* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),s(y)) -> f(x,s(c(s(y)))) - Signature: {f/2} / {c/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {c,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(f) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(c) = [1] x1 + [14] p(f) = [1] x2 + [0] p(s) = [1] x1 + [2] Following rules are strictly oriented: f(x,c(y)) = [1] y + [14] > [1] y + [2] = f(x,s(f(y,y))) Following rules are (at-least) weakly oriented: f(s(x),s(y)) = [1] y + [2] >= [1] y + [18] = f(x,s(c(s(y)))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(s(x),s(y)) -> f(x,s(c(s(y)))) - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) - Signature: {f/2} / {c/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {c,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))): The following argument positions are considered usable: uargs(f) = {2}, uargs(s) = {1} Following symbols are considered usable: {f} TcT has computed the following interpretation: p(c) = [1 2 0] [0] [0 0 2] x_1 + [1] [0 0 0] [4] p(f) = [0 0 3] [1 2 0] [0] [4 0 0] x_1 + [0 0 0] x_2 + [0] [0 0 0] [0 0 0] [0] p(s) = [1 0 0] [2] [0 0 1] x_1 + [0] [0 0 1] [4] Following rules are strictly oriented: f(s(x),s(y)) = [0 0 3] [1 0 2] [14] [4 0 0] x + [0 0 0] y + [8] [0 0 0] [0 0 0] [0] > [0 0 3] [1 0 2] [12] [4 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = f(x,s(c(s(y)))) Following rules are (at-least) weakly oriented: f(x,c(y)) = [0 0 3] [1 2 4] [2] [4 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] >= [0 0 3] [1 2 3] [2] [4 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = f(x,s(f(y,y))) * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(x,c(y)) -> f(x,s(f(y,y))) f(s(x),s(y)) -> f(x,s(c(s(y)))) - Signature: {f/2} / {c/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {c,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))