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

WORST_CASE(?,O(1))