MAYBE
* Step 1: InnermostRuleRemoval MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            p(s(X)) -> X
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            prod(s(X),Y) -> add(Y,prod(X,Y))
            s(X) -> n__s(X)
            zero(0()) -> true()
            zero(s(X)) -> false()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0
            ,false,n__prod,n__s,true}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          add(s(X),Y) -> s(add(X,Y))
          p(s(X)) -> X
          prod(s(X),Y) -> add(Y,prod(X,Y))
          zero(s(X)) -> false()
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            add(0(),X) -> X
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {0/0,false/0,n__prod/2,n__s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,fact,if,p,prod,s,zero} and constructors {0
            ,false,n__prod,n__s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          activate#(n__s(X)) -> c_3(s#(X))
          add#(0(),X) -> c_4()
          fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
          if#(false(),X,Y) -> c_6(activate#(Y))
          if#(true(),X,Y) -> c_7(activate#(X))
          prod#(X1,X2) -> c_8()
          prod#(0(),X) -> c_9()
          s#(X) -> c_10()
          zero#(0()) -> c_11()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
            activate#(n__s(X)) -> c_3(s#(X))
            add#(0(),X) -> c_4()
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            prod#(X1,X2) -> c_8()
            prod#(0(),X) -> c_9()
            s#(X) -> c_10()
            zero#(0()) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            add(0(),X) -> X
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__prod(X1,X2)) -> prod(X1,X2)
          activate(n__s(X)) -> s(X)
          fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
          if(false(),X,Y) -> activate(Y)
          if(true(),X,Y) -> activate(X)
          prod(X1,X2) -> n__prod(X1,X2)
          prod(0(),X) -> 0()
          s(X) -> n__s(X)
          zero(0()) -> true()
          activate#(X) -> c_1()
          activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          activate#(n__s(X)) -> c_3(s#(X))
          add#(0(),X) -> c_4()
          fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
          if#(false(),X,Y) -> c_6(activate#(Y))
          if#(true(),X,Y) -> c_7(activate#(X))
          prod#(X1,X2) -> c_8()
          prod#(0(),X) -> c_9()
          s#(X) -> c_10()
          zero#(0()) -> c_11()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
            activate#(n__s(X)) -> c_3(s#(X))
            add#(0(),X) -> c_4()
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            prod#(X1,X2) -> c_8()
            prod#(0(),X) -> c_9()
            s#(X) -> c_10()
            zero#(0()) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,8,9,10,11}
        by application of
          Pre({1,4,8,9,10,11}) = {2,3,5,6,7}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          3: activate#(n__s(X)) -> c_3(s#(X))
          4: add#(0(),X) -> c_4()
          5: fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
          6: if#(false(),X,Y) -> c_6(activate#(Y))
          7: if#(true(),X,Y) -> c_7(activate#(X))
          8: prod#(X1,X2) -> c_8()
          9: prod#(0(),X) -> c_9()
          10: s#(X) -> c_10()
          11: zero#(0()) -> c_11()
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
            activate#(n__s(X)) -> c_3(s#(X))
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            add#(0(),X) -> c_4()
            prod#(X1,X2) -> c_8()
            prod#(0(),X) -> c_9()
            s#(X) -> c_10()
            zero#(0()) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {4,5}.
        Here rules are labelled as follows:
          1: activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          2: activate#(n__s(X)) -> c_3(s#(X))
          3: fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
          4: if#(false(),X,Y) -> c_6(activate#(Y))
          5: if#(true(),X,Y) -> c_7(activate#(X))
          6: activate#(X) -> c_1()
          7: add#(0(),X) -> c_4()
          8: prod#(X1,X2) -> c_8()
          9: prod#(0(),X) -> c_9()
          10: s#(X) -> c_10()
          11: zero#(0()) -> c_11()
* Step 6: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
            activate#(n__s(X)) -> c_3(s#(X))
            add#(0(),X) -> c_4()
            prod#(X1,X2) -> c_8()
            prod#(0(),X) -> c_9()
            s#(X) -> c_10()
            zero#(0()) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3}
        by application of
          Pre({2,3}) = {1}.
        Here rules are labelled as follows:
          1: fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
          2: if#(false(),X,Y) -> c_6(activate#(Y))
          3: if#(true(),X,Y) -> c_7(activate#(X))
          4: activate#(X) -> c_1()
          5: activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          6: activate#(n__s(X)) -> c_3(s#(X))
          7: add#(0(),X) -> c_4()
          8: prod#(X1,X2) -> c_8()
          9: prod#(0(),X) -> c_9()
          10: s#(X) -> c_10()
          11: zero#(0()) -> c_11()
* Step 7: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
            activate#(n__s(X)) -> c_3(s#(X))
            add#(0(),X) -> c_4()
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            prod#(X1,X2) -> c_8()
            prod#(0(),X) -> c_9()
            s#(X) -> c_10()
            zero#(0()) -> c_11()
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
             -->_1 if#(true(),X,Y) -> c_7(activate#(X)):7
             -->_1 if#(false(),X,Y) -> c_6(activate#(Y)):6
             -->_2 zero#(0()) -> c_11():11
             -->_3 fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X)):1
          
          2:W:activate#(X) -> c_1()
             
          
          3:W:activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
             -->_1 prod#(0(),X) -> c_9():9
             -->_1 prod#(X1,X2) -> c_8():8
          
          4:W:activate#(n__s(X)) -> c_3(s#(X))
             -->_1 s#(X) -> c_10():10
          
          5:W:add#(0(),X) -> c_4()
             
          
          6:W:if#(false(),X,Y) -> c_6(activate#(Y))
             -->_1 activate#(n__s(X)) -> c_3(s#(X)):4
             -->_1 activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2)):3
             -->_1 activate#(X) -> c_1():2
          
          7:W:if#(true(),X,Y) -> c_7(activate#(X))
             -->_1 activate#(n__s(X)) -> c_3(s#(X)):4
             -->_1 activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2)):3
             -->_1 activate#(X) -> c_1():2
          
          8:W:prod#(X1,X2) -> c_8()
             
          
          9:W:prod#(0(),X) -> c_9()
             
          
          10:W:s#(X) -> c_10()
             
          
          11:W:zero#(0()) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: add#(0(),X) -> c_4()
          11: zero#(0()) -> c_11()
          6: if#(false(),X,Y) -> c_6(activate#(Y))
          7: if#(true(),X,Y) -> c_7(activate#(X))
          2: activate#(X) -> c_1()
          3: activate#(n__prod(X1,X2)) -> c_2(prod#(X1,X2))
          8: prod#(X1,X2) -> c_8()
          9: prod#(0(),X) -> c_9()
          4: activate#(n__s(X)) -> c_3(s#(X))
          10: s#(X) -> c_10()
* Step 8: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/4,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X))
             -->_3 fact#(X) -> c_5(if#(zero(X),n__s(0()),n__prod(X,fact(p(X)))),zero#(X),fact#(p(X)),p#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          fact#(X) -> c_5(fact#(p(X)))
* Step 9: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fact#(X) -> c_5(fact#(p(X)))
        - Weak TRS:
            activate(X) -> X
            activate(n__prod(X1,X2)) -> prod(X1,X2)
            activate(n__s(X)) -> s(X)
            fact(X) -> if(zero(X),n__s(0()),n__prod(X,fact(p(X))))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            prod(X1,X2) -> n__prod(X1,X2)
            prod(0(),X) -> 0()
            s(X) -> n__s(X)
            zero(0()) -> true()
        - Signature:
            {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
            ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0
            ,c_10/0,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
            ,zero#} and constructors {0,false,n__prod,n__s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          fact#(X) -> c_5(fact#(p(X)))
* Step 10: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fact#(X) -> c_5(fact#(p(X)))
      - Signature:
          {activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2,s#/1
          ,zero#/1} / {0/0,false/0,n__prod/2,n__s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0
          ,c_10/0,c_11/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {activate#,add#,fact#,if#,p#,prod#,s#
          ,zero#} and constructors {0,false,n__prod,n__s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE