False/0,Nil/0,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {and,even,goal,lte,notEmpty} 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(and) = {1,2} Following symbols are considered usable: {and,even,goal,lte,notEmpty} TcT has computed the following interpretation: p(Cons) = [1] x2 + [2] p(False) = [0] p(Nil) = [8] p(True) = [0] p(and) = [2] x1 + [8] x2 + [0] p(even) = [1] x1 + [0] p(goal) = [10] x1 + [8] x2 + [12] p(lte) = [1] x1 + [5] p(notEmpty) = [2] Following rules are strictly oriented: even(Cons(x,Nil())) = [10] > [0] = False() even(Cons(x',Cons(x,xs))) = [1] xs + [4] > [1] xs + [0] = even(xs) even(Nil()) = [8] > [0] = True() goal(x,y) = [10] x + [8] y + [12] > [10] x + [10] = and(lte(x,y),even(x)) lte(Cons(x,xs),Nil()) = [1] xs + [7] > [0] = False() lte(Cons(x',xs'),Cons(x,xs)) = [1] xs' + [7] > [1] xs' + [5] = lte(xs',xs) lte(Nil(),y) = [13] > [0] = True() notEmpty(Cons(x,xs)) = [2] > [0] = True() notEmpty(Nil()) = [2] > [0] = False() Following rules are (at-least) weakly oriented: and(False(),False()) = [0] >= [0] = False() and(False(),True()) = [0] >= [0] = False() and(True(),False()) = [0] >= [0] = False() and(True(),True()) = [0] >= [0] = True() WORST_CASE(?,O(n^1))