*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(g(x),s(0())) -> f(g(x),g(x))
        g(0()) -> 0()
        g(s(x)) -> s(g(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {f,g}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(g(x),s(0())) -> c_1(f#(g(x),g(x)))
        g#(0()) -> c_2()
        g#(s(x)) -> c_3(g#(x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(g(x),s(0())) -> c_1(f#(g(x),g(x)))
        g#(0()) -> c_2()
        g#(s(x)) -> c_3(g#(x))
      Strict TRS Rules:
        f(g(x),s(0())) -> f(g(x),g(x))
        g(0()) -> 0()
        g(s(x)) -> s(g(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g#(0()) -> c_2()
        g#(s(x)) -> c_3(g#(x))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0()) -> c_2()
        g#(s(x)) -> c_3(g#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      Succeeding
    Proof:
      ()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0()) -> c_2()
        g#(s(x)) -> c_3(g#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1}
      by application of
        Pre({1}) = {2}.
      Here rules are labelled as follows:
        1: g#(0()) -> c_2()      
        2: g#(s(x)) -> c_3(g#(x))
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(s(x)) -> c_3(g#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(0()) -> c_2()
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:g#(s(x)) -> c_3(g#(x))
           -->_1 g#(0()) -> c_2():2
           -->_1 g#(s(x)) -> c_3(g#(x)):1
        
        2:W:g#(0()) -> c_2()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        2: g#(0()) -> c_2()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(s(x)) -> c_3(g#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
      Obligation:
        Full
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: g#(s(x)) -> c_3(g#(x))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          g#(s(x)) -> c_3(g#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        Obligation:
          Full
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(0) = [0]                  
            p(f) = [2] x2 + [1]         
            p(g) = [8] x1 + [1]         
            p(s) = [1] x1 + [8]         
           p(f#) = [1] x1 + [1] x2 + [0]
           p(g#) = [1] x1 + [8]         
          p(c_1) = [2] x1 + [1]         
          p(c_2) = [1]                  
          p(c_3) = [1] x1 + [7]         
        
        Following rules are strictly oriented:
        g#(s(x)) = [1] x + [16]
                 > [1] x + [15]
                 = c_3(g#(x))  
        
        
        Following rules are (at-least) weakly oriented:
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(s(x)) -> c_3(g#(x))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        Obligation:
          Full
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(s(x)) -> c_3(g#(x))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        Obligation:
          Full
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:g#(s(x)) -> c_3(g#(x))
             -->_1 g#(s(x)) -> c_3(g#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(s(x)) -> c_3(g#(x))
  *** 1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        Obligation:
          Full
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).