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(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(f(a())) -> c(n__f(n__g(n__f(n__a())))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,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(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(f(a())) -> c(n__f(n__g(n__f(n__a())))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,n__a,n__f,n__g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__f(x)} = activate(n__f(x)) ->^+ f(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(f(a())) -> c(n__f(n__g(n__f(n__a())))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,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(f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = 10 p(activate) = 13 + x1 p(c) = 0 p(f) = 8 + x1 p(g) = x1 p(n__a) = 6 p(n__f) = 8 + x1 p(n__g) = x1 Following rules are strictly oriented: a() = 10 > 6 = n__a() activate(X) = 13 + X > X = X activate(n__a()) = 19 > 10 = a() f(f(a())) = 26 > 0 = c(n__f(n__g(n__f(n__a())))) Following rules are (at-least) weakly oriented: activate(n__f(X)) = 21 + X >= 21 + X = f(activate(X)) activate(n__g(X)) = 13 + X >= 13 + X = g(activate(X)) f(X) = 8 + X >= 8 + X = n__f(X) g(X) = X >= X = n__g(X) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) g(X) -> n__g(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() f(f(a())) -> c(n__f(n__g(n__f(n__a())))) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,n__a,n__f,n__g} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = 0 p(activate) = 2*x1 p(c) = 0 p(f) = 3 + x1 p(g) = 4 + x1 p(n__a) = 0 p(n__f) = 3 + x1 p(n__g) = 2 + x1 Following rules are strictly oriented: activate(n__f(X)) = 6 + 2*X > 3 + 2*X = f(activate(X)) g(X) = 4 + X > 2 + X = n__g(X) Following rules are (at-least) weakly oriented: a() = 0 >= 0 = n__a() activate(X) = 2*X >= X = X activate(n__a()) = 0 >= 0 = a() activate(n__g(X)) = 4 + 2*X >= 4 + 2*X = g(activate(X)) f(X) = 3 + X >= 3 + X = n__f(X) f(f(a())) = 6 >= 0 = c(n__f(n__g(n__f(n__a())))) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(activate(X)) f(f(a())) -> c(n__f(n__g(n__f(n__a())))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,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(f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a,activate,f,g} TcT has computed the following interpretation: p(a) = 9 p(activate) = 8 + 2*x1 p(c) = x1 p(f) = 2 + x1 p(g) = 8 + x1 p(n__a) = 3 p(n__f) = 1 + x1 p(n__g) = 8 + x1 Following rules are strictly oriented: activate(n__g(X)) = 24 + 2*X > 16 + 2*X = g(activate(X)) f(X) = 2 + X > 1 + X = n__f(X) Following rules are (at-least) weakly oriented: a() = 9 >= 3 = n__a() activate(X) = 8 + 2*X >= X = X activate(n__a()) = 14 >= 9 = a() activate(n__f(X)) = 10 + 2*X >= 10 + 2*X = f(activate(X)) f(f(a())) = 13 >= 13 = c(n__f(n__g(n__f(n__a())))) g(X) = 8 + X >= 8 + X = n__g(X) ** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a() -> n__a() activate(X) -> X activate(n__a()) -> a() activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(activate(X)) f(X) -> n__f(X) f(f(a())) -> c(n__f(n__g(n__f(n__a())))) g(X) -> n__g(X) - Signature: {a/0,activate/1,f/1,g/1} / {c/1,n__a/0,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {c,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))