WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1} / {0/0,1/0,c/1,d/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1} / {0/0,1/0,c/1,d/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          g#(c(x)) -> c_3()
          g#(c(0())) -> c_4(g#(d(1())))
          g#(c(1())) -> c_5(g#(d(0())))
          g#(d(x)) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(c(x)) -> c_3()
            g#(c(0())) -> c_4(g#(d(1())))
            g#(c(1())) -> c_5(g#(d(0())))
            g#(d(x)) -> c_6()
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,6}
        by application of
          Pre({3,6}) = {4,5}.
        Here rules are labelled as follows:
          1: f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          2: f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          3: g#(c(x)) -> c_3()
          4: g#(c(0())) -> c_4(g#(d(1())))
          5: g#(c(1())) -> c_5(g#(d(0())))
          6: g#(d(x)) -> c_6()
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(c(0())) -> c_4(g#(d(1())))
            g#(c(1())) -> c_5(g#(d(0())))
        - Weak DPs:
            g#(c(x)) -> c_3()
            g#(d(x)) -> c_6()
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4}
        by application of
          Pre({3,4}) = {}.
        Here rules are labelled as follows:
          1: f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          2: f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          3: g#(c(0())) -> c_4(g#(d(1())))
          4: g#(c(1())) -> c_5(g#(d(0())))
          5: g#(c(x)) -> c_3()
          6: g#(d(x)) -> c_6()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
        - Weak DPs:
            g#(c(x)) -> c_3()
            g#(c(0())) -> c_4(g#(d(1())))
            g#(c(1())) -> c_5(g#(d(0())))
            g#(d(x)) -> c_6()
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          2:S:f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          3:W:g#(c(x)) -> c_3()
             
          
          4:W:g#(c(0())) -> c_4(g#(d(1())))
             -->_1 g#(d(x)) -> c_6():6
          
          5:W:g#(c(1())) -> c_5(g#(d(0())))
             -->_1 g#(d(x)) -> c_6():6
          
          6:W:g#(d(x)) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: g#(c(1())) -> c_5(g#(d(0())))
          4: g#(c(0())) -> c_4(g#(d(1())))
          6: g#(d(x)) -> c_6()
          3: g#(c(x)) -> c_3()
* Step 6: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          2:S:f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(f(x)) -> c_1(f#(x))
          f#(f(x)) -> c_2(f#(x))
* Step 7: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(f(x)) -> c_2(f#(x))
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(0())) -> g(d(1()))
            g(c(1())) -> g(d(0()))
            g(d(x)) -> x
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        UsableRules
    + Details:
        No rule is usable, rules are removed from the input problem.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))