```* 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))
```