*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(x,0()) -> x
        +(x,s(y)) -> s(+(x,y))
        +(s(x),y) -> s(+(x,y))
        double(x) -> +(x,x)
        double(0()) -> 0()
        double(s(x)) -> s(s(double(x)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {+/2,double/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,double}/{0,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(+) = [6] x1 + [0]
               p(0) = [1]         
          p(double) = [6] x1 + [8]
               p(s) = [1] x1 + [3]
        
        Following rules are strictly oriented:
           +(s(x),y) = [6] x + [18]   
                     > [6] x + [3]    
                     = s(+(x,y))      
        
           double(x) = [6] x + [8]    
                     > [6] x + [0]    
                     = +(x,x)         
        
         double(0()) = [14]           
                     > [1]            
                     = 0()            
        
        double(s(x)) = [6] x + [26]   
                     > [6] x + [14]   
                     = s(s(double(x)))
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  [6] x + [0]
                  >= [1] x + [0]
                  =  x          
        
        +(x,s(y)) =  [6] x + [0]
                  >= [6] x + [3]
                  =  s(+(x,y))  
        
      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:
        +(x,0()) -> x
        +(x,s(y)) -> s(+(x,y))
      Weak DP Rules:
        
      Weak TRS Rules:
        +(s(x),y) -> s(+(x,y))
        double(x) -> +(x,x)
        double(0()) -> 0()
        double(s(x)) -> s(s(double(x)))
      Signature:
        {+/2,double/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,double}/{0,s}
    Applied Processor:
      NaturalMI {miDimension = 1, 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:
      The following argument positions are considered usable:
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
             p(+) = [2] x1 + [2] x2 + [2]
             p(0) = [0]                  
        p(double) = [8] x1 + [4]         
             p(s) = [1] x1 + [3]         
      
      Following rules are strictly oriented:
       +(x,0()) = [2] x + [2]        
                > [1] x + [0]        
                = x                  
      
      +(x,s(y)) = [2] x + [2] y + [8]
                > [2] x + [2] y + [5]
                = s(+(x,y))          
      
      
      Following rules are (at-least) weakly oriented:
         +(s(x),y) =  [2] x + [2] y + [8]
                   >= [2] x + [2] y + [5]
                   =  s(+(x,y))          
      
         double(x) =  [8] x + [4]        
                   >= [4] x + [2]        
                   =  +(x,x)             
      
       double(0()) =  [4]                
                   >= [0]                
                   =  0()                
      
      double(s(x)) =  [8] x + [28]       
                   >= [8] x + [10]       
                   =  s(s(double(x)))    
      
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        +(x,0()) -> x
        +(x,s(y)) -> s(+(x,y))
        +(s(x),y) -> s(+(x,y))
        double(x) -> +(x,x)
        double(0()) -> 0()
        double(s(x)) -> s(s(double(x)))
      Signature:
        {+/2,double/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,double}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).