* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,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: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,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: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. f(f(X)) -> c(n__f(n__g(n__f(X)))) All above mentioned rules can be savely removed. ** Step 1.b:2: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 4. The enriched problem is compatible with follwoing automaton. activate_0(2) -> 1 activate_0(2) -> 4 activate_1(2) -> 3 activate_1(2) -> 6 activate_2(1) -> 4 activate_2(2) -> 5 activate_3(3) -> 6 c_0(2) -> 1 c_0(2) -> 4 c_1(1) -> 1 c_1(1) -> 4 d_0(2) -> 1 d_0(2) -> 4 d_1(2) -> 1 d_1(2) -> 3 d_1(2) -> 4 d_1(2) -> 5 d_1(2) -> 6 d_1(3) -> 1 d_1(3) -> 4 d_2(2) -> 4 d_2(4) -> 1 d_2(4) -> 4 d_3(2) -> 4 d_3(2) -> 6 d_3(3) -> 4 d_3(4) -> 4 f_0(2) -> 1 f_0(2) -> 4 f_1(3) -> 1 f_1(3) -> 3 f_1(3) -> 4 f_1(3) -> 6 f_1(6) -> 5 f_2(5) -> 4 f_3(6) -> 4 f_3(6) -> 6 g_0(2) -> 1 g_0(2) -> 4 g_1(2) -> 1 g_1(2) -> 3 g_1(2) -> 4 g_1(2) -> 5 g_1(2) -> 6 g_2(2) -> 4 g_3(2) -> 4 g_3(2) -> 6 h_0(2) -> 1 h_0(2) -> 4 n__d_0(2) -> 1 n__d_0(2) -> 2 n__d_0(2) -> 3 n__d_0(2) -> 4 n__d_0(2) -> 5 n__d_0(2) -> 6 n__d_1(2) -> 1 n__d_1(2) -> 4 n__d_2(2) -> 1 n__d_2(2) -> 3 n__d_2(2) -> 4 n__d_2(2) -> 5 n__d_2(2) -> 6 n__d_2(3) -> 1 n__d_2(3) -> 4 n__d_3(2) -> 4 n__d_3(4) -> 1 n__d_3(4) -> 4 n__d_4(2) -> 4 n__d_4(2) -> 6 n__d_4(3) -> 4 n__d_4(4) -> 4 n__f_0(2) -> 1 n__f_0(2) -> 2 n__f_0(2) -> 3 n__f_0(2) -> 4 n__f_0(2) -> 5 n__f_0(2) -> 6 n__f_1(2) -> 1 n__f_1(2) -> 4 n__f_2(3) -> 1 n__f_2(3) -> 3 n__f_2(3) -> 4 n__f_2(3) -> 6 n__f_2(6) -> 5 n__f_3(5) -> 4 n__f_4(6) -> 4 n__f_4(6) -> 6 n__g_0(2) -> 1 n__g_0(2) -> 2 n__g_0(2) -> 3 n__g_0(2) -> 4 n__g_0(2) -> 5 n__g_0(2) -> 6 n__g_1(2) -> 1 n__g_1(2) -> 4 n__g_2(2) -> 1 n__g_2(2) -> 3 n__g_2(2) -> 4 n__g_2(2) -> 5 n__g_2(2) -> 6 n__g_3(2) -> 4 n__g_4(2) -> 4 n__g_4(2) -> 6 1 -> 4 2 -> 1 2 -> 3 2 -> 4 2 -> 5 2 -> 6 3 -> 6 ** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,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))