* Step 1: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,cons,from} and constructors {n__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: none Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [2] x_1 + [3] p(activate) = [8] x_1 + [10] p(cons) = [1] x_1 + [8] x_2 + [4] p(from) = [1] x_1 + [4] p(n__cons) = [1] x_1 + [1] x_2 + [0] p(n__from) = [1] x_1 + [0] p(s) = [0] Following rules are strictly oriented: 2nd(cons(X,n__cons(Y,Z))) = [2] X + [16] Y + [16] Z + [11] > [8] Y + [10] = activate(Y) activate(X) = [8] X + [10] > [1] X + [0] = X activate(n__cons(X1,X2)) = [8] X1 + [8] X2 + [10] > [1] X1 + [8] X2 + [4] = cons(X1,X2) activate(n__from(X)) = [8] X + [10] > [1] X + [4] = from(X) cons(X1,X2) = [1] X1 + [8] X2 + [4] > [1] X1 + [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] X + [4] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: from(X) = [1] X + [4] >= [1] X + [4] = cons(X,n__from(s(X))) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(s(X))) - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,cons,from} and constructors {n__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: none Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [9] x1 + [7] p(activate) = [8] x1 + [1] p(cons) = [4] x1 + [2] x2 + [1] p(from) = [8] x1 + [15] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [2] p(s) = [0] Following rules are strictly oriented: from(X) = [8] X + [15] > [4] X + [5] = cons(X,n__from(s(X))) Following rules are (at-least) weakly oriented: 2nd(cons(X,n__cons(Y,Z))) = [36] X + [18] Y + [18] Z + [16] >= [8] Y + [1] = activate(Y) activate(X) = [8] X + [1] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [8] X1 + [8] X2 + [1] >= [4] X1 + [2] X2 + [1] = cons(X1,X2) activate(n__from(X)) = [8] X + [17] >= [8] X + [15] = from(X) cons(X1,X2) = [4] X1 + [2] X2 + [1] >= [1] X1 + [1] X2 + [0] = n__cons(X1,X2) from(X) = [8] X + [15] >= [1] X + [2] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {2nd,activate,cons,from} and constructors {n__cons,n__from,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))