*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(s(x),y,y) -> f(y,x,s(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3} / {s/1}
      Obligation:
        Full
        basic terms: {f}/{s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
      Strict TRS Rules:
        f(s(x),y,y) -> f(y,x,s(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {s/1,c_1/1}
      Obligation:
        Full
        basic terms: {f#}/{s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {s/1,c_1/1}
      Obligation:
        Full
        basic terms: {f#}/{s}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      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:
        none
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
          p(f) = [0]
          p(s) = [0]
         p(f#) = [4]
        p(c_1) = [0]
      
      Following rules are strictly oriented:
      f#(s(x),y,y) = [4]              
                   > [0]              
                   = c_1(f#(y,x,s(x)))
      
      
      Following rules are (at-least) weakly oriented:
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(s(x),y,y) -> c_1(f#(y,x,s(x)))
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {s/1,c_1/1}
      Obligation:
        Full
        basic terms: {f#}/{s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).