* Step 1: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(s(x),y,y) -> f(y,x,s(x)) g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {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(f) = [5] p(g) = [1] x_1 + [2] x_2 + [2] p(s) = [0] Following rules are strictly oriented: g(x,y) = [1] x + [2] y + [2] > [1] x + [0] = x g(x,y) = [1] x + [2] y + [2] > [1] y + [0] = y Following rules are (at-least) weakly oriented: f(s(x),y,y) = [5] >= [5] = f(y,x,s(x)) * Step 2: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(s(x),y,y) -> f(y,x,s(x)) - Weak TRS: g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {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(f) = [2] x_1 + [1] x_2 + [1] x_3 + [12] p(g) = [2] x_1 + [2] x_2 + [1] p(s) = [1] x_1 + [8] Following rules are strictly oriented: f(s(x),y,y) = [2] x + [2] y + [28] > [2] x + [2] y + [20] = f(y,x,s(x)) Following rules are (at-least) weakly oriented: g(x,y) = [2] x + [2] y + [1] >= [1] x + [0] = x g(x,y) = [2] x + [2] y + [1] >= [1] y + [0] = y * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(s(x),y,y) -> f(y,x,s(x)) g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))