*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0(),y) -> 0()
        f(s(x),y) -> f(f(x,y),y)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {f}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        f#(0(),y) -> c_1()
        f#(s(x),y) -> c_2(f#(f(x,y),y))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(0(),y) -> c_1()
        f#(s(x),y) -> c_2(f#(f(x,y),y))
      Strict TRS Rules:
        f(0(),y) -> 0()
        f(s(x),y) -> f(f(x,y),y)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
      Obligation:
        Full
        basic terms: {f#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1}
      by application of
        Pre({1}) = {2}.
      Here rules are labelled as follows:
        1: f#(0(),y) -> c_1()             
        2: f#(s(x),y) -> c_2(f#(f(x,y),y))
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(x),y) -> c_2(f#(f(x,y),y))
      Strict TRS Rules:
        f(0(),y) -> 0()
        f(s(x),y) -> f(f(x,y),y)
      Weak DP Rules:
        f#(0(),y) -> c_1()
      Weak TRS Rules:
        
      Signature:
        {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
      Obligation:
        Full
        basic terms: {f#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, 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(f) = {1},
          uargs(f#) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(0) = [0]         
            p(f) = [1] x1 + [0]
            p(s) = [1] x1 + [3]
           p(f#) = [1] x1 + [0]
          p(c_1) = [0]         
          p(c_2) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        f#(s(x),y) = [1] x + [3]      
                   > [1] x + [0]      
                   = c_2(f#(f(x,y),y))
        
         f(s(x),y) = [1] x + [3]      
                   > [1] x + [0]      
                   = f(f(x,y),y)      
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),y) =  [0]  
                  >= [0]  
                  =  c_1()
        
         f(0(),y) =  [0]  
                  >= [0]  
                  =  0()  
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(0(),y) -> 0()
      Weak DP Rules:
        f#(0(),y) -> c_1()
        f#(s(x),y) -> c_2(f#(f(x,y),y))
      Weak TRS Rules:
        f(s(x),y) -> f(f(x,y),y)
      Signature:
        {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
      Obligation:
        Full
        basic terms: {f#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, 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(f) = {1},
          uargs(f#) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(0) = [0]          
            p(f) = [1] x1 + [10]
            p(s) = [1] x1 + [10]
           p(f#) = [1] x1 + [0] 
          p(c_1) = [0]          
          p(c_2) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
        f(0(),y) = [10]
                 > [0] 
                 = 0() 
        
        
        Following rules are (at-least) weakly oriented:
         f#(0(),y) =  [0]              
                   >= [0]              
                   =  c_1()            
        
        f#(s(x),y) =  [1] x + [10]     
                   >= [1] x + [10]     
                   =  c_2(f#(f(x,y),y))
        
         f(s(x),y) =  [1] x + [20]     
                   >= [1] x + [20]     
                   =  f(f(x,y),y)      
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(0(),y) -> c_1()
        f#(s(x),y) -> c_2(f#(f(x,y),y))
      Weak TRS Rules:
        f(0(),y) -> 0()
        f(s(x),y) -> f(f(x,y),y)
      Signature:
        {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
      Obligation:
        Full
        basic terms: {f#}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).