```* Step 1: ToInnermost WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
a__c() -> a__f(g(c()))
a__c() -> c()
a__f(X) -> f(X)
a__f(g(X)) -> g(X)
mark(c()) -> a__c()
mark(f(X)) -> a__f(X)
mark(g(X)) -> g(X)
- Signature:
{a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
- Obligation:
runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {c,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: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
a__c() -> a__f(g(c()))
a__c() -> c()
a__f(X) -> f(X)
a__f(g(X)) -> g(X)
mark(c()) -> a__c()
mark(f(X)) -> a__f(X)
mark(g(X)) -> g(X)
- Signature:
{a__c/0,a__f/1,mark/1} / {c/0,f/1,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {c,f,g}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
a__c#() -> c_1(a__f#(g(c())))
a__c#() -> c_2()
a__f#(X) -> c_3()
a__f#(g(X)) -> c_4()
mark#(c()) -> c_5(a__c#())
mark#(f(X)) -> c_6(a__f#(X))
mark#(g(X)) -> c_7()
Weak DPs

and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__c#() -> c_1(a__f#(g(c())))
a__c#() -> c_2()
a__f#(X) -> c_3()
a__f#(g(X)) -> c_4()
mark#(c()) -> c_5(a__c#())
mark#(f(X)) -> c_6(a__f#(X))
mark#(g(X)) -> c_7()
- Weak TRS:
a__c() -> a__f(g(c()))
a__c() -> c()
a__f(X) -> f(X)
a__f(g(X)) -> g(X)
mark(c()) -> a__c()
mark(f(X)) -> a__f(X)
mark(g(X)) -> g(X)
- Signature:
{a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {c,f,g}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
a__c#() -> c_1(a__f#(g(c())))
a__c#() -> c_2()
a__f#(X) -> c_3()
a__f#(g(X)) -> c_4()
mark#(c()) -> c_5(a__c#())
mark#(f(X)) -> c_6(a__f#(X))
mark#(g(X)) -> c_7()
* Step 4: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a__c#() -> c_1(a__f#(g(c())))
a__c#() -> c_2()
a__f#(X) -> c_3()
a__f#(g(X)) -> c_4()
mark#(c()) -> c_5(a__c#())
mark#(f(X)) -> c_6(a__f#(X))
mark#(g(X)) -> c_7()
- Signature:
{a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {c,f,g}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:a__c#() -> c_1(a__f#(g(c())))
-->_1 a__f#(g(X)) -> c_4():4
-->_1 a__f#(X) -> c_3():3

2:S:a__c#() -> c_2()

3:S:a__f#(X) -> c_3()

4:S:a__f#(g(X)) -> c_4()

5:S:mark#(c()) -> c_5(a__c#())
-->_1 a__c#() -> c_2():2
-->_1 a__c#() -> c_1(a__f#(g(c()))):1

6:S:mark#(f(X)) -> c_6(a__f#(X))
-->_1 a__f#(g(X)) -> c_4():4
-->_1 a__f#(X) -> c_3():3

7:S:mark#(g(X)) -> c_7()

The dependency graph contains no loops, we remove all dependency pairs.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{a__c/0,a__f/1,mark/1,a__c#/0,a__f#/1,mark#/1} / {c/0,f/1,g/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {c,f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))
```