MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fact(X) -> if(zero(X),s(0()),prod(X,fact(p(X))))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            p(s(X)) -> X
            prod(0(),X) -> 0()
            prod(s(X),Y) -> add(Y,prod(X,Y))
            zero(0()) -> true()
            zero(s(X)) -> false()
        - Signature:
            {add/2,fact/1,if/3,p/1,prod/2,zero/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,fact,if,p,prod,zero} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
          if#(false(),X,Y) -> c_4()
          if#(true(),X,Y) -> c_5()
          p#(s(X)) -> c_6()
          prod#(0(),X) -> c_7()
          prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
          zero#(0()) -> c_9()
          zero#(s(X)) -> c_10()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
            if#(false(),X,Y) -> c_4()
            if#(true(),X,Y) -> c_5()
            p#(s(X)) -> c_6()
            prod#(0(),X) -> c_7()
            prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
            zero#(0()) -> c_9()
            zero#(s(X)) -> c_10()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fact(X) -> if(zero(X),s(0()),prod(X,fact(p(X))))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            p(s(X)) -> X
            prod(0(),X) -> 0()
            prod(s(X),Y) -> add(Y,prod(X,Y))
            zero(0()) -> true()
            zero(s(X)) -> false()
        - Signature:
            {add/2,fact/1,if/3,p/1,prod/2,zero/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,zero#/1} / {0/0,false/0,s/1,true/0
            ,c_1/0,c_2/1,c_3/5,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fact#,if#,p#,prod#,zero#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6,7,9,10}
        by application of
          Pre({1,4,5,6,7,9,10}) = {2,3,8}.
        Here rules are labelled as follows:
          1: add#(0(),X) -> c_1()
          2: add#(s(X),Y) -> c_2(add#(X,Y))
          3: fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
          4: if#(false(),X,Y) -> c_4()
          5: if#(true(),X,Y) -> c_5()
          6: p#(s(X)) -> c_6()
          7: prod#(0(),X) -> c_7()
          8: prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
          9: zero#(0()) -> c_9()
          10: zero#(s(X)) -> c_10()
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
            prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
        - Weak DPs:
            add#(0(),X) -> c_1()
            if#(false(),X,Y) -> c_4()
            if#(true(),X,Y) -> c_5()
            p#(s(X)) -> c_6()
            prod#(0(),X) -> c_7()
            zero#(0()) -> c_9()
            zero#(s(X)) -> c_10()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fact(X) -> if(zero(X),s(0()),prod(X,fact(p(X))))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            p(s(X)) -> X
            prod(0(),X) -> 0()
            prod(s(X),Y) -> add(Y,prod(X,Y))
            zero(0()) -> true()
            zero(s(X)) -> false()
        - Signature:
            {add/2,fact/1,if/3,p/1,prod/2,zero/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,zero#/1} / {0/0,false/0,s/1,true/0
            ,c_1/0,c_2/1,c_3/5,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fact#,if#,p#,prod#,zero#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(0(),X) -> c_1():4
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:S:fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
             -->_3 prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y)):3
             -->_2 zero#(s(X)) -> c_10():10
             -->_2 zero#(0()) -> c_9():9
             -->_3 prod#(0(),X) -> c_7():8
             -->_5 p#(s(X)) -> c_6():7
             -->_1 if#(true(),X,Y) -> c_5():6
             -->_1 if#(false(),X,Y) -> c_4():5
             -->_4 fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X))))
                                  ,zero#(X)
                                  ,prod#(X,fact(p(X)))
                                  ,fact#(p(X))
                                  ,p#(X)):2
          
          3:S:prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
             -->_2 prod#(0(),X) -> c_7():8
             -->_1 add#(0(),X) -> c_1():4
             -->_2 prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y)):3
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          4:W:add#(0(),X) -> c_1()
             
          
          5:W:if#(false(),X,Y) -> c_4()
             
          
          6:W:if#(true(),X,Y) -> c_5()
             
          
          7:W:p#(s(X)) -> c_6()
             
          
          8:W:prod#(0(),X) -> c_7()
             
          
          9:W:zero#(0()) -> c_9()
             
          
          10:W:zero#(s(X)) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: if#(false(),X,Y) -> c_4()
          6: if#(true(),X,Y) -> c_5()
          7: p#(s(X)) -> c_6()
          9: zero#(0()) -> c_9()
          10: zero#(s(X)) -> c_10()
          8: prod#(0(),X) -> c_7()
          4: add#(0(),X) -> c_1()
* Step 4: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
            prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fact(X) -> if(zero(X),s(0()),prod(X,fact(p(X))))
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            p(s(X)) -> X
            prod(0(),X) -> 0()
            prod(s(X),Y) -> add(Y,prod(X,Y))
            zero(0()) -> true()
            zero(s(X)) -> false()
        - Signature:
            {add/2,fact/1,if/3,p/1,prod/2,zero/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,zero#/1} / {0/0,false/0,s/1,true/0
            ,c_1/0,c_2/1,c_3/5,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,fact#,if#,p#,prod#,zero#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:S:fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X)))),zero#(X),prod#(X,fact(p(X))),fact#(p(X)),p#(X))
             -->_3 prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y)):3
             -->_4 fact#(X) -> c_3(if#(zero(X),s(0()),prod(X,fact(p(X))))
                                  ,zero#(X)
                                  ,prod#(X,fact(p(X)))
                                  ,fact#(p(X))
                                  ,p#(X)):2
          
          3:S:prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
             -->_2 prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y)):3
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fact#(X) -> c_3(prod#(X,fact(p(X))),fact#(p(X)))
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          add#(s(X),Y) -> c_2(add#(X,Y))
          fact#(X) -> c_3(prod#(X,fact(p(X))),fact#(p(X)))
          prod#(s(X),Y) -> c_8(add#(Y,prod(X,Y)),prod#(X,Y))
      - Weak TRS:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          fact(X) -> if(zero(X),s(0()),prod(X,fact(p(X))))
          if(false(),X,Y) -> Y
          if(true(),X,Y) -> X
          p(s(X)) -> X
          prod(0(),X) -> 0()
          prod(s(X),Y) -> add(Y,prod(X,Y))
          zero(0()) -> true()
          zero(s(X)) -> false()
      - Signature:
          {add/2,fact/1,if/3,p/1,prod/2,zero/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,zero#/1} / {0/0,false/0,s/1,true/0
          ,c_1/0,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/0,c_8/2,c_9/0,c_10/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,fact#,if#,p#,prod#,zero#} and constructors {0,false
          ,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE