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