*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        f(0(),1(),x) -> f(s(x),x,x)
        g(x,y) -> x
        g(x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g}/{0,1,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(0) = [0]                  
          p(1) = [0]                  
          p(f) = [0]                  
          p(g) = [2] x1 + [2] x2 + [1]
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        g(x,y) = [2] x + [2] y + [1]
               > [1] x + [0]        
               = x                  
        
        g(x,y) = [2] x + [2] y + [1]
               > [1] y + [0]        
               = y                  
        
        
        Following rules are (at-least) weakly oriented:
         f(x,y,s(z)) =  [0]            
                     >= [0]            
                     =  s(f(0(),1(),z))
        
        f(0(),1(),x) =  [0]            
                     >= [0]            
                     =  f(s(x),x,x)    
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        f(0(),1(),x) -> f(s(x),x,x)
      Weak DP Rules:
        
      Weak TRS Rules:
        g(x,y) -> x
        g(x,y) -> y
      Signature:
        {f/3,g/2} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g}/{0,1,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(0) = [1]                  
          p(1) = [1]                  
          p(f) = [8] x3 + [0]         
          p(g) = [1] x1 + [2] x2 + [8]
          p(s) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        f(x,y,s(z)) = [8] z + [16]   
                    > [8] z + [2]    
                    = s(f(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f(0(),1(),x) =  [8] x + [0]        
                     >= [8] x + [0]        
                     =  f(s(x),x,x)        
        
              g(x,y) =  [1] x + [2] y + [8]
                     >= [1] x + [0]        
                     =  x                  
        
              g(x,y) =  [1] x + [2] y + [8]
                     >= [1] y + [0]        
                     =  y                  
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0(),1(),x) -> f(s(x),x,x)
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        g(x,y) -> x
        g(x,y) -> y
      Signature:
        {f/3,g/2} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g}/{0,1,s}
    Applied Processor:
      NaturalMI {miDimension = 2, miDegree = 1, 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 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
      The following argument positions are considered usable:
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {f,g}
      TcT has computed the following interpretation:
        p(0) = [1]                      
               [2]                      
        p(1) = [1]                      
               [0]                      
        p(f) = [0 1] x1 + [2 0] x3 + [0]
               [0 0]      [0 0]      [0]
        p(g) = [4 1] x1 + [2 4] x2 + [1]
               [1 4]      [0 4]      [0]
        p(s) = [1 0] x1 + [4]           
               [0 0]      [0]           
      
      Following rules are strictly oriented:
      f(0(),1(),x) = [2 0] x + [2]
                     [0 0]     [0]
                   > [2 0] x + [0]
                     [0 0]     [0]
                   = f(s(x),x,x)  
      
      
      Following rules are (at-least) weakly oriented:
      f(x,y,s(z)) =  [0 1] x + [2 0] z + [8]
                     [0 0]     [0 0]     [0]
                  >= [2 0] z + [6]          
                     [0 0]     [0]          
                  =  s(f(0(),1(),z))        
      
           g(x,y) =  [4 1] x + [2 4] y + [1]
                     [1 4]     [0 4]     [0]
                  >= [1 0] x + [0]          
                     [0 1]     [0]          
                  =  x                      
      
           g(x,y) =  [4 1] x + [2 4] y + [1]
                     [1 4]     [0 4]     [0]
                  >= [1 0] y + [0]          
                     [0 1]     [0]          
                  =  y                      
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,y,s(z)) -> s(f(0(),1(),z))
        f(0(),1(),x) -> f(s(x),x,x)
        g(x,y) -> x
        g(x,y) -> y
      Signature:
        {f/3,g/2} / {0/0,1/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g}/{0,1,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).