* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) after(0(),XS) -> XS after(s(N),cons(X,XS)) -> after(N,activate(XS)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,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(after) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(after) = [1] x2 + [11] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [1] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: after(0(),XS) = [1] XS + [11] > [1] XS + [0] = XS from(X) = [1] X + [1] > [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [1] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [1] = from(X) after(s(N),cons(X,XS)) = [1] XS + [11] >= [1] XS + [11] = after(N,activate(XS)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) after(s(N),cons(X,XS)) -> after(N,activate(XS)) - Weak TRS: after(0(),XS) -> XS from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(after) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x_1 + [4] p(after) = [7] x_1 + [1] x_2 + [12] p(cons) = [1] x_2 + [0] p(from) = [0] p(n__from) = [0] p(s) = [1] x_1 + [1] Following rules are strictly oriented: activate(X) = [1] X + [4] > [1] X + [0] = X activate(n__from(X)) = [4] > [0] = from(X) after(s(N),cons(X,XS)) = [7] N + [1] XS + [19] > [7] N + [1] XS + [16] = after(N,activate(XS)) Following rules are (at-least) weakly oriented: after(0(),XS) = [1] XS + [12] >= [1] XS + [0] = XS from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) after(0(),XS) -> XS after(s(N),cons(X,XS)) -> after(N,activate(XS)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,after,from} and constructors {0,cons,n__from,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))