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