WORST_CASE(?,O(1))
* Step 1: Sum 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(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            f(g(x),x,y) -> y
        - Signature:
            {f/3} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: 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(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            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#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
          f#(g(x),x,y) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation 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#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
            f#(g(x),x,y) -> c_5()
        - Weak TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            f(g(x),x,y) -> y
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,5}
        by application of
          Pre({1,2,3,5}) = {4}.
        Here rules are labelled as follows:
          1: f#(x,x,y) -> c_1()
          2: f#(x,y,y) -> c_2()
          3: f#(x,y,g(y)) -> c_3()
          4: f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
          5: f#(g(x),x,y) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
        - Weak 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_5()
        - Weak TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            f(g(x),x,y) -> y
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
             -->_2 f#(g(x),x,y) -> c_5():5
             -->_1 f#(g(x),x,y) -> c_5():5
             -->_2 f#(x,y,g(y)) -> c_3():4
             -->_1 f#(x,y,g(y)) -> c_3():4
             -->_2 f#(x,y,y) -> c_2():3
             -->_1 f#(x,y,y) -> c_2():3
             -->_2 f#(x,x,y) -> c_1():2
             -->_1 f#(x,x,y) -> c_1():2
             -->_2 f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v)):1
             -->_1 f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v)):1
          
          2:W:f#(x,x,y) -> c_1()
             
          
          3:W:f#(x,y,y) -> c_2()
             
          
          4:W:f#(x,y,g(y)) -> c_3()
             
          
          5:W:f#(g(x),x,y) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(x,x,y) -> c_1()
          3: f#(x,y,y) -> c_2()
          4: f#(x,y,g(y)) -> c_3()
          5: f#(g(x),x,y) -> c_5()
* Step 5: RemoveInapplicable WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
        - Weak TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            f(g(x),x,y) -> y
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
    + Applied Processor:
        RemoveInapplicable
    + Details:
        Only the nodes
          {}
        are reachable from nodes
          {}
        that start derivation from marked basic terms.
        The nodes not reachable are removed from the problem.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,x,y) -> x
            f(x,y,y) -> y
            f(x,y,g(y)) -> x
            f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
            f(g(x),x,y) -> y
        - Signature:
            {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
        - 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))