WORST_CASE(?,O(1))
* 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:
            innermost runtime complexity wrt. defined symbols {f,g,p} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(g(x),g(y)) -> c_1(f#(p(f(g(x),s(y))),g(s(p(x))))
                              ,p#(f(g(x),s(y)))
                              ,f#(g(x),s(y))
                              ,g#(x)
                              ,g#(s(p(x)))
                              ,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)))),p#(f(g(x),s(y))),f#(g(x),s(y)),g#(x),g#(s(p(x))),p#(x))
            g#(s(p(x))) -> c_2(p#(x))
            p#(0()) -> c_3(g#(0()))
        - Weak 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_1/6,c_2/1,c_3/1}
        - Obligation:
            innermost 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_1/6,c_2/1,c_3/1}
        - Obligation:
            innermost 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_1/6,c_2/1,c_3/1}
        - Obligation:
            innermost 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))