MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            ack(0(),x) -> s(x)
            ack(s(x),0()) -> ack(x,s(0()))
            ack(s(x),s(y)) -> ack(x,ack(s(x),y))
            d(x) -> if(le(x,nr()),x)
            if(false(),x) -> nil()
            if(true(),x) -> cons(x,d(s(x)))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            nr() -> ack(s(s(s(s(s(s(0())))))),0())
            numbers() -> d(0())
        - Signature:
            {ack/2,d/1,if/2,le/2,nr/0,numbers/0} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack,d,if,le,nr,numbers} and constructors {0,cons,false
            ,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          ack#(0(),x) -> c_1()
          ack#(s(x),0()) -> c_2(ack#(x,s(0())))
          ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
          d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
          if#(false(),x) -> c_5()
          if#(true(),x) -> c_6(d#(s(x)))
          le#(0(),y) -> c_7()
          le#(s(x),0()) -> c_8()
          le#(s(x),s(y)) -> c_9(le#(x,y))
          nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
          numbers#() -> c_11(d#(0()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            ack#(0(),x) -> c_1()
            ack#(s(x),0()) -> c_2(ack#(x,s(0())))
            ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
            d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
            if#(false(),x) -> c_5()
            if#(true(),x) -> c_6(d#(s(x)))
            le#(0(),y) -> c_7()
            le#(s(x),0()) -> c_8()
            le#(s(x),s(y)) -> c_9(le#(x,y))
            nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
            numbers#() -> c_11(d#(0()))
        - Weak TRS:
            ack(0(),x) -> s(x)
            ack(s(x),0()) -> ack(x,s(0()))
            ack(s(x),s(y)) -> ack(x,ack(s(x),y))
            d(x) -> if(le(x,nr()),x)
            if(false(),x) -> nil()
            if(true(),x) -> cons(x,d(s(x)))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            nr() -> ack(s(s(s(s(s(s(0())))))),0())
            numbers() -> d(0())
        - Signature:
            {ack/2,d/1,if/2,le/2,nr/0,numbers/0,ack#/2,d#/1,if#/2,le#/2,nr#/0,numbers#/0} / {0/0,cons/2,false/0,nil/0
            ,s/1,true/0,c_1/0,c_2/1,c_3/2,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack#,d#,if#,le#,nr#,numbers#} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ack(0(),x) -> s(x)
          ack(s(x),0()) -> ack(x,s(0()))
          ack(s(x),s(y)) -> ack(x,ack(s(x),y))
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          nr() -> ack(s(s(s(s(s(s(0())))))),0())
          ack#(0(),x) -> c_1()
          ack#(s(x),0()) -> c_2(ack#(x,s(0())))
          ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
          d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
          if#(false(),x) -> c_5()
          if#(true(),x) -> c_6(d#(s(x)))
          le#(0(),y) -> c_7()
          le#(s(x),0()) -> c_8()
          le#(s(x),s(y)) -> c_9(le#(x,y))
          nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
          numbers#() -> c_11(d#(0()))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            ack#(0(),x) -> c_1()
            ack#(s(x),0()) -> c_2(ack#(x,s(0())))
            ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
            d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
            if#(false(),x) -> c_5()
            if#(true(),x) -> c_6(d#(s(x)))
            le#(0(),y) -> c_7()
            le#(s(x),0()) -> c_8()
            le#(s(x),s(y)) -> c_9(le#(x,y))
            nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
            numbers#() -> c_11(d#(0()))
        - Weak TRS:
            ack(0(),x) -> s(x)
            ack(s(x),0()) -> ack(x,s(0()))
            ack(s(x),s(y)) -> ack(x,ack(s(x),y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            nr() -> ack(s(s(s(s(s(s(0())))))),0())
        - Signature:
            {ack/2,d/1,if/2,le/2,nr/0,numbers/0,ack#/2,d#/1,if#/2,le#/2,nr#/0,numbers#/0} / {0/0,cons/2,false/0,nil/0
            ,s/1,true/0,c_1/0,c_2/1,c_3/2,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack#,d#,if#,le#,nr#,numbers#} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,5,7,8}
        by application of
          Pre({1,5,7,8}) = {2,3,4,9}.
        Here rules are labelled as follows:
          1: ack#(0(),x) -> c_1()
          2: ack#(s(x),0()) -> c_2(ack#(x,s(0())))
          3: ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
          4: d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
          5: if#(false(),x) -> c_5()
          6: if#(true(),x) -> c_6(d#(s(x)))
          7: le#(0(),y) -> c_7()
          8: le#(s(x),0()) -> c_8()
          9: le#(s(x),s(y)) -> c_9(le#(x,y))
          10: nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
          11: numbers#() -> c_11(d#(0()))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            ack#(s(x),0()) -> c_2(ack#(x,s(0())))
            ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
            d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
            if#(true(),x) -> c_6(d#(s(x)))
            le#(s(x),s(y)) -> c_9(le#(x,y))
            nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
            numbers#() -> c_11(d#(0()))
        - Weak DPs:
            ack#(0(),x) -> c_1()
            if#(false(),x) -> c_5()
            le#(0(),y) -> c_7()
            le#(s(x),0()) -> c_8()
        - Weak TRS:
            ack(0(),x) -> s(x)
            ack(s(x),0()) -> ack(x,s(0()))
            ack(s(x),s(y)) -> ack(x,ack(s(x),y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            nr() -> ack(s(s(s(s(s(s(0())))))),0())
        - Signature:
            {ack/2,d/1,if/2,le/2,nr/0,numbers/0,ack#/2,d#/1,if#/2,le#/2,nr#/0,numbers#/0} / {0/0,cons/2,false/0,nil/0
            ,s/1,true/0,c_1/0,c_2/1,c_3/2,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack#,d#,if#,le#,nr#,numbers#} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ack#(s(x),0()) -> c_2(ack#(x,s(0())))
             -->_1 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
             -->_1 ack#(0(),x) -> c_1():8
          
          2:S:ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
             -->_1 ack#(0(),x) -> c_1():8
             -->_2 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
             -->_1 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
             -->_2 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
             -->_1 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
          
          3:S:d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
             -->_3 nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0())):6
             -->_2 le#(s(x),s(y)) -> c_9(le#(x,y)):5
             -->_1 if#(true(),x) -> c_6(d#(s(x))):4
             -->_2 le#(s(x),0()) -> c_8():11
             -->_2 le#(0(),y) -> c_7():10
             -->_1 if#(false(),x) -> c_5():9
          
          4:S:if#(true(),x) -> c_6(d#(s(x)))
             -->_1 d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#()):3
          
          5:S:le#(s(x),s(y)) -> c_9(le#(x,y))
             -->_1 le#(s(x),0()) -> c_8():11
             -->_1 le#(0(),y) -> c_7():10
             -->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):5
          
          6:S:nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
             -->_1 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
          
          7:S:numbers#() -> c_11(d#(0()))
             -->_1 d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#()):3
          
          8:W:ack#(0(),x) -> c_1()
             
          
          9:W:if#(false(),x) -> c_5()
             
          
          10:W:le#(0(),y) -> c_7()
             
          
          11:W:le#(s(x),0()) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: if#(false(),x) -> c_5()
          10: le#(0(),y) -> c_7()
          11: le#(s(x),0()) -> c_8()
          8: ack#(0(),x) -> c_1()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            ack#(s(x),0()) -> c_2(ack#(x,s(0())))
            ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
            d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
            if#(true(),x) -> c_6(d#(s(x)))
            le#(s(x),s(y)) -> c_9(le#(x,y))
            nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
            numbers#() -> c_11(d#(0()))
        - Weak TRS:
            ack(0(),x) -> s(x)
            ack(s(x),0()) -> ack(x,s(0()))
            ack(s(x),s(y)) -> ack(x,ack(s(x),y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            nr() -> ack(s(s(s(s(s(s(0())))))),0())
        - Signature:
            {ack/2,d/1,if/2,le/2,nr/0,numbers/0,ack#/2,d#/1,if#/2,le#/2,nr#/0,numbers#/0} / {0/0,cons/2,false/0,nil/0
            ,s/1,true/0,c_1/0,c_2/1,c_3/2,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ack#,d#,if#,le#,nr#,numbers#} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:ack#(s(x),0()) -> c_2(ack#(x,s(0())))
           -->_1 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
        
        2:S:ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
           -->_2 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
           -->_1 ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y)):2
           -->_2 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
           -->_1 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
        
        3:S:d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
           -->_3 nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0())):6
           -->_2 le#(s(x),s(y)) -> c_9(le#(x,y)):5
           -->_1 if#(true(),x) -> c_6(d#(s(x))):4
        
        4:S:if#(true(),x) -> c_6(d#(s(x)))
           -->_1 d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#()):3
        
        5:S:le#(s(x),s(y)) -> c_9(le#(x,y))
           -->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):5
        
        6:S:nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
           -->_1 ack#(s(x),0()) -> c_2(ack#(x,s(0()))):1
        
        7:S:numbers#() -> c_11(d#(0()))
           -->_1 d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#()):3
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(7,numbers#() -> c_11(d#(0())))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          ack#(s(x),0()) -> c_2(ack#(x,s(0())))
          ack#(s(x),s(y)) -> c_3(ack#(x,ack(s(x),y)),ack#(s(x),y))
          d#(x) -> c_4(if#(le(x,nr()),x),le#(x,nr()),nr#())
          if#(true(),x) -> c_6(d#(s(x)))
          le#(s(x),s(y)) -> c_9(le#(x,y))
          nr#() -> c_10(ack#(s(s(s(s(s(s(0())))))),0()))
      - Weak TRS:
          ack(0(),x) -> s(x)
          ack(s(x),0()) -> ack(x,s(0()))
          ack(s(x),s(y)) -> ack(x,ack(s(x),y))
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          nr() -> ack(s(s(s(s(s(s(0())))))),0())
      - Signature:
          {ack/2,d/1,if/2,le/2,nr/0,numbers/0,ack#/2,d#/1,if#/2,le#/2,nr#/0,numbers#/0} / {0/0,cons/2,false/0,nil/0
          ,s/1,true/0,c_1/0,c_2/1,c_3/2,c_4/3,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {ack#,d#,if#,le#,nr#,numbers#} and constructors {0,cons
          ,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE