WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__g(x)} = activate(n__g(x)) ->^+ g(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + 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(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = [10] p(activate) = [1] x1 + [9] p(f) = [1] x1 + [10] p(g) = [1] x1 + [0] p(n__a) = [1] p(n__f) = [1] x1 + [7] p(n__g) = [1] x1 + [0] Following rules are strictly oriented: a() = [10] > [1] = n__a() activate(X) = [1] X + [9] > [1] X + [0] = X activate(n__f(X)) = [1] X + [16] > [1] X + [10] = f(X) f(X) = [1] X + [10] > [1] X + [7] = n__f(X) Following rules are (at-least) weakly oriented: activate(n__a()) = [10] >= [10] = a() activate(n__g(X)) = [1] X + [9] >= [1] X + [9] = g(activate(X)) f(n__f(n__a())) = [18] >= [18] = f(n__g(n__f(n__a()))) g(X) = [1] X + [0] >= [1] X + [0] = n__g(X) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__a()) -> a() activate(n__g(X)) -> g(activate(X)) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__f(X)) -> f(X) f(X) -> n__f(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = 2 p(activate) = 8*x1 p(f) = 0 p(g) = x1 p(n__a) = 2 p(n__f) = 0 p(n__g) = x1 Following rules are strictly oriented: activate(n__a()) = 16 > 2 = a() Following rules are (at-least) weakly oriented: a() = 2 >= 2 = n__a() activate(X) = 8*X >= X = X activate(n__f(X)) = 0 >= 0 = f(X) activate(n__g(X)) = 8*X >= 8*X = g(activate(X)) f(X) = 0 >= 0 = n__f(X) f(n__f(n__a())) = 0 >= 0 = f(n__g(n__f(n__a()))) g(X) = X >= X = n__g(X) ** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__g(X)) -> g(activate(X)) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) f(X) -> n__f(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + 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(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = [8] p(activate) = [4] x1 + [0] p(f) = [0] p(g) = [1] x1 + [2] p(n__a) = [2] p(n__f) = [0] p(n__g) = [1] x1 + [2] Following rules are strictly oriented: activate(n__g(X)) = [4] X + [8] > [4] X + [2] = g(activate(X)) Following rules are (at-least) weakly oriented: a() = [8] >= [2] = n__a() activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__a()) = [8] >= [8] = a() activate(n__f(X)) = [0] >= [0] = f(X) f(X) = [0] >= [0] = n__f(X) f(n__f(n__a())) = [0] >= [0] = f(n__g(n__f(n__a()))) g(X) = [1] X + [2] >= [1] X + [2] = n__g(X) ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + 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(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = [6] p(activate) = [4] x1 + [2] p(f) = [0] p(g) = [1] x1 + [5] p(n__a) = [1] p(n__f) = [0] p(n__g) = [1] x1 + [4] Following rules are strictly oriented: g(X) = [1] X + [5] > [1] X + [4] = n__g(X) Following rules are (at-least) weakly oriented: a() = [6] >= [1] = n__a() activate(X) = [4] X + [2] >= [1] X + [0] = X activate(n__a()) = [6] >= [6] = a() activate(n__f(X)) = [2] >= [0] = f(X) activate(n__g(X)) = [4] X + [18] >= [4] X + [7] = g(activate(X)) f(X) = [0] >= [0] = n__f(X) f(n__f(n__a())) = [0] >= [0] = f(n__g(n__f(n__a()))) ** Step 1.b:5: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(n__f(n__a())) -> f(n__g(n__f(n__a()))) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = [0] [1] [4] p(activate) = [4 0 0] [7] [0 1 0] x1 + [1] [5 0 1] [5] p(f) = [0 4 2] [0] [0 0 0] x1 + [0] [0 0 1] [2] p(g) = [1 3 0] [5] [0 0 0] x1 + [0] [1 0 0] [4] p(n__a) = [0] [1] [2] p(n__f) = [0 2 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(n__g) = [1 2 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: f(n__f(n__a())) = [2] [0] [3] > [0] [0] [2] = f(n__g(n__f(n__a()))) Following rules are (at-least) weakly oriented: a() = [0] [1] [4] >= [0] [1] [2] = n__a() activate(X) = [4 0 0] [7] [0 1 0] X + [1] [5 0 1] [5] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X activate(n__a()) = [7] [2] [7] >= [0] [1] [4] = a() activate(n__f(X)) = [0 8 4] [7] [0 0 0] X + [1] [0 10 5] [6] >= [0 4 2] [0] [0 0 0] X + [0] [0 0 1] [2] = f(X) activate(n__g(X)) = [4 8 0] [15] [0 0 0] X + [1] [5 10 0] [15] >= [4 3 0] [15] [0 0 0] X + [0] [4 0 0] [11] = g(activate(X)) f(X) = [0 4 2] [0] [0 0 0] X + [0] [0 0 1] [2] >= [0 2 1] [0] [0 0 0] X + [0] [0 0 0] [1] = n__f(X) g(X) = [1 3 0] [5] [0 0 0] X + [0] [1 0 0] [4] >= [1 2 0] [2] [0 0 0] X + [0] [0 0 0] [0] = n__g(X) ** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(X) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(n__f(n__a())) -> f(n__g(n__f(n__a()))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))