```* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
f(g(x),g(y)) -> f(p(f(g(x),s(y))),g(s(p(x))))
g(s(p(x))) -> p(x)
p(0()) -> g(0())
- Signature:
{f/2,g/1,p/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f,g,p} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following weak dependency pairs:

Strict DPs
f#(g(x),g(y)) -> c_1(f#(p(f(g(x),s(y))),g(s(p(x)))))
g#(s(p(x))) -> c_2(p#(x))
p#(0()) -> c_3(g#(0()))
Weak DPs

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

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

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

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