* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: g(x,s(y)) -> g(f(x,y),0()) g(0(),f(x,x)) -> x g(f(x,y),0()) -> f(g(x,0()),g(y,0())) g(s(x),y) -> g(f(x,y),0()) - Signature: {g/2} / {0/0,f/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: g(x,s(y)) -> g(f(x,y),0()) g(0(),f(x,x)) -> x g(f(x,y),0()) -> f(g(x,0()),g(y,0())) g(s(x),y) -> g(f(x,y),0()) - Signature: {g/2} / {0/0,f/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: g(x,0()){x -> f(x,y)} = g(f(x,y),0()) ->^+ f(g(x,0()),g(y,0())) = C[g(x,0()) = g(x,0()){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(x,s(y)) -> g(f(x,y),0()) g(0(),f(x,x)) -> x g(f(x,y),0()) -> f(g(x,0()),g(y,0())) g(s(x),y) -> g(f(x,y),0()) - Signature: {g/2} / {0/0,f/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. 0_0() -> 1 0_0() -> 2 0_1() -> 4 0_2() -> 9 0_2() -> 10 f_0(2,2) -> 1 f_0(2,2) -> 2 f_1(2,2) -> 3 f_1(2,4) -> 1 f_1(2,4) -> 2 f_1(2,9) -> 11 f_1(2,10) -> 12 f_1(5,6) -> 1 f_1(6,6) -> 5 f_1(6,6) -> 6 f_1(6,6) -> 7 f_1(6,6) -> 8 f_2(7,8) -> 1 f_2(8,13) -> 5 f_2(8,13) -> 6 f_2(8,13) -> 7 f_2(8,13) -> 8 g_0(2,2) -> 1 g_1(2,4) -> 5 g_1(2,4) -> 6 g_1(3,4) -> 1 g_1(4,4) -> 6 g_1(11,4) -> 7 g_1(12,4) -> 8 g_2(2,9) -> 7 g_2(2,10) -> 8 g_2(4,10) -> 13 g_2(9,10) -> 13 g_2(10,10) -> 13 s_0(2) -> 1 s_0(2) -> 2 2 -> 1 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: g(x,s(y)) -> g(f(x,y),0()) g(0(),f(x,x)) -> x g(f(x,y),0()) -> f(g(x,0()),g(y,0())) g(s(x),y) -> g(f(x,y),0()) - Signature: {g/2} / {0/0,f/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {g} and constructors {0,f,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))