MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(X) -> cons(X,f(g(X)))
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,Z)
        - Signature:
            {f/1,g/1,sel/2} / {0/0,cons/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,sel} and constructors {0,cons,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(X) -> c_1(f#(g(X)),g#(X))
          g#(0()) -> c_2()
          g#(s(X)) -> c_3(g#(X))
          sel#(0(),cons(X,Y)) -> c_4()
          sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(0()) -> c_2()
            g#(s(X)) -> c_3(g#(X))
            sel#(0(),cons(X,Y)) -> c_4()
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak TRS:
            f(X) -> cons(X,f(g(X)))
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,Z)
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g(0()) -> s(0())
          g(s(X)) -> s(s(g(X)))
          f#(X) -> c_1(f#(g(X)),g#(X))
          g#(0()) -> c_2()
          g#(s(X)) -> c_3(g#(X))
          sel#(0(),cons(X,Y)) -> c_4()
          sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(0()) -> c_2()
            g#(s(X)) -> c_3(g#(X))
            sel#(0(),cons(X,Y)) -> c_4()
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + 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}) = {1,3,5}.
        Here rules are labelled as follows:
          1: f#(X) -> c_1(f#(g(X)),g#(X))
          2: g#(0()) -> c_2()
          3: g#(s(X)) -> c_3(g#(X))
          4: sel#(0(),cons(X,Y)) -> c_4()
          5: sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(s(X)) -> c_3(g#(X))
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak DPs:
            g#(0()) -> c_2()
            sel#(0(),cons(X,Y)) -> c_4()
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(X) -> c_1(f#(g(X)),g#(X))
             -->_2 g#(s(X)) -> c_3(g#(X)):2
             -->_2 g#(0()) -> c_2():4
             -->_1 f#(X) -> c_1(f#(g(X)),g#(X)):1
          
          2:S:g#(s(X)) -> c_3(g#(X))
             -->_1 g#(0()) -> c_2():4
             -->_1 g#(s(X)) -> c_3(g#(X)):2
          
          3:S:sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
             -->_1 sel#(0(),cons(X,Y)) -> c_4():5
             -->_1 sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z)):3
          
          4:W:g#(0()) -> c_2()
             
          
          5:W:sel#(0(),cons(X,Y)) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: sel#(0(),cons(X,Y)) -> c_4()
          4: g#(0()) -> c_2()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(s(X)) -> c_3(g#(X))
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              f#(X) -> c_1(f#(g(X)),g#(X))
              g#(s(X)) -> c_3(g#(X))
          - Weak DPs:
              sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
          - Weak TRS:
              g(0()) -> s(0())
              g(s(X)) -> s(s(g(X)))
          - Signature:
              {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
        
        Problem (S)
          - Strict DPs:
              sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
          - Weak DPs:
              f#(X) -> c_1(f#(g(X)),g#(X))
              g#(s(X)) -> c_3(g#(X))
          - Weak TRS:
              g(0()) -> s(0())
              g(s(X)) -> s(s(g(X)))
          - Signature:
              {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(s(X)) -> c_3(g#(X))
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(X) -> c_1(f#(g(X)),g#(X))
             -->_2 g#(s(X)) -> c_3(g#(X)):2
             -->_1 f#(X) -> c_1(f#(g(X)),g#(X)):1
          
          2:S:g#(s(X)) -> c_3(g#(X))
             -->_1 g#(s(X)) -> c_3(g#(X)):2
          
          3:W:sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
** Step 5.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(X) -> c_1(f#(g(X)),g#(X))
          g#(s(X)) -> c_3(g#(X))
      - Weak TRS:
          g(0()) -> s(0())
          g(s(X)) -> s(s(g(X)))
      - Signature:
          {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak DPs:
            f#(X) -> c_1(f#(g(X)),g#(X))
            g#(s(X)) -> c_3(g#(X))
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z)):1
          
          2:W:f#(X) -> c_1(f#(g(X)),g#(X))
             -->_2 g#(s(X)) -> c_3(g#(X)):3
             -->_1 f#(X) -> c_1(f#(g(X)),g#(X)):2
          
          3:W:g#(s(X)) -> c_3(g#(X))
             -->_1 g#(s(X)) -> c_3(g#(X)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(X) -> c_1(f#(g(X)),g#(X))
          3: g#(s(X)) -> c_3(g#(X))
** Step 5.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Weak TRS:
            g(0()) -> s(0())
            g(s(X)) -> s(s(g(X)))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,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: sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,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_5) = {1}
        
        Following symbols are considered usable:
          {f#,g#,sel#}
        TcT has computed the following interpretation:
             p(0) = [0]         
          p(cons) = [0]         
             p(f) = [1] x1 + [2]
             p(g) = [1]         
             p(s) = [1] x1 + [1]
           p(sel) = [2]         
            p(f#) = [8] x1 + [1]
            p(g#) = [1]         
          p(sel#) = [5] x1 + [0]
           p(c_1) = [2] x2 + [2]
           p(c_2) = [1]         
           p(c_3) = [2]         
           p(c_4) = [0]         
           p(c_5) = [1] x1 + [3]
        
        Following rules are strictly oriented:
        sel#(s(X),cons(Y,Z)) = [5] X + [5]   
                             > [5] X + [3]   
                             = c_5(sel#(X,Z))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sel#(s(X),cons(Y,Z)) -> c_5(sel#(X,Z))
*** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/1,g/1,sel/2,f#/1,g#/1,sel#/2} / {0/0,cons/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,sel#} and constructors {0,cons,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE