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