*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(0(),y) -> y
        +(s(x),0()) -> s(x)
        +(s(x),s(y)) -> s(+(s(x),+(y,0())))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {+/2} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+}/{0,s}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(0(),y) -> y
        +(s(x),0()) -> s(x)
        +(s(x),s(y)) -> s(+(s(x),+(y,0())))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {+/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {+}/{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(+) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(+) = [1] x1 + [1] x2 + [0]
          p(0) = [1]                  
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
           +(0(),y) = [1] y + [1]
                    > [1] y + [0]
                    = y          
        
        +(s(x),0()) = [1] x + [1]
                    > [1] x + [0]
                    = s(x)       
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),s(y)) =  [1] x + [1] y + [0]
                     >= [1] x + [1] y + [1]
                     =  s(+(s(x),+(y,0())))
        
      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:
        +(s(x),s(y)) -> s(+(s(x),+(y,0())))
      Weak DP Rules:
        
      Weak TRS Rules:
        +(0(),y) -> y
        +(s(x),0()) -> s(x)
      Signature:
        {+/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {+}/{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(+) = {2},
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {+}
      TcT has computed the following interpretation:
        p(+) = [1] x1 + [2] x2 + [0]
        p(0) = [0]                  
        p(s) = [1] x1 + [8]         
      
      Following rules are strictly oriented:
      +(s(x),s(y)) = [1] x + [2] y + [24]
                   > [1] x + [2] y + [16]
                   = s(+(s(x),+(y,0()))) 
      
      
      Following rules are (at-least) weakly oriented:
         +(0(),y) =  [2] y + [0]
                  >= [1] y + [0]
                  =  y          
      
      +(s(x),0()) =  [1] x + [8]
                  >= [1] x + [8]
                  =  s(x)       
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        +(0(),y) -> y
        +(s(x),0()) -> s(x)
        +(s(x),s(y)) -> s(+(s(x),+(y,0())))
      Signature:
        {+/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {+}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).