```* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
f(X,X) -> c(X)
f(X,c(X)) -> f(s(X),X)
f(s(X),X) -> f(X,a(X))
- Signature:
{f/2} / {a/1,c/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {a,c,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
f#(X,X) -> c_1()
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,X) -> c_1()
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
- Weak TRS:
f(X,X) -> c(X)
f(X,c(X)) -> f(s(X),X)
f(s(X),X) -> f(X,a(X))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(X,X) -> c_1()
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
* Step 3: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X,X) -> c_1()
f#(X,c(X)) -> c_2(f#(s(X),X))
f#(s(X),X) -> c_3(f#(X,a(X)))
- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:f#(X,X) -> c_1()

2:S:f#(X,c(X)) -> c_2(f#(s(X),X))
-->_1 f#(s(X),X) -> c_3(f#(X,a(X))):3

3:S:f#(s(X),X) -> c_3(f#(X,a(X)))

The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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