MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            double(x) -> g(d(),x)
            f(s(x),y) -> f(half(s(x)),double(y))
            f(s(0()),y) -> y
            g(x,0()) -> 0()
            g(d(),s(x)) -> s(s(g(d(),x)))
            g(h(),s(0())) -> 0()
            g(h(),s(s(x))) -> s(g(h(),x))
            half(x) -> g(h(),x)
            id(x) -> f(x,s(0()))
        - Signature:
            {double/1,f/2,g/2,half/1,id/1} / {0/0,d/0,h/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {double,f,g,half,id} and constructors {0,d,h,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          double#(x) -> c_1(g#(d(),x))
          f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
          f#(s(0()),y) -> c_3()
          g#(x,0()) -> c_4()
          g#(d(),s(x)) -> c_5(g#(d(),x))
          g#(h(),s(0())) -> c_6()
          g#(h(),s(s(x))) -> c_7(g#(h(),x))
          half#(x) -> c_8(g#(h(),x))
          id#(x) -> c_9(f#(x,s(0())))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(x) -> c_1(g#(d(),x))
            f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
            f#(s(0()),y) -> c_3()
            g#(x,0()) -> c_4()
            g#(d(),s(x)) -> c_5(g#(d(),x))
            g#(h(),s(0())) -> c_6()
            g#(h(),s(s(x))) -> c_7(g#(h(),x))
            half#(x) -> c_8(g#(h(),x))
            id#(x) -> c_9(f#(x,s(0())))
        - Weak TRS:
            double(x) -> g(d(),x)
            f(s(x),y) -> f(half(s(x)),double(y))
            f(s(0()),y) -> y
            g(x,0()) -> 0()
            g(d(),s(x)) -> s(s(g(d(),x)))
            g(h(),s(0())) -> 0()
            g(h(),s(s(x))) -> s(g(h(),x))
            half(x) -> g(h(),x)
            id(x) -> f(x,s(0()))
        - Signature:
            {double/1,f/2,g/2,half/1,id/1,double#/1,f#/2,g#/2,half#/1,id#/1} / {0/0,d/0,h/0,s/1,c_1/1,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {double#,f#,g#,half#,id#} and constructors {0,d,h,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          double(x) -> g(d(),x)
          g(x,0()) -> 0()
          g(d(),s(x)) -> s(s(g(d(),x)))
          g(h(),s(0())) -> 0()
          g(h(),s(s(x))) -> s(g(h(),x))
          half(x) -> g(h(),x)
          double#(x) -> c_1(g#(d(),x))
          f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
          f#(s(0()),y) -> c_3()
          g#(x,0()) -> c_4()
          g#(d(),s(x)) -> c_5(g#(d(),x))
          g#(h(),s(0())) -> c_6()
          g#(h(),s(s(x))) -> c_7(g#(h(),x))
          half#(x) -> c_8(g#(h(),x))
          id#(x) -> c_9(f#(x,s(0())))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(x) -> c_1(g#(d(),x))
            f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
            f#(s(0()),y) -> c_3()
            g#(x,0()) -> c_4()
            g#(d(),s(x)) -> c_5(g#(d(),x))
            g#(h(),s(0())) -> c_6()
            g#(h(),s(s(x))) -> c_7(g#(h(),x))
            half#(x) -> c_8(g#(h(),x))
            id#(x) -> c_9(f#(x,s(0())))
        - Weak TRS:
            double(x) -> g(d(),x)
            g(x,0()) -> 0()
            g(d(),s(x)) -> s(s(g(d(),x)))
            g(h(),s(0())) -> 0()
            g(h(),s(s(x))) -> s(g(h(),x))
            half(x) -> g(h(),x)
        - Signature:
            {double/1,f/2,g/2,half/1,id/1,double#/1,f#/2,g#/2,half#/1,id#/1} / {0/0,d/0,h/0,s/1,c_1/1,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {double#,f#,g#,half#,id#} and constructors {0,d,h,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4,6}
        by application of
          Pre({3,4,6}) = {1,2,5,7,8,9}.
        Here rules are labelled as follows:
          1: double#(x) -> c_1(g#(d(),x))
          2: f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
          3: f#(s(0()),y) -> c_3()
          4: g#(x,0()) -> c_4()
          5: g#(d(),s(x)) -> c_5(g#(d(),x))
          6: g#(h(),s(0())) -> c_6()
          7: g#(h(),s(s(x))) -> c_7(g#(h(),x))
          8: half#(x) -> c_8(g#(h(),x))
          9: id#(x) -> c_9(f#(x,s(0())))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(x) -> c_1(g#(d(),x))
            f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
            g#(d(),s(x)) -> c_5(g#(d(),x))
            g#(h(),s(s(x))) -> c_7(g#(h(),x))
            half#(x) -> c_8(g#(h(),x))
            id#(x) -> c_9(f#(x,s(0())))
        - Weak DPs:
            f#(s(0()),y) -> c_3()
            g#(x,0()) -> c_4()
            g#(h(),s(0())) -> c_6()
        - Weak TRS:
            double(x) -> g(d(),x)
            g(x,0()) -> 0()
            g(d(),s(x)) -> s(s(g(d(),x)))
            g(h(),s(0())) -> 0()
            g(h(),s(s(x))) -> s(g(h(),x))
            half(x) -> g(h(),x)
        - Signature:
            {double/1,f/2,g/2,half/1,id/1,double#/1,f#/2,g#/2,half#/1,id#/1} / {0/0,d/0,h/0,s/1,c_1/1,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {double#,f#,g#,half#,id#} and constructors {0,d,h,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:double#(x) -> c_1(g#(d(),x))
             -->_1 g#(d(),s(x)) -> c_5(g#(d(),x)):3
             -->_1 g#(x,0()) -> c_4():8
          
          2:S:f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
             -->_2 half#(x) -> c_8(g#(h(),x)):5
             -->_1 f#(s(0()),y) -> c_3():7
             -->_1 f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y)):2
             -->_3 double#(x) -> c_1(g#(d(),x)):1
          
          3:S:g#(d(),s(x)) -> c_5(g#(d(),x))
             -->_1 g#(x,0()) -> c_4():8
             -->_1 g#(d(),s(x)) -> c_5(g#(d(),x)):3
          
          4:S:g#(h(),s(s(x))) -> c_7(g#(h(),x))
             -->_1 g#(h(),s(0())) -> c_6():9
             -->_1 g#(x,0()) -> c_4():8
             -->_1 g#(h(),s(s(x))) -> c_7(g#(h(),x)):4
          
          5:S:half#(x) -> c_8(g#(h(),x))
             -->_1 g#(h(),s(0())) -> c_6():9
             -->_1 g#(x,0()) -> c_4():8
             -->_1 g#(h(),s(s(x))) -> c_7(g#(h(),x)):4
          
          6:S:id#(x) -> c_9(f#(x,s(0())))
             -->_1 f#(s(0()),y) -> c_3():7
             -->_1 f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y)):2
          
          7:W:f#(s(0()),y) -> c_3()
             
          
          8:W:g#(x,0()) -> c_4()
             
          
          9:W:g#(h(),s(0())) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: f#(s(0()),y) -> c_3()
          9: g#(h(),s(0())) -> c_6()
          8: g#(x,0()) -> c_4()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            double#(x) -> c_1(g#(d(),x))
            f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
            g#(d(),s(x)) -> c_5(g#(d(),x))
            g#(h(),s(s(x))) -> c_7(g#(h(),x))
            half#(x) -> c_8(g#(h(),x))
            id#(x) -> c_9(f#(x,s(0())))
        - Weak TRS:
            double(x) -> g(d(),x)
            g(x,0()) -> 0()
            g(d(),s(x)) -> s(s(g(d(),x)))
            g(h(),s(0())) -> 0()
            g(h(),s(s(x))) -> s(g(h(),x))
            half(x) -> g(h(),x)
        - Signature:
            {double/1,f/2,g/2,half/1,id/1,double#/1,f#/2,g#/2,half#/1,id#/1} / {0/0,d/0,h/0,s/1,c_1/1,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {double#,f#,g#,half#,id#} and constructors {0,d,h,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:double#(x) -> c_1(g#(d(),x))
           -->_1 g#(d(),s(x)) -> c_5(g#(d(),x)):3
        
        2:S:f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
           -->_2 half#(x) -> c_8(g#(h(),x)):5
           -->_1 f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y)):2
           -->_3 double#(x) -> c_1(g#(d(),x)):1
        
        3:S:g#(d(),s(x)) -> c_5(g#(d(),x))
           -->_1 g#(d(),s(x)) -> c_5(g#(d(),x)):3
        
        4:S:g#(h(),s(s(x))) -> c_7(g#(h(),x))
           -->_1 g#(h(),s(s(x))) -> c_7(g#(h(),x)):4
        
        5:S:half#(x) -> c_8(g#(h(),x))
           -->_1 g#(h(),s(s(x))) -> c_7(g#(h(),x)):4
        
        6:S:id#(x) -> c_9(f#(x,s(0())))
           -->_1 f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y)):2
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(6,id#(x) -> c_9(f#(x,s(0()))))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          double#(x) -> c_1(g#(d(),x))
          f#(s(x),y) -> c_2(f#(half(s(x)),double(y)),half#(s(x)),double#(y))
          g#(d(),s(x)) -> c_5(g#(d(),x))
          g#(h(),s(s(x))) -> c_7(g#(h(),x))
          half#(x) -> c_8(g#(h(),x))
      - Weak TRS:
          double(x) -> g(d(),x)
          g(x,0()) -> 0()
          g(d(),s(x)) -> s(s(g(d(),x)))
          g(h(),s(0())) -> 0()
          g(h(),s(s(x))) -> s(g(h(),x))
          half(x) -> g(h(),x)
      - Signature:
          {double/1,f/2,g/2,half/1,id/1,double#/1,f#/2,g#/2,half#/1,id#/1} / {0/0,d/0,h/0,s/1,c_1/1,c_2/3,c_3/0,c_4/0
          ,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {double#,f#,g#,half#,id#} and constructors {0,d,h,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE