MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            add0(x,0()) -> x
            add0(x',S(x)) -> +(S(0()),add0(x',x))
            mult(x,0()) -> 0()
            mult(x',S(x)) -> add0(x',mult(x',x))
            power(x,0()) -> S(0())
            power(x',S(x)) -> mult(x',power(x',x))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
        - Signature:
            {+/2,add0/2,mult/2,power/2} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,add0,mult,power} and constructors {0,S}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add0#(x,0()) -> c_1()
          add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
          mult#(x,0()) -> c_3()
          mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
          power#(x,0()) -> c_5()
          power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
        Weak DPs
          +#(x,S(0())) -> c_7()
          +#(S(0()),y) -> c_8()
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            add0#(x,0()) -> c_1()
            add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
            mult#(x,0()) -> c_3()
            mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
            power#(x,0()) -> c_5()
            power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
        - Weak DPs:
            +#(x,S(0())) -> c_7()
            +#(S(0()),y) -> c_8()
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            add0(x,0()) -> x
            add0(x',S(x)) -> +(S(0()),add0(x',x))
            mult(x,0()) -> 0()
            mult(x',S(x)) -> add0(x',mult(x',x))
            power(x,0()) -> S(0())
            power(x',S(x)) -> mult(x',power(x',x))
        - Signature:
            {+/2,add0/2,mult/2,power/2,+#/2,add0#/2,mult#/2,power#/2} / {0/0,S/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/2
            ,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,add0#,mult#,power#} and constructors {0,S}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5}
        by application of
          Pre({1,3,5}) = {2,4,6}.
        Here rules are labelled as follows:
          1: add0#(x,0()) -> c_1()
          2: add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
          3: mult#(x,0()) -> c_3()
          4: mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
          5: power#(x,0()) -> c_5()
          6: power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
          7: +#(x,S(0())) -> c_7()
          8: +#(S(0()),y) -> c_8()
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
            mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
            power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
        - Weak DPs:
            +#(x,S(0())) -> c_7()
            +#(S(0()),y) -> c_8()
            add0#(x,0()) -> c_1()
            mult#(x,0()) -> c_3()
            power#(x,0()) -> c_5()
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            add0(x,0()) -> x
            add0(x',S(x)) -> +(S(0()),add0(x',x))
            mult(x,0()) -> 0()
            mult(x',S(x)) -> add0(x',mult(x',x))
            power(x,0()) -> S(0())
            power(x',S(x)) -> mult(x',power(x',x))
        - Signature:
            {+/2,add0/2,mult/2,power/2,+#/2,add0#/2,mult#/2,power#/2} / {0/0,S/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/2
            ,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,add0#,mult#,power#} and constructors {0,S}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
             -->_2 add0#(x,0()) -> c_1():6
             -->_1 +#(S(0()),y) -> c_8():5
             -->_1 +#(x,S(0())) -> c_7():4
             -->_2 add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x)):1
          
          2:S:mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
             -->_2 mult#(x,0()) -> c_3():7
             -->_1 add0#(x,0()) -> c_1():6
             -->_2 mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x)):2
             -->_1 add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x)):1
          
          3:S:power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
             -->_2 power#(x,0()) -> c_5():8
             -->_1 mult#(x,0()) -> c_3():7
             -->_2 power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x)):3
             -->_1 mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x)):2
          
          4:W:+#(x,S(0())) -> c_7()
             
          
          5:W:+#(S(0()),y) -> c_8()
             
          
          6:W:add0#(x,0()) -> c_1()
             
          
          7:W:mult#(x,0()) -> c_3()
             
          
          8:W:power#(x,0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: power#(x,0()) -> c_5()
          7: mult#(x,0()) -> c_3()
          4: +#(x,S(0())) -> c_7()
          5: +#(S(0()),y) -> c_8()
          6: add0#(x,0()) -> c_1()
* Step 4: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
            mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
            power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            add0(x,0()) -> x
            add0(x',S(x)) -> +(S(0()),add0(x',x))
            mult(x,0()) -> 0()
            mult(x',S(x)) -> add0(x',mult(x',x))
            power(x,0()) -> S(0())
            power(x',S(x)) -> mult(x',power(x',x))
        - Signature:
            {+/2,add0/2,mult/2,power/2,+#/2,add0#/2,mult#/2,power#/2} / {0/0,S/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/2
            ,c_7/0,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,add0#,mult#,power#} and constructors {0,S}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x))
             -->_2 add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x)):1
          
          2:S:mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
             -->_2 mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x)):2
             -->_1 add0#(x',S(x)) -> c_2(+#(S(0()),add0(x',x)),add0#(x',x)):1
          
          3:S:power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
             -->_2 power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x)):3
             -->_1 mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          add0#(x',S(x)) -> c_2(add0#(x',x))
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          add0#(x',S(x)) -> c_2(add0#(x',x))
          mult#(x',S(x)) -> c_4(add0#(x',mult(x',x)),mult#(x',x))
          power#(x',S(x)) -> c_6(mult#(x',power(x',x)),power#(x',x))
      - Weak TRS:
          +(x,S(0())) -> S(x)
          +(S(0()),y) -> S(y)
          add0(x,0()) -> x
          add0(x',S(x)) -> +(S(0()),add0(x',x))
          mult(x,0()) -> 0()
          mult(x',S(x)) -> add0(x',mult(x',x))
          power(x,0()) -> S(0())
          power(x',S(x)) -> mult(x',power(x',x))
      - Signature:
          {+/2,add0/2,mult/2,power/2,+#/2,add0#/2,mult#/2,power#/2} / {0/0,S/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/0,c_6/2
          ,c_7/0,c_8/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {+#,add0#,mult#,power#} and constructors {0,S}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE