*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0()) -> cons(0())
        f(s(0())) -> f(p(s(0())))
        p(s(0())) -> 0()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,p/1} / {0/0,cons/1,s/1}
      Obligation:
        Full
        basic terms: {f,p}/{0,cons,s}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0()) -> cons(0())
        f(s(0())) -> f(p(s(0())))
        p(s(0())) -> 0()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/1,p/1} / {0/0,cons/1,s/1}
      Obligation:
        Innermost
        basic terms: {f,p}/{0,cons,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        f#(0()) -> c_1()
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        p#(s(0())) -> c_3()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(0()) -> c_1()
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        p#(s(0())) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(0()) -> cons(0())
        f(s(0())) -> f(p(s(0())))
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        p(s(0())) -> 0()
        f#(0()) -> c_1()
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        p#(s(0())) -> c_3()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(0()) -> c_1()
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
        p#(s(0())) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3}
      by application of
        Pre({1,3}) = {2}.
      Here rules are labelled as follows:
        1: f#(0()) -> c_1()               
        2: f#(s(0())) -> c_2(f#(p(s(0())))
                            ,p#(s(0())))  
        3: p#(s(0())) -> c_3()            
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(0()) -> c_1()
        p#(s(0())) -> c_3()
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
           -->_2 p#(s(0())) -> c_3():3
           -->_1 f#(0()) -> c_1():2
           -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1
        
        2:W:f#(0()) -> c_1()
           
        
        3:W:p#(s(0())) -> c_3()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        2: f#(0()) -> c_1()   
        3: p#(s(0())) -> c_3()
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/2,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,s}
    Applied Processor:
      SimplifyRHS
    Proof:
      Consider the dependency graph
        1:S:f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0())))
           -->_1 f#(s(0())) -> c_2(f#(p(s(0()))),p#(s(0()))):1
        
      Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
        f#(s(0())) -> c_2(f#(p(s(0()))))
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(0())) -> c_2(f#(p(s(0()))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,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_2) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
             p(0) = [0]         
          p(cons) = [0]         
             p(f) = [0]         
             p(p) = [0]         
             p(s) = [1]         
            p(f#) = [1] x1 + [0]
            p(p#) = [0]         
           p(c_1) = [0]         
           p(c_2) = [1] x1 + [0]
           p(c_3) = [0]         
        
        Following rules are strictly oriented:
        f#(s(0())) = [1]               
                   > [0]               
                   = c_2(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_2(f#(p(s(0()))))
      Weak TRS Rules:
        p(s(0())) -> 0()
      Signature:
        {f/1,p/1,f#/1,p#/1} / {0/0,cons/1,s/1,c_1/0,c_2/1,c_3/0}
      Obligation:
        Innermost
        basic terms: {f#,p#}/{0,cons,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).