MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            eq(X,Y) -> false()
            eq(0(),0()) -> true()
            eq(s(X),s(Y)) -> eq(X,Y)
            inf(X) -> cons(X,inf(s(X)))
            length(cons(X,L)) -> s(length(L))
            length(nil()) -> 0()
            take(0(),X) -> nil()
            take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
        - Signature:
            {eq/2,inf/1,length/1,take/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq,inf,length,take} and constructors {0,cons,false,nil,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          eq#(X,Y) -> c_1()
          eq#(0(),0()) -> c_2()
          eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
          inf#(X) -> c_4(inf#(s(X)))
          length#(cons(X,L)) -> c_5(length#(L))
          length#(nil()) -> c_6()
          take#(0(),X) -> c_7()
          take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            eq#(X,Y) -> c_1()
            eq#(0(),0()) -> c_2()
            eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
            inf#(X) -> c_4(inf#(s(X)))
            length#(cons(X,L)) -> c_5(length#(L))
            length#(nil()) -> c_6()
            take#(0(),X) -> c_7()
            take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
        - Weak TRS:
            eq(X,Y) -> false()
            eq(0(),0()) -> true()
            eq(s(X),s(Y)) -> eq(X,Y)
            inf(X) -> cons(X,inf(s(X)))
            length(cons(X,L)) -> s(length(L))
            length(nil()) -> 0()
            take(0(),X) -> nil()
            take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
        - Signature:
            {eq/2,inf/1,length/1,take/2,eq#/2,inf#/1,length#/1,take#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0
            ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,inf#,length#,take#} and constructors {0,cons,false
            ,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          eq#(X,Y) -> c_1()
          eq#(0(),0()) -> c_2()
          eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
          inf#(X) -> c_4(inf#(s(X)))
          length#(cons(X,L)) -> c_5(length#(L))
          length#(nil()) -> c_6()
          take#(0(),X) -> c_7()
          take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            eq#(X,Y) -> c_1()
            eq#(0(),0()) -> c_2()
            eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
            inf#(X) -> c_4(inf#(s(X)))
            length#(cons(X,L)) -> c_5(length#(L))
            length#(nil()) -> c_6()
            take#(0(),X) -> c_7()
            take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
        - Signature:
            {eq/2,inf/1,length/1,take/2,eq#/2,inf#/1,length#/1,take#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0
            ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,inf#,length#,take#} and constructors {0,cons,false
            ,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,6,7}
        by application of
          Pre({1,2,6,7}) = {3,5,8}.
        Here rules are labelled as follows:
          1: eq#(X,Y) -> c_1()
          2: eq#(0(),0()) -> c_2()
          3: eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
          4: inf#(X) -> c_4(inf#(s(X)))
          5: length#(cons(X,L)) -> c_5(length#(L))
          6: length#(nil()) -> c_6()
          7: take#(0(),X) -> c_7()
          8: take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
            inf#(X) -> c_4(inf#(s(X)))
            length#(cons(X,L)) -> c_5(length#(L))
            take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
        - Weak DPs:
            eq#(X,Y) -> c_1()
            eq#(0(),0()) -> c_2()
            length#(nil()) -> c_6()
            take#(0(),X) -> c_7()
        - Signature:
            {eq/2,inf/1,length/1,take/2,eq#/2,inf#/1,length#/1,take#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0
            ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,inf#,length#,take#} and constructors {0,cons,false
            ,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
             -->_1 eq#(0(),0()) -> c_2():6
             -->_1 eq#(X,Y) -> c_1():5
             -->_1 eq#(s(X),s(Y)) -> c_3(eq#(X,Y)):1
          
          2:S:inf#(X) -> c_4(inf#(s(X)))
             -->_1 inf#(X) -> c_4(inf#(s(X))):2
          
          3:S:length#(cons(X,L)) -> c_5(length#(L))
             -->_1 length#(nil()) -> c_6():7
             -->_1 length#(cons(X,L)) -> c_5(length#(L)):3
          
          4:S:take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
             -->_1 take#(0(),X) -> c_7():8
             -->_1 take#(s(X),cons(Y,L)) -> c_8(take#(X,L)):4
          
          5:W:eq#(X,Y) -> c_1()
             
          
          6:W:eq#(0(),0()) -> c_2()
             
          
          7:W:length#(nil()) -> c_6()
             
          
          8:W:take#(0(),X) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: take#(0(),X) -> c_7()
          7: length#(nil()) -> c_6()
          5: eq#(X,Y) -> c_1()
          6: eq#(0(),0()) -> c_2()
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          eq#(s(X),s(Y)) -> c_3(eq#(X,Y))
          inf#(X) -> c_4(inf#(s(X)))
          length#(cons(X,L)) -> c_5(length#(L))
          take#(s(X),cons(Y,L)) -> c_8(take#(X,L))
      - Signature:
          {eq/2,inf/1,length/1,take/2,eq#/2,inf#/1,length#/1,take#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0
          ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {eq#,inf#,length#,take#} and constructors {0,cons,false
          ,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE