WORST_CASE(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + 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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + 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:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "0") :: [] -(0)-> "A"(2, 2, 1)
          F (TrsFun "activate") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "cons") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "f") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "n__f") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "p") :: ["A"(1, 1, 1)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(2, 2, 1)] -(1)-> "A"(1, 1, 1)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          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()
          main(x1) -> p(x1)
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^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 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^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:
        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 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^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:
            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:
        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 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^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:
            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:
        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 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_2(f#(X))
            f#(s(0())) -> c_5(f#(p(s(0()))))
        - 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_5/1,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#(n__f(X)) -> c_2(f#(X))
          f#(s(0())) -> c_5(f#(p(s(0()))))
** Step 1.b:6: RemoveHeads.  WORST_CASE(?,O(n^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 1.b:7: NaturalMI.  WORST_CASE(?,O(n^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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {p,activate#,f#,p#}
        TcT has computed the following interpretation:
                  p(0) = [3]                  
           p(activate) = [0]                  
               p(cons) = [1] x1 + [1] x2 + [0]
                  p(f) = [0]                  
               p(n__f) = [0]                  
                  p(p) = [3]                  
                  p(s) = [8]                  
          p(activate#) = [2]                  
                 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) = [2] x1 + [0]         
                p(c_6) = [0]                  
        
        Following rules are strictly oriented:
        f#(s(0())) = [8]               
                   > [6]               
                   = c_5(f#(p(s(0()))))
        
        
        Following rules are (at-least) weakly oriented:
        p(s(0())) =  [3]
                  >= [3]
                  =  0()
        
** Step 1.b: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(n^1))