```* 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))
```