WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1} / {b/1,c/1,d/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,u,v,w} and constructors {b,c,d}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1} / {b/1,c/1,d/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,u,v,w} and constructors {b,c,d}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#(c(d(x))) -> c_1()
          u#(b(d(d(x)))) -> c_2()
          v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
          v#(a(c(x))) -> c_4(u#(b(d(x))))
          v#(c(x)) -> c_5()
          w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
          w#(a(c(x))) -> c_7(u#(b(d(x))))
          w#(c(x)) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            a#(c(d(x))) -> c_1()
            u#(b(d(d(x)))) -> c_2()
            v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
            v#(a(c(x))) -> c_4(u#(b(d(x))))
            v#(c(x)) -> c_5()
            w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
            w#(a(c(x))) -> c_7(u#(b(d(x))))
            w#(c(x)) -> c_8()
        - Weak TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/2,c_4/1,c_5/0,c_6/2,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,c,d}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,5,8}
        by application of
          Pre({1,2,5,8}) = {3,4,6,7}.
        Here rules are labelled as follows:
          1: a#(c(d(x))) -> c_1()
          2: u#(b(d(d(x)))) -> c_2()
          3: v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
          4: v#(a(c(x))) -> c_4(u#(b(d(x))))
          5: v#(c(x)) -> c_5()
          6: w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
          7: w#(a(c(x))) -> c_7(u#(b(d(x))))
          8: w#(c(x)) -> c_8()
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
            v#(a(c(x))) -> c_4(u#(b(d(x))))
            w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
            w#(a(c(x))) -> c_7(u#(b(d(x))))
        - Weak DPs:
            a#(c(d(x))) -> c_1()
            u#(b(d(d(x)))) -> c_2()
            v#(c(x)) -> c_5()
            w#(c(x)) -> c_8()
        - Weak TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/2,c_4/1,c_5/0,c_6/2,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,c,d}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3}.
        Here rules are labelled as follows:
          1: v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
          2: v#(a(c(x))) -> c_4(u#(b(d(x))))
          3: w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
          4: w#(a(c(x))) -> c_7(u#(b(d(x))))
          5: a#(c(d(x))) -> c_1()
          6: u#(b(d(d(x)))) -> c_2()
          7: v#(c(x)) -> c_5()
          8: w#(c(x)) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
            w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
        - Weak DPs:
            a#(c(d(x))) -> c_1()
            u#(b(d(d(x)))) -> c_2()
            v#(a(c(x))) -> c_4(u#(b(d(x))))
            v#(c(x)) -> c_5()
            w#(a(c(x))) -> c_7(u#(b(d(x))))
            w#(c(x)) -> c_8()
        - Weak TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/2,c_4/1,c_5/0,c_6/2,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,c,d}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
             -->_2 v#(a(c(x))) -> c_4(u#(b(d(x)))):5
             -->_2 v#(c(x)) -> c_5():6
             -->_1 u#(b(d(d(x)))) -> c_2():4
             -->_2 v#(a(a(x))) -> c_3(u#(v(x)),v#(x)):1
          
          2:S:w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
             -->_2 w#(a(c(x))) -> c_7(u#(b(d(x)))):7
             -->_2 w#(c(x)) -> c_8():8
             -->_1 u#(b(d(d(x)))) -> c_2():4
             -->_2 w#(a(a(x))) -> c_6(u#(w(x)),w#(x)):2
          
          3:W:a#(c(d(x))) -> c_1()
             
          
          4:W:u#(b(d(d(x)))) -> c_2()
             
          
          5:W:v#(a(c(x))) -> c_4(u#(b(d(x))))
             -->_1 u#(b(d(d(x)))) -> c_2():4
          
          6:W:v#(c(x)) -> c_5()
             
          
          7:W:w#(a(c(x))) -> c_7(u#(b(d(x))))
             -->_1 u#(b(d(d(x)))) -> c_2():4
          
          8:W:w#(c(x)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: a#(c(d(x))) -> c_1()
          8: w#(c(x)) -> c_8()
          7: w#(a(c(x))) -> c_7(u#(b(d(x))))
          6: v#(c(x)) -> c_5()
          5: v#(a(c(x))) -> c_4(u#(b(d(x))))
          4: u#(b(d(d(x)))) -> c_2()
* Step 6: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
            w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
        - Weak TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/2,c_4/1,c_5/0,c_6/2,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,c,d}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:v#(a(a(x))) -> c_3(u#(v(x)),v#(x))
             -->_2 v#(a(a(x))) -> c_3(u#(v(x)),v#(x)):1
          
          2:S:w#(a(a(x))) -> c_6(u#(w(x)),w#(x))
             -->_2 w#(a(a(x))) -> c_6(u#(w(x)),w#(x)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          v#(a(a(x))) -> c_3(v#(x))
          w#(a(a(x))) -> c_6(w#(x))
* Step 7: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            v#(a(a(x))) -> c_3(v#(x))
            w#(a(a(x))) -> c_6(w#(x))
        - Weak TRS:
            a(c(d(x))) -> c(x)
            u(b(d(d(x)))) -> b(x)
            v(a(a(x))) -> u(v(x))
            v(a(c(x))) -> u(b(d(x)))
            v(c(x)) -> b(x)
            w(a(a(x))) -> u(w(x))
            w(a(c(x))) -> u(b(d(x)))
            w(c(x)) -> b(x)
        - Signature:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,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:
            {a/1,u/1,v/1,w/1,a#/1,u#/1,v#/1,w#/1} / {b/1,c/1,d/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,u#,v#,w#} and constructors {b,c,d}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))