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

Strict DPs
p#(f(f(x))) -> c_1(q#(f(g(x))))
p#(g(g(x))) -> c_2(q#(g(f(x))))
q#(f(f(x))) -> c_3(p#(f(g(x))))
q#(g(g(x))) -> c_4(p#(g(f(x))))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
p#(f(f(x))) -> c_1(q#(f(g(x))))
p#(g(g(x))) -> c_2(q#(g(f(x))))
q#(f(f(x))) -> c_3(p#(f(g(x))))
q#(g(g(x))) -> c_4(p#(g(f(x))))
- Weak TRS:
p(f(f(x))) -> q(f(g(x)))
p(g(g(x))) -> q(g(f(x)))
q(f(f(x))) -> p(f(g(x)))
q(g(g(x))) -> p(g(f(x)))
- Signature:
{p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
p#(f(f(x))) -> c_1(q#(f(g(x))))
p#(g(g(x))) -> c_2(q#(g(f(x))))
q#(f(f(x))) -> c_3(p#(f(g(x))))
q#(g(g(x))) -> c_4(p#(g(f(x))))
* Step 3: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
p#(f(f(x))) -> c_1(q#(f(g(x))))
p#(g(g(x))) -> c_2(q#(g(f(x))))
q#(f(f(x))) -> c_3(p#(f(g(x))))
q#(g(g(x))) -> c_4(p#(g(f(x))))
- Signature:
{p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:p#(f(f(x))) -> c_1(q#(f(g(x))))

2:S:p#(g(g(x))) -> c_2(q#(g(f(x))))

3:S:q#(f(f(x))) -> c_3(p#(f(g(x))))

4:S:q#(g(g(x))) -> c_4(p#(g(f(x))))

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

- Signature:
{p/1,q/1,p#/1,q#/1} / {f/1,g/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {p#,q#} and constructors {f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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