WORST_CASE(?,O(1))
* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(0()) -> cons(0(),n__f(s(0())))
            f(s(0())) -> f(p(s(0())))
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__f(X)) -> c_2(f#(X))
          f#(X) -> c_3()
          f#(0()) -> c_4()
          f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
          p#(s(0())) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(X))
            f#(X) -> c_3()
            f#(0()) -> c_4()
            f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
            p#(s(0())) -> c_6()
        - Weak TRS:
            activate(X) -> X
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(0()) -> cons(0(),n__f(s(0())))
            f(s(0())) -> f(p(s(0())))
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(0())) -> 0()
          activate#(X) -> c_1()
          activate#(n__f(X)) -> c_2(f#(X))
          f#(X) -> c_3()
          f#(0()) -> c_4()
          f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
          p#(s(0())) -> c_6()
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__f(X)) -> c_2(f#(X))
            f#(X) -> c_3()
            f#(0()) -> c_4()
            f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
            p#(s(0())) -> c_6()
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,6}
        by application of
          Pre({1,3,4,6}) = {2,5}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__f(X)) -> c_2(f#(X))
          3: f#(X) -> c_3()
          4: f#(0()) -> c_4()
          5: f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
          6: p#(s(0())) -> c_6()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(X))
            f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
        - Weak DPs:
            activate#(X) -> c_1()
            f#(X) -> c_3()
            f#(0()) -> c_4()
            p#(s(0())) -> c_6()
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X)) -> c_2(f#(X))
             -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
             -->_1 f#(0()) -> c_4():5
             -->_1 f#(X) -> c_3():4
          
          2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
             -->_2 p#(s(0())) -> c_6():6
             -->_1 f#(0()) -> c_4():5
             -->_1 f#(X) -> c_3():4
             -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
          
          3:W:activate#(X) -> c_1()
             
          
          4:W:f#(X) -> c_3()
             
          
          5:W:f#(0()) -> c_4()
             
          
          6:W:p#(s(0())) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: activate#(X) -> c_1()
          4: f#(X) -> c_3()
          5: f#(0()) -> c_4()
          6: p#(s(0())) -> c_6()
* Step 5: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(X))
            f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__f(X)) -> c_2(f#(X))
             -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
          
          2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
             -->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(0())) -> c_5(f#(p(s(0()))))
* Step 6: RemoveHeads WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(X))
            f#(s(0())) -> c_5(f#(p(s(0()))))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:activate#(n__f(X)) -> c_2(f#(X))
           -->_1 f#(s(0())) -> c_5(f#(p(s(0())))):2
        
        2:S:f#(s(0())) -> c_5(f#(p(s(0()))))
           -->_1 f#(s(0())) -> c_5(f#(p(s(0())))):2
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(1,activate#(n__f(X)) -> c_2(f#(X)))]
* Step 7: WeightGap WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(s(0())) -> c_5(f#(p(s(0()))))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]         
             p(activate) = [0]         
                 p(cons) = [0]         
                    p(f) = [0]         
                 p(n__f) = [0]         
                    p(p) = [0]         
                    p(s) = [5]         
            p(activate#) = [0]         
                   p(f#) = [1] x1 + [0]
                   p(p#) = [0]         
                  p(c_1) = [0]         
                  p(c_2) = [0]         
                  p(c_3) = [0]         
                  p(c_4) = [0]         
                  p(c_5) = [1] x1 + [0]
                  p(c_6) = [0]         
          
          Following rules are strictly oriented:
          f#(s(0())) = [5]               
                     > [0]               
                     = c_5(f#(p(s(0()))))
          
          
          Following rules are (at-least) weakly oriented:
          p(s(0())) =  [0]
                    >= [0]
                    =  0()
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(0())) -> c_5(f#(p(s(0()))))
        - Weak TRS:
            p(s(0())) -> 0()
        - Signature:
            {activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))