* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(X) -> g(h(f(X))) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> h(mark(X)) - Signature: {a__f/1,mark/1} / {f/1,g/1,h/1} - Obligation: runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h} + 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(X) -> g(h(f(X))) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> h(mark(X)) - Signature: {a__f/1,mark/1} / {f/1,g/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h} + Applied Processor: Bounds {initialAutomaton = perSymbol, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. a__f_0(2) -> 1 a__f_0(3) -> 1 a__f_0(4) -> 1 a__f_1(7) -> 5 a__f_1(7) -> 7 f_0(2) -> 2 f_0(3) -> 2 f_0(4) -> 2 f_1(2) -> 1 f_1(3) -> 1 f_1(4) -> 1 f_2(7) -> 5 f_2(7) -> 7 g_0(2) -> 3 g_0(3) -> 3 g_0(4) -> 3 g_1(2) -> 5 g_1(2) -> 7 g_1(3) -> 5 g_1(3) -> 7 g_1(4) -> 5 g_1(4) -> 7 g_1(6) -> 1 g_2(8) -> 5 g_2(8) -> 7 h_0(2) -> 4 h_0(3) -> 4 h_0(4) -> 4 h_1(1) -> 6 h_1(7) -> 5 h_1(7) -> 7 h_2(5) -> 8 mark_0(2) -> 5 mark_0(3) -> 5 mark_0(4) -> 5 mark_1(2) -> 7 mark_1(3) -> 7 mark_1(4) -> 7 * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X) -> f(X) a__f(X) -> g(h(f(X))) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> h(mark(X)) - Signature: {a__f/1,mark/1} / {f/1,g/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))