* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules * Step 2: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} + Applied Processor: Bounds {initialAutomaton = perSymbol, enrichment = match} + Details: The problem is match-bounded by 3. The enriched problem is compatible with follwoing automaton. a_0() -> 1 a_1() -> 5 a_1() -> 8 a_1() -> 9 a_2() -> 12 a__f_0(1) -> 2 a__f_0(3) -> 2 a__f_0(4) -> 2 a__f_1(6) -> 2 a__f_1(9) -> 5 a__f_1(9) -> 9 a__f_2(10) -> 5 a__f_2(10) -> 9 f_0(1) -> 3 f_0(3) -> 3 f_0(4) -> 3 f_1(1) -> 2 f_1(3) -> 2 f_1(4) -> 2 f_1(8) -> 7 f_2(6) -> 2 f_2(9) -> 5 f_2(9) -> 9 f_2(12) -> 11 f_3(10) -> 5 f_3(10) -> 9 g_0(1) -> 4 g_0(3) -> 4 g_0(4) -> 4 g_1(1) -> 5 g_1(1) -> 9 g_1(3) -> 5 g_1(3) -> 9 g_1(4) -> 5 g_1(4) -> 9 g_1(7) -> 6 g_2(11) -> 10 mark_0(1) -> 5 mark_0(3) -> 5 mark_0(4) -> 5 mark_1(1) -> 9 mark_1(3) -> 9 mark_1(4) -> 9 * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))