WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first ,n__from,n__s,nil} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first ,n__from,n__s,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [5] x1 + [2] p(cons) = [1] x1 + [5] p(first) = [1] x1 + [1] x2 + [8] p(from) = [1] x1 + [6] p(n__first) = [1] x1 + [1] x2 + [5] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(nil) = [0] p(s) = [1] x1 + [3] Following rules are strictly oriented: activate(X) = [5] X + [2] > [1] X + [0] = X activate(n__first(X1,X2)) = [5] X1 + [5] X2 + [27] > [5] X1 + [5] X2 + [12] = first(activate(X1),activate(X2)) activate(n__s(X)) = [5] X + [7] > [5] X + [5] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [5] = n__first(X1,X2) first(0(),X) = [1] X + [8] > [0] = nil() from(X) = [1] X + [6] > [1] X + [5] = cons(X,n__from(n__s(X))) from(X) = [1] X + [6] > [1] X + [0] = n__from(X) s(X) = [1] X + [3] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: activate(n__from(X)) = [5] X + [2] >= [5] X + [8] = from(activate(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__from(X)) -> from(activate(X)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first ,n__from,n__s,nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,first,from,s} TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [0] p(cons) = [1] x2 + [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [2] p(n__first) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(n__from(X)) = [2] X + [4] > [2] X + [2] = from(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = first(activate(X1),activate(X2)) activate(n__s(X)) = [2] X + [0] >= [2] X + [0] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [0] >= [0] = nil() from(X) = [1] X + [2] >= [1] X + [2] = cons(X,n__from(n__s(X))) from(X) = [1] X + [2] >= [1] X + [2] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first ,n__from,n__s,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))