MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            ack_in(0(),n) -> ack_out(s(n))
            ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
            ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
            u11(ack_out(n)) -> ack_out(n)
            u21(ack_out(n),m) -> u22(ack_in(m,n))
            u22(ack_out(n)) -> ack_out(n)
        - Signature:
            {ack_in/2,u11/1,u21/2,u22/1} / {0/0,ack_out/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack_in,u11,u21,u22} and constructors {0,ack_out,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          ack_in#(0(),n) -> c_1()
          ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
          ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
          u11#(ack_out(n)) -> c_4()
          u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
          u22#(ack_out(n)) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            ack_in#(0(),n) -> c_1()
            ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
            ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
            u11#(ack_out(n)) -> c_4()
            u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
            u22#(ack_out(n)) -> c_6()
        - Weak TRS:
            ack_in(0(),n) -> ack_out(s(n))
            ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
            ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
            u11(ack_out(n)) -> ack_out(n)
            u21(ack_out(n),m) -> u22(ack_in(m,n))
            u22(ack_out(n)) -> ack_out(n)
        - Signature:
            {ack_in/2,u11/1,u21/2,u22/1,ack_in#/2,u11#/1,u21#/2,u22#/1} / {0/0,ack_out/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0
            ,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack_in#,u11#,u21#,u22#} and constructors {0,ack_out,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,6}
        by application of
          Pre({1,4,6}) = {2,5}.
        Here rules are labelled as follows:
          1: ack_in#(0(),n) -> c_1()
          2: ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
          3: ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
          4: u11#(ack_out(n)) -> c_4()
          5: u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
          6: u22#(ack_out(n)) -> c_6()
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
            ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
            u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
        - Weak DPs:
            ack_in#(0(),n) -> c_1()
            u11#(ack_out(n)) -> c_4()
            u22#(ack_out(n)) -> c_6()
        - Weak TRS:
            ack_in(0(),n) -> ack_out(s(n))
            ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
            ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
            u11(ack_out(n)) -> ack_out(n)
            u21(ack_out(n),m) -> u22(ack_in(m,n))
            u22(ack_out(n)) -> ack_out(n)
        - Signature:
            {ack_in/2,u11/1,u21/2,u22/1,ack_in#/2,u11#/1,u21#/2,u22#/1} / {0/0,ack_out/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0
            ,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack_in#,u11#,u21#,u22#} and constructors {0,ack_out,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
             -->_1 u11#(ack_out(n)) -> c_4():5
             -->_2 ack_in#(0(),n) -> c_1():4
          
          2:S:ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
             -->_1 u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n)):3
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
             -->_2 ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0()))):1
          
          3:S:u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
             -->_1 u22#(ack_out(n)) -> c_6():6
             -->_2 ack_in#(0(),n) -> c_1():4
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
             -->_2 ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0()))):1
          
          4:W:ack_in#(0(),n) -> c_1()
             
          
          5:W:u11#(ack_out(n)) -> c_4()
             
          
          6:W:u22#(ack_out(n)) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: u11#(ack_out(n)) -> c_4()
          4: ack_in#(0(),n) -> c_1()
          6: u22#(ack_out(n)) -> c_6()
* Step 4: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
            ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
            u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
        - Weak TRS:
            ack_in(0(),n) -> ack_out(s(n))
            ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
            ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
            u11(ack_out(n)) -> ack_out(n)
            u21(ack_out(n),m) -> u22(ack_in(m,n))
            u22(ack_out(n)) -> ack_out(n)
        - Signature:
            {ack_in/2,u11/1,u21/2,u22/1,ack_in#/2,u11#/1,u21#/2,u22#/1} / {0/0,ack_out/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0
            ,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack_in#,u11#,u21#,u22#} and constructors {0,ack_out,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0())))
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
          
          2:S:ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
             -->_1 u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n)):3
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
             -->_2 ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0()))):1
          
          3:S:u21#(ack_out(n),m) -> c_5(u22#(ack_in(m,n)),ack_in#(m,n))
             -->_2 ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n)):2
             -->_2 ack_in#(s(m),0()) -> c_2(u11#(ack_in(m,s(0()))),ack_in#(m,s(0()))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          ack_in#(s(m),0()) -> c_2(ack_in#(m,s(0())))
          u21#(ack_out(n),m) -> c_5(ack_in#(m,n))
* Step 5: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            ack_in#(s(m),0()) -> c_2(ack_in#(m,s(0())))
            ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
            u21#(ack_out(n),m) -> c_5(ack_in#(m,n))
        - Weak TRS:
            ack_in(0(),n) -> ack_out(s(n))
            ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
            ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
            u11(ack_out(n)) -> ack_out(n)
            u21(ack_out(n),m) -> u22(ack_in(m,n))
            u22(ack_out(n)) -> ack_out(n)
        - Signature:
            {ack_in/2,u11/1,u21/2,u22/1,ack_in#/2,u11#/1,u21#/2,u22#/1} / {0/0,ack_out/1,s/1,c_1/0,c_2/1,c_3/2,c_4/0
            ,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack_in#,u11#,u21#,u22#} and constructors {0,ack_out,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(u11) = {1},
            uargs(u21) = {1},
            uargs(u22) = {1},
            uargs(u21#) = {1},
            uargs(c_2) = {1},
            uargs(c_3) = {1,2},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(0) = [0]                  
             p(ack_in) = [3]                  
            p(ack_out) = [3]                  
                  p(s) = [0]                  
                p(u11) = [1] x1 + [0]         
                p(u21) = [1] x1 + [0]         
                p(u22) = [1] x1 + [0]         
            p(ack_in#) = [0]                  
               p(u11#) = [0]                  
               p(u21#) = [1] x1 + [0]         
               p(u22#) = [0]                  
                p(c_1) = [0]                  
                p(c_2) = [1] x1 + [0]         
                p(c_3) = [1] x1 + [1] x2 + [0]
                p(c_4) = [0]                  
                p(c_5) = [1] x1 + [0]         
                p(c_6) = [0]                  
          
          Following rules are strictly oriented:
          u21#(ack_out(n),m) = [3]              
                             > [0]              
                             = c_5(ack_in#(m,n))
          
          
          Following rules are (at-least) weakly oriented:
           ack_in#(s(m),0()) =  [0]                                        
                             >= [0]                                        
                             =  c_2(ack_in#(m,s(0())))                     
          
          ack_in#(s(m),s(n)) =  [0]                                        
                             >= [3]                                        
                             =  c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
          
               ack_in(0(),n) =  [3]                                        
                             >= [3]                                        
                             =  ack_out(s(n))                              
          
            ack_in(s(m),0()) =  [3]                                        
                             >= [3]                                        
                             =  u11(ack_in(m,s(0())))                      
          
           ack_in(s(m),s(n)) =  [3]                                        
                             >= [3]                                        
                             =  u21(ack_in(s(m),n),m)                      
          
             u11(ack_out(n)) =  [3]                                        
                             >= [3]                                        
                             =  ack_out(n)                                 
          
           u21(ack_out(n),m) =  [3]                                        
                             >= [3]                                        
                             =  u22(ack_in(m,n))                           
          
             u22(ack_out(n)) =  [3]                                        
                             >= [3]                                        
                             =  ack_out(n)                                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ack_in#(s(m),0()) -> c_2(ack_in#(m,s(0())))
          ack_in#(s(m),s(n)) -> c_3(u21#(ack_in(s(m),n),m),ack_in#(s(m),n))
      - Weak DPs:
          u21#(ack_out(n),m) -> c_5(ack_in#(m,n))
      - Weak TRS:
          ack_in(0(),n) -> ack_out(s(n))
          ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
          ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
          u11(ack_out(n)) -> ack_out(n)
          u21(ack_out(n),m) -> u22(ack_in(m,n))
          u22(ack_out(n)) -> ack_out(n)
      - Signature:
          {ack_in/2,u11/1,u21/2,u22/1,ack_in#/2,u11#/1,u21#/2,u22#/1} / {0/0,ack_out/1,s/1,c_1/0,c_2/1,c_3/2,c_4/0
          ,c_5/1,c_6/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {ack_in#,u11#,u21#,u22#} and constructors {0,ack_out,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE