*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(s(0()),g(x)) -> f(x,g(x))
        g(s(x)) -> g(x)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        f#(s(0()),g(x)) -> c_1(f#(x,g(x)))
        g#(s(x)) -> c_2(g#(x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(0()),g(x)) -> c_1(f#(x,g(x)))
        g#(s(x)) -> c_2(g#(x))
      Strict TRS Rules:
        f(s(0()),g(x)) -> f(x,g(x))
        g(s(x)) -> 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_2/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g#(s(x)) -> c_2(g#(x))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(s(x)) -> c_2(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_2/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{0,s}
    Applied Processor:
      Succeeding
    Proof:
      ()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(s(x)) -> c_2(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_2/1}
      Obligation:
        Innermost
        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_2(g#(x))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          g#(s(x)) -> c_2(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_2/1}
        Obligation:
          Innermost
          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_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [2]          
            p(f) = [1] x2 + [1] 
            p(g) = [1] x1 + [0] 
            p(s) = [1] x1 + [2] 
           p(f#) = [0]          
           p(g#) = [15] x1 + [0]
          p(c_1) = [8]          
          p(c_2) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
        g#(s(x)) = [15] x + [30]
                 > [15] x + [0] 
                 = c_2(g#(x))   
        
        
        Following rules are (at-least) weakly oriented:
        
  *** 1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(s(x)) -> c_2(g#(x))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 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_2(g#(x))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:g#(s(x)) -> c_2(g#(x))
             -->_1 g#(s(x)) -> c_2(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_2(g#(x))
  *** 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_2/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{0,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).