* Step 1: WeightGap 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: runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil ,s} + 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(cons) = {2}, uargs(n__first) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [4] p(cons) = [1] x2 + [0] p(first) = [1] x2 + [0] p(from) = [0] p(n__first) = [1] x2 + [0] p(n__from) = [0] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [4] > [1] X + [0] = X activate(n__first(X1,X2)) = [1] X2 + [4] > [1] X2 + [0] = first(X1,X2) activate(n__from(X)) = [4] > [0] = from(X) Following rules are (at-least) weakly oriented: first(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [0] >= [0] = nil() first(s(X),cons(Y,Z)) = [1] Z + [0] >= [1] Z + [4] = cons(Y,n__first(X,activate(Z))) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 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(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) - Signature: {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil ,s} + 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(cons) = {2}, uargs(n__first) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [10] x1 + [8] p(cons) = [1] x2 + [0] p(first) = [8] x1 + [10] x2 + [0] p(from) = [2] p(n__first) = [1] x1 + [1] x2 + [1] p(n__from) = [0] p(nil) = [0] p(s) = [1] x1 + [2] Following rules are strictly oriented: first(s(X),cons(Y,Z)) = [8] X + [10] Z + [16] > [1] X + [10] Z + [9] = cons(Y,n__first(X,activate(Z))) from(X) = [2] > [0] = cons(X,n__from(s(X))) from(X) = [2] > [0] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [10] X + [8] >= [1] X + [0] = X activate(n__first(X1,X2)) = [10] X1 + [10] X2 + [18] >= [8] X1 + [10] X2 + [0] = first(X1,X2) activate(n__from(X)) = [8] >= [2] = from(X) first(X1,X2) = [8] X1 + [10] X2 + [0] >= [1] X1 + [1] X2 + [1] = n__first(X1,X2) first(0(),X) = [10] X + [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__from(X)) -> from(X) 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: runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil ,s} + 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(cons) = {2}, uargs(n__first) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [2] x1 + [8] p(cons) = [1] x2 + [1] p(first) = [2] x2 + [7] p(from) = [4] p(n__first) = [1] x2 + [0] p(n__from) = [0] p(nil) = [1] p(s) = [1] x1 + [8] Following rules are strictly oriented: first(X1,X2) = [2] X2 + [7] > [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [2] X + [7] > [1] = nil() Following rules are (at-least) weakly oriented: activate(X) = [2] X + [8] >= [1] X + [0] = X activate(n__first(X1,X2)) = [2] X2 + [8] >= [2] X2 + [7] = first(X1,X2) activate(n__from(X)) = [8] >= [4] = from(X) first(s(X),cons(Y,Z)) = [2] Z + [9] >= [2] Z + [9] = cons(Y,n__first(X,activate(Z))) from(X) = [4] >= [1] = cons(X,n__from(s(X))) from(X) = [4] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * 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: 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))