*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1}
      Obligation:
        Innermost
        basic terms: {activate,f,p}/{0,cons,n__f,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      UsableRules
    Proof:
      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()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      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()             
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(n__f(X)) -> c_2(f#(X))
        f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
      Strict TRS Rules:
        
      Weak DP Rules:
        activate#(X) -> c_1()
        f#(X) -> c_3()
        f#(0()) -> c_4()
        p#(s(0())) -> c_6()
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      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()  
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(n__f(X)) -> c_2(f#(X))
        f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      SimplifyRHS
    Proof:
      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()))))
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        activate#(n__f(X)) -> c_2(f#(X))
        f#(s(0())) -> c_5(f#(p(s(0()))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      RemoveHeads
    Proof:
      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)))]
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(0())) -> c_5(f#(p(s(0()))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        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) = [3]         
          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())) = [3]               
                   > [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.
*** 1.1.1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(s(0())) -> c_5(f#(p(s(0()))))
      Weak TRS Rules:
        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
        basic terms: {activate#,f#,p#}/{0,cons,n__f,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).