* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: f(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(c(1())) -> g(d(h(0()))) g(c(h(0()))) -> g(d(1())) g(d(x)) -> x g(h(x)) -> g(x) - Signature: {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(c(1())) -> g(d(h(0()))) g(c(h(0()))) -> g(d(1())) g(d(x)) -> x g(h(x)) -> g(x) - Signature: {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: g(x){x -> h(x)} = g(h(x)) ->^+ g(x) = C[g(x) = g(x){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(c(1())) -> g(d(h(0()))) g(c(h(0()))) -> g(d(1())) g(d(x)) -> x g(h(x)) -> g(x) - Signature: {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. 0_0() -> 1 0_0() -> 2 0_1() -> 5 1_0() -> 1 1_0() -> 2 1_1() -> 1 1_1() -> 4 c_0(2) -> 1 c_0(2) -> 2 d_0(2) -> 1 d_0(2) -> 2 d_1(4) -> 3 f_0(2) -> 1 g_0(2) -> 1 g_1(2) -> 1 g_1(3) -> 1 h_0(2) -> 1 h_0(2) -> 2 h_1(5) -> 1 h_1(5) -> 4 2 -> 1 4 -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(f(x)) -> f(c(f(x))) f(f(x)) -> f(d(f(x))) g(c(x)) -> x g(c(1())) -> g(d(h(0()))) g(c(h(0()))) -> g(d(1())) g(d(x)) -> x g(h(x)) -> g(x) - Signature: {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))