*** 1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        max(L(x)) -> x
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {max/1} / {0/0,L/1,N/2,s/1}
      Obligation:
        Innermost
        basic terms: {max}/{0,L,N,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        max#(L(x)) -> c_1()
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(0()),L(y))) -> c_3()
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        max#(L(x)) -> c_1()
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(0()),L(y))) -> c_3()
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        max(L(x)) -> x
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
      Signature:
        {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {max#}/{0,L,N,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        max#(L(x)) -> c_1()
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(0()),L(y))) -> c_3()
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
*** 1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        max#(L(x)) -> c_1()
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(0()),L(y))) -> c_3()
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
      Signature:
        {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {max#}/{0,L,N,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3}
      by application of
        Pre({1,3}) = {2,4}.
      Here rules are labelled as follows:
        1: max#(L(x)) -> c_1()               
        2: max#(N(L(x),N(y,z))) ->           
             c_2(max#(N(L(x),L(max(N(y,z)))))
                ,max#(N(y,z)))               
        3: max#(N(L(0()),L(y))) -> c_3()     
        4: max#(N(L(s(x)),L(s(y)))) ->       
             c_4(max#(N(L(x),L(y))))         
*** 1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        max#(L(x)) -> c_1()
        max#(N(L(0()),L(y))) -> c_3()
      Weak TRS Rules:
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
      Signature:
        {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {max#}/{0,L,N,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
           -->_2 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
           -->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
           -->_2 max#(N(L(0()),L(y))) -> c_3():4
           -->_1 max#(N(L(0()),L(y))) -> c_3():4
           -->_2 max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z))):1
        
        2:S:max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
           -->_1 max#(N(L(0()),L(y))) -> c_3():4
           -->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
        
        3:W:max#(L(x)) -> c_1()
           
        
        4:W:max#(N(L(0()),L(y))) -> c_3()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        3: max#(L(x)) -> c_1()          
        4: max#(N(L(0()),L(y))) -> c_3()
*** 1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        max(N(L(0()),L(y))) -> y
        max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
      Signature:
        {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {max#}/{0,L,N,s}
    Applied Processor:
      DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    Proof:
      We decompose the input problem according to the dependency graph into the upper component
        max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
      and a lower component
        max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
      Further, following extension rules are added to the lower component.
        max#(N(L(x),N(y,z))) -> max#(N(y,z))
        max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
  *** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
          max(N(L(0()),L(y))) -> y
          max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        Signature:
          {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {max#}/{0,L,N,s}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: max#(N(L(x),N(y,z))) ->           
               c_2(max#(N(L(x),L(max(N(y,z)))))
                  ,max#(N(y,z)))               
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and 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(c_2) = {1,2}
          
          Following symbols are considered usable:
            {max#}
          TcT has computed the following interpretation:
               p(0) = [1]                      
                      [1]                      
               p(L) = [0]                      
                      [0]                      
               p(N) = [0 1] x2 + [0]           
                      [0 1]      [2]           
             p(max) = [0 0] x1 + [0]           
                      [1 0]      [0]           
               p(s) = [0]                      
                      [0]                      
            p(max#) = [1 0] x1 + [0]           
                      [0 0]      [1]           
             p(c_1) = [1]                      
                      [2]                      
             p(c_2) = [1 0] x1 + [1 0] x2 + [1]
                      [0 0]      [0 1]      [0]
             p(c_3) = [2]                      
                      [0]                      
             p(c_4) = [2]                      
                      [2]                      
          
          Following rules are strictly oriented:
          max#(N(L(x),N(y,z))) = [0 1] z + [2]                   
                                 [0 0]     [1]                   
                               > [0 1] z + [1]                   
                                 [0 0]     [1]                   
                               = c_2(max#(N(L(x),L(max(N(y,z)))))
                                    ,max#(N(y,z)))               
          
          
          Following rules are (at-least) weakly oriented:
          
    *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
               -->_2 max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z))):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: max#(N(L(x),N(y,z))) ->           
                 c_2(max#(N(L(x),L(max(N(y,z)))))
                    ,max#(N(y,z)))               
    *** 1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).
    
  *** 1.1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
        Strict TRS Rules:
          
        Weak DP Rules:
          max#(N(L(x),N(y,z))) -> max#(N(y,z))
          max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
        Weak TRS Rules:
          max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
          max(N(L(0()),L(y))) -> y
          max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        Signature:
          {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {max#}/{0,L,N,s}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: max#(N(L(s(x)),L(s(y)))) ->
               c_4(max#(N(L(x),L(y))))  
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
          Strict TRS Rules:
            
          Weak DP Rules:
            max#(N(L(x),N(y,z))) -> max#(N(y,z))
            max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
        Proof:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            {max,max#}
          TcT has computed the following interpretation:
               p(0) = [2]                  
               p(L) = [1] x1 + [0]         
               p(N) = [1] x1 + [1] x2 + [0]
             p(max) = [1] x1 + [0]         
               p(s) = [1] x1 + [1]         
            p(max#) = [8] x1 + [4]         
             p(c_1) = [0]                  
             p(c_2) = [1] x1 + [1] x2 + [1]
             p(c_3) = [1]                  
             p(c_4) = [1] x1 + [15]        
          
          Following rules are strictly oriented:
          max#(N(L(s(x)),L(s(y)))) = [8] x + [8] y + [20]   
                                   > [8] x + [8] y + [19]   
                                   = c_4(max#(N(L(x),L(y))))
          
          
          Following rules are (at-least) weakly oriented:
             max#(N(L(x),N(y,z))) =  [8] x + [8] y + [8] z + [4] 
                                  >= [8] y + [8] z + [4]         
                                  =  max#(N(y,z))                
          
             max#(N(L(x),N(y,z))) =  [8] x + [8] y + [8] z + [4] 
                                  >= [8] x + [8] y + [8] z + [4] 
                                  =  max#(N(L(x),L(max(N(y,z)))))
          
              max(N(L(x),N(y,z))) =  [1] x + [1] y + [1] z + [0] 
                                  >= [1] x + [1] y + [1] z + [0] 
                                  =  max(N(L(x),L(max(N(y,z))))) 
          
              max(N(L(0()),L(y))) =  [1] y + [2]                 
                                  >= [1] y + [0]                 
                                  =  y                           
          
          max(N(L(s(x)),L(s(y)))) =  [1] x + [1] y + [2]         
                                  >= [1] x + [1] y + [1]         
                                  =  s(max(N(L(x),L(y))))        
          
    *** 1.1.1.1.1.2.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            max#(N(L(x),N(y,z))) -> max#(N(y,z))
            max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
            max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.2.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            max#(N(L(x),N(y,z))) -> max#(N(y,z))
            max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
            max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:max#(N(L(x),N(y,z))) -> max#(N(y,z))
               -->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
               -->_1 max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z))))):2
               -->_1 max#(N(L(x),N(y,z))) -> max#(N(y,z)):1
            
            2:W:max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
               -->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
            
            3:W:max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
               -->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: max#(N(L(x),N(y,z))) -> max#(N(y  
                                             ,z))
            2: max#(N(L(x),N(y,z))) ->           
                 max#(N(L(x),L(max(N(y,z)))))    
            3: max#(N(L(s(x)),L(s(y)))) ->       
                 c_4(max#(N(L(x),L(y))))         
    *** 1.1.1.1.1.2.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
          Signature:
            {max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
          Obligation:
            Innermost
            basic terms: {max#}/{0,L,N,s}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).