*** 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:
        Innermost
        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:
        Innermost
        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(+) = [3] x1 + [2] x2 + [7]
               p(0) = [1]                  
          p(double) = [5] x1 + [9]         
               p(s) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
         +(x,0()) = [3] x + [9]        
                  > [1] x + [0]        
                  = x                  
        
        +(x,s(y)) = [3] x + [2] y + [9]
                  > [3] x + [2] y + [8]
                  = s(+(x,y))          
        
        
        Following rules are (at-least) weakly oriented:
           +(s(x),y) =  [3] x + [2] y + [10]
                     >= [3] x + [2] y + [8] 
                     =  s(+(x,y))           
        
           double(x) =  [5] x + [9]         
                     >= [5] x + [7]         
                     =  +(x,x)              
        
         double(0()) =  [14]                
                     >= [1]                 
                     =  0()                 
        
        double(s(x)) =  [5] x + [14]        
                     >= [5] x + [11]        
                     =  s(s(double(x)))     
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 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:
        Innermost
        basic terms: {+,double}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).