WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: g(x,h(y,z)) -> h(g(x,y),z) g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: g(x,h(y,z)) -> h(g(x,y),z) g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: g(x,h(y,z)) -> h(g(x,y),z) g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: g(x,y){y -> h(y,z)} = g(x,h(y,z)) ->^+ h(g(x,y),z) = C[g(x,y) = g(x,y){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(x,h(y,z)) -> h(g(x,y),z) g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(f) = {2}, uargs(h) = {1} Following symbols are considered usable: {g} TcT has computed the following interpretation: p(f) = [1] x2 + [0] p(g) = [5] x2 + [6] p(h) = [1] x1 + [1] Following rules are strictly oriented: g(x,h(y,z)) = [5] y + [11] > [5] y + [7] = h(g(x,y),z) Following rules are (at-least) weakly oriented: g(f(x,y),z) = [5] z + [6] >= [5] z + [6] = f(x,g(y,z)) g(h(x,y),z) = [5] z + [6] >= [5] z + [6] = g(x,f(y,z)) ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Weak TRS: g(x,h(y,z)) -> h(g(x,y),z) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(f) = {2}, uargs(h) = {1} Following symbols are considered usable: {g} TcT has computed the following interpretation: p(f) = [1] x2 + [8] p(g) = [2] x1 + [1] x2 + [0] p(h) = [1] x1 + [8] Following rules are strictly oriented: g(f(x,y),z) = [2] y + [1] z + [16] > [2] y + [1] z + [8] = f(x,g(y,z)) g(h(x,y),z) = [2] x + [1] z + [16] > [2] x + [1] z + [8] = g(x,f(y,z)) Following rules are (at-least) weakly oriented: g(x,h(y,z)) = [2] x + [1] y + [8] >= [2] x + [1] y + [8] = h(g(x,y),z) ** Step 1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: g(x,h(y,z)) -> h(g(x,y),z) g(f(x,y),z) -> f(x,g(y,z)) g(h(x,y),z) -> g(x,f(y,z)) - Signature: {g/2} / {f/2,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {f,h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))