* Step 1: ToInnermost WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            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} / {0/0,L/1,N/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            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} / {0/0,L/1,N/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {max} and constructors {0,L,N,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        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.
* Step 3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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 TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        UsableRules
    + Details:
        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))))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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 TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        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))))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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))))
        - Weak DPs:
            max#(L(x)) -> c_1()
            max#(N(L(0()),L(y))) -> c_3()
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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()
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            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))))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        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)))))
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} 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.
*** Step 6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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]                      
                    [0]                      
             p(L) = [0]                      
                    [0]                      
             p(N) = [0 2] x2 + [0]           
                    [0 1]      [1]           
           p(max) = [1 1] x1 + [3]           
                    [1 2]      [0]           
             p(s) = [0]                      
                    [0]                      
          p(max#) = [2 0] x1 + [0]           
                    [0 3]      [0]           
           p(c_1) = [0]                      
                    [1]                      
           p(c_2) = [1 0] x1 + [1 0] x2 + [0]
                    [0 0]      [0 0]      [2]
           p(c_3) = [0]                      
                    [0]                      
           p(c_4) = [0 0] x1 + [0]           
                    [1 0]      [0]           
        
        Following rules are strictly oriented:
        max#(N(L(x),N(y,z))) = [0 4] z + [4]                                 
                               [0 3]     [6]                                 
                             > [0 4] z + [0]                                 
                               [0 0]     [2]                                 
                             = c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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)))
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {max#} and constructors {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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} 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.
*** Step 6.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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) = [5]         
             p(L) = [1] x1 + [0]
             p(N) = [1] x2 + [0]
           p(max) = [1] x1 + [0]
             p(s) = [1] x1 + [4]
          p(max#) = [1] x1 + [1]
           p(c_1) = [0]         
           p(c_2) = [4] x1 + [0]
           p(c_3) = [8]         
           p(c_4) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        max#(N(L(s(x)),L(s(y)))) = [1] y + [5]            
                                 > [1] y + [2]            
                                 = c_4(max#(N(L(x),L(y))))
        
        
        Following rules are (at-least) weakly oriented:
           max#(N(L(x),N(y,z))) =  [1] z + [1]                 
                                >= [1] z + [1]                 
                                =  max#(N(y,z))                
        
           max#(N(L(x),N(y,z))) =  [1] z + [1]                 
                                >= [1] z + [1]                 
                                =  max#(N(L(x),L(max(N(y,z)))))
        
            max(N(L(x),N(y,z))) =  [1] z + [0]                 
                                >= [1] z + [0]                 
                                =  max(N(L(x),L(max(N(y,z))))) 
        
            max(N(L(0()),L(y))) =  [1] y + [0]                 
                                >= [1] y + [0]                 
                                =  y                           
        
        max(N(L(s(x)),L(s(y)))) =  [1] y + [4]                 
                                >= [1] y + [4]                 
                                =  s(max(N(L(x),L(y))))        
        
*** Step 6.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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))))
*** Step 6.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {max#} and constructors {0,L,N,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))