* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(g(x),x,y) -> y
        - Signature:
            {f/3} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,x,y) -> c_1()
          f#(x,y,y) -> c_2()
          f#(x,y,g(y)) -> c_3()
          f#(g(x),x,y) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,x,y) -> c_1()
            f#(x,y,y) -> c_2()
            f#(x,y,g(y)) -> c_3()
            f#(g(x),x,y) -> c_4()
        - Weak TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(g(x),x,y) -> y
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,x,y) -> c_1()
          f#(x,y,y) -> c_2()
          f#(x,y,g(y)) -> c_3()
          f#(g(x),x,y) -> c_4()
* Step 3: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,x,y) -> c_1()
            f#(x,y,y) -> c_2()
            f#(x,y,g(y)) -> c_3()
            f#(g(x),x,y) -> c_4()
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:f#(x,x,y) -> c_1()
             
          
          2:S:f#(x,y,y) -> c_2()
             
          
          3:S:f#(x,y,g(y)) -> c_3()
             
          
          4:S:f#(g(x),x,y) -> c_4()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))