WORST_CASE(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,c,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,c,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(X,X) -> c_1()
          f#(X,c(X)) -> c_2(f#(s(X),X))
          f#(s(X),X) -> c_3(f#(X,a(X)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,X) -> c_1()
            f#(X,c(X)) -> c_2(f#(s(X),X))
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Weak TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2}.
        Here rules are labelled as follows:
          1: f#(X,X) -> c_1()
          2: f#(X,c(X)) -> c_2(f#(s(X),X))
          3: f#(s(X),X) -> c_3(f#(X,a(X)))
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,c(X)) -> c_2(f#(s(X),X))
        - Weak DPs:
            f#(X,X) -> c_1()
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Weak TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: f#(X,c(X)) -> c_2(f#(s(X),X))
          2: f#(X,X) -> c_1()
          3: f#(s(X),X) -> c_3(f#(X,a(X)))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(X,X) -> c_1()
            f#(X,c(X)) -> c_2(f#(s(X),X))
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Weak TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(X,X) -> c_1()
             
          
          2:W:f#(X,c(X)) -> c_2(f#(s(X),X))
             -->_1 f#(s(X),X) -> c_3(f#(X,a(X))):3
          
          3:W:f#(s(X),X) -> c_3(f#(X,a(X)))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(X,c(X)) -> c_2(f#(s(X),X))
          3: f#(s(X),X) -> c_3(f#(X,a(X)))
          1: f#(X,X) -> c_1()
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))