MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            purge(.(x,y)) -> .(x,purge(remove(x,y)))
            purge(nil()) -> nil()
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2} / {./2,=/2,if/3,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge,remove} and constructors {.,=,if,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
          purge#(nil()) -> c_2()
          remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
          remove#(x,nil()) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
            purge#(nil()) -> c_2()
            remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
            remove#(x,nil()) -> c_4()
        - Weak TRS:
            purge(.(x,y)) -> .(x,purge(remove(x,y)))
            purge(nil()) -> nil()
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
          remove(x,nil()) -> nil()
          purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
          purge#(nil()) -> c_2()
          remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
          remove#(x,nil()) -> c_4()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
            purge#(nil()) -> c_2()
            remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
            remove#(x,nil()) -> c_4()
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3}.
        Here rules are labelled as follows:
          1: purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
          2: purge#(nil()) -> c_2()
          3: remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
          4: remove#(x,nil()) -> c_4()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
            remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
        - Weak DPs:
            purge#(nil()) -> c_2()
            remove#(x,nil()) -> c_4()
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
             -->_2 remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z)):2
             -->_2 remove#(x,nil()) -> c_4():4
             -->_1 purge#(nil()) -> c_2():3
             -->_1 purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y)):1
          
          2:S:remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
             -->_2 remove#(x,nil()) -> c_4():4
             -->_1 remove#(x,nil()) -> c_4():4
             -->_2 remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z)):2
             -->_1 remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z)):2
          
          3:W:purge#(nil()) -> c_2()
             
          
          4:W:remove#(x,nil()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: purge#(nil()) -> c_2()
          4: remove#(x,nil()) -> c_4()
* Step 5: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
            remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + 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
          purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
        and a lower component
          remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
        Further, following extension rules are added to the lower component.
          purge#(.(x,y)) -> purge#(remove(x,y))
          purge#(.(x,y)) -> remove#(x,y)
** Step 5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + 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: purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + 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_1) = {1}
        
        Following symbols are considered usable:
          {remove,purge#,remove#}
        TcT has computed the following interpretation:
                p(.) = [5]                  
                p(=) = [1] x1 + [8]         
               p(if) = [1] x2 + [0]         
              p(nil) = [0]                  
            p(purge) = [1] x1 + [2]         
           p(remove) = [0]                  
           p(purge#) = [5] x1 + [4]         
          p(remove#) = [4]                  
              p(c_1) = [4] x1 + [1] x2 + [8]
              p(c_2) = [1]                  
              p(c_3) = [1]                  
              p(c_4) = [0]                  
        
        Following rules are strictly oriented:
        purge#(.(x,y)) = [29]                                 
                       > [28]                                 
                       = c_1(purge#(remove(x,y)),remove#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        remove(x,.(y,z)) =  [0]                                    
                         >= [0]                                    
                         =  if(=(x,y),remove(x,z),.(y,remove(x,z)))
        
         remove(x,nil()) =  [0]                                    
                         >= [0]                                    
                         =  nil()                                  
        
*** Step 5.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
             -->_1 purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: purge#(.(x,y)) -> c_1(purge#(remove(x,y)),remove#(x,y))
*** Step 5.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
            remove(x,nil()) -> nil()
        - Signature:
            {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          remove#(x,.(y,z)) -> c_3(remove#(x,z),remove#(x,z))
      - Weak DPs:
          purge#(.(x,y)) -> purge#(remove(x,y))
          purge#(.(x,y)) -> remove#(x,y)
      - Weak TRS:
          remove(x,.(y,z)) -> if(=(x,y),remove(x,z),.(y,remove(x,z)))
          remove(x,nil()) -> nil()
      - Signature:
          {purge/1,remove/2,purge#/1,remove#/2} / {./2,=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2,c_4/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {purge#,remove#} and constructors {.,=,if,nil}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE