WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(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(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first ,n__from,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(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(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first ,n__from,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__first) = {2} Following symbols are considered usable: {activate,first,from} TcT has computed the following interpretation: p(0) = 4 p(activate) = 8*x1 p(cons) = x2 p(first) = 2 + 8*x2 p(from) = 0 p(n__first) = 2 + x2 p(n__from) = 0 p(nil) = 2 p(s) = 2 Following rules are strictly oriented: activate(n__first(X1,X2)) = 16 + 8*X2 > 2 + 8*X2 = first(X1,X2) Following rules are (at-least) weakly oriented: activate(X) = 8*X >= X = X activate(n__from(X)) = 0 >= 0 = from(X) first(X1,X2) = 2 + 8*X2 >= 2 + X2 = n__first(X1,X2) first(0(),X) = 2 + 8*X >= 2 = nil() first(s(X),cons(Y,Z)) = 2 + 8*Z >= 2 + 8*Z = cons(Y,n__first(X,activate(Z))) from(X) = 0 >= 0 = cons(X,n__from(s(X))) from(X) = 0 >= 0 = n__from(X) * Step 3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(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(s(X))) from(X) -> n__from(X) - Weak TRS: activate(n__first(X1,X2)) -> first(X1,X2) - Signature: {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first ,n__from,nil,s} + 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(cons) = {2}, uargs(n__first) = {2} Following symbols are considered usable: {activate,first,from} TcT has computed the following interpretation: p(0) = [2] p(activate) = [1] x1 + [6] p(cons) = [1] x2 + [0] p(first) = [1] x1 + [1] x2 + [5] p(from) = [6] p(n__first) = [1] x1 + [1] x2 + [0] p(n__from) = [1] p(nil) = [0] p(s) = [1] x1 + [2] Following rules are strictly oriented: activate(X) = [1] X + [6] > [1] X + [0] = X activate(n__from(X)) = [7] > [6] = from(X) first(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [7] > [0] = nil() first(s(X),cons(Y,Z)) = [1] X + [1] Z + [7] > [1] X + [1] Z + [6] = cons(Y,n__first(X,activate(Z))) from(X) = [6] > [1] = cons(X,n__from(s(X))) from(X) = [6] > [1] = n__from(X) Following rules are (at-least) weakly oriented: activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [5] = first(X1,X2) * Step 4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(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(s(X))) from(X) -> n__from(X) - Signature: {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first ,n__from,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))