MAYBE
* Step 1: InnermostRuleRemoval MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            activate(n__s(X)) -> s(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> activate(X)
            add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
            and(false(),Y) -> false()
            and(true(),X) -> activate(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
            from(X) -> cons(activate(X),n__from(n__s(activate(X))))
            from(X) -> n__from(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1
            ,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,and,first,from,if,s} and constructors {0
            ,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
          first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            activate(n__s(X)) -> s(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> activate(X)
            and(false(),Y) -> false()
            and(true(),X) -> activate(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            from(X) -> cons(activate(X),n__from(n__s(activate(X))))
            from(X) -> n__from(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1
            ,nil/0,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,add,and,first,from,if,s} and constructors {0
            ,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          activate#(n__s(X)) -> c_5(s#(X))
          add#(X1,X2) -> c_6()
          add#(0(),X) -> c_7(activate#(X))
          and#(false(),Y) -> c_8()
          and#(true(),X) -> c_9(activate#(X))
          first#(X1,X2) -> c_10()
          first#(0(),X) -> c_11()
          from#(X) -> c_12(activate#(X),activate#(X))
          from#(X) -> c_13()
          if#(false(),X,Y) -> c_14(activate#(Y))
          if#(true(),X,Y) -> c_15(activate#(X))
          s#(X) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            activate#(n__s(X)) -> c_5(s#(X))
            add#(X1,X2) -> c_6()
            add#(0(),X) -> c_7(activate#(X))
            and#(false(),Y) -> c_8()
            and#(true(),X) -> c_9(activate#(X))
            first#(X1,X2) -> c_10()
            first#(0(),X) -> c_11()
            from#(X) -> c_12(activate#(X),activate#(X))
            from#(X) -> c_13()
            if#(false(),X,Y) -> c_14(activate#(Y))
            if#(true(),X,Y) -> c_15(activate#(X))
            s#(X) -> c_16()
        - Strict TRS:
            activate(X) -> X
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            activate(n__s(X)) -> s(X)
            add(X1,X2) -> n__add(X1,X2)
            add(0(),X) -> activate(X)
            and(false(),Y) -> false()
            and(true(),X) -> activate(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            from(X) -> cons(activate(X),n__from(n__s(activate(X))))
            from(X) -> n__from(X)
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            s(X) -> n__s(X)
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
            ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
            ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(X) -> c_1()
          activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          activate#(n__s(X)) -> c_5(s#(X))
          add#(X1,X2) -> c_6()
          add#(0(),X) -> c_7(activate#(X))
          and#(false(),Y) -> c_8()
          and#(true(),X) -> c_9(activate#(X))
          first#(X1,X2) -> c_10()
          first#(0(),X) -> c_11()
          from#(X) -> c_12(activate#(X),activate#(X))
          from#(X) -> c_13()
          if#(false(),X,Y) -> c_14(activate#(Y))
          if#(true(),X,Y) -> c_15(activate#(X))
          s#(X) -> c_16()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            activate#(n__s(X)) -> c_5(s#(X))
            add#(X1,X2) -> c_6()
            add#(0(),X) -> c_7(activate#(X))
            and#(false(),Y) -> c_8()
            and#(true(),X) -> c_9(activate#(X))
            first#(X1,X2) -> c_10()
            first#(0(),X) -> c_11()
            from#(X) -> c_12(activate#(X),activate#(X))
            from#(X) -> c_13()
            if#(false(),X,Y) -> c_14(activate#(Y))
            if#(true(),X,Y) -> c_15(activate#(X))
            s#(X) -> c_16()
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
            ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
            ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,6,8,10,11,13,16}
        by application of
          Pre({1,6,8,10,11,13,16}) = {2,3,4,5,7,9,12,14,15}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          3: activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
          4: activate#(n__from(X)) -> c_4(from#(X))
          5: activate#(n__s(X)) -> c_5(s#(X))
          6: add#(X1,X2) -> c_6()
          7: add#(0(),X) -> c_7(activate#(X))
          8: and#(false(),Y) -> c_8()
          9: and#(true(),X) -> c_9(activate#(X))
          10: first#(X1,X2) -> c_10()
          11: first#(0(),X) -> c_11()
          12: from#(X) -> c_12(activate#(X),activate#(X))
          13: from#(X) -> c_13()
          14: if#(false(),X,Y) -> c_14(activate#(Y))
          15: if#(true(),X,Y) -> c_15(activate#(X))
          16: s#(X) -> c_16()
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            activate#(n__s(X)) -> c_5(s#(X))
            add#(0(),X) -> c_7(activate#(X))
            and#(true(),X) -> c_9(activate#(X))
            from#(X) -> c_12(activate#(X),activate#(X))
            if#(false(),X,Y) -> c_14(activate#(Y))
            if#(true(),X,Y) -> c_15(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            add#(X1,X2) -> c_6()
            and#(false(),Y) -> c_8()
            first#(X1,X2) -> c_10()
            first#(0(),X) -> c_11()
            from#(X) -> c_13()
            s#(X) -> c_16()
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
            ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
            ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {5,6,7,8,9}.
        Here rules are labelled as follows:
          1: activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          2: activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
          3: activate#(n__from(X)) -> c_4(from#(X))
          4: activate#(n__s(X)) -> c_5(s#(X))
          5: add#(0(),X) -> c_7(activate#(X))
          6: and#(true(),X) -> c_9(activate#(X))
          7: from#(X) -> c_12(activate#(X),activate#(X))
          8: if#(false(),X,Y) -> c_14(activate#(Y))
          9: if#(true(),X,Y) -> c_15(activate#(X))
          10: activate#(X) -> c_1()
          11: add#(X1,X2) -> c_6()
          12: and#(false(),Y) -> c_8()
          13: first#(X1,X2) -> c_10()
          14: first#(0(),X) -> c_11()
          15: from#(X) -> c_13()
          16: s#(X) -> c_16()
* Step 6: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            add#(0(),X) -> c_7(activate#(X))
            and#(true(),X) -> c_9(activate#(X))
            from#(X) -> c_12(activate#(X),activate#(X))
            if#(false(),X,Y) -> c_14(activate#(Y))
            if#(true(),X,Y) -> c_15(activate#(X))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
            activate#(n__s(X)) -> c_5(s#(X))
            add#(X1,X2) -> c_6()
            and#(false(),Y) -> c_8()
            first#(X1,X2) -> c_10()
            first#(0(),X) -> c_11()
            from#(X) -> c_13()
            s#(X) -> c_16()
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
            ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
            ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
             -->_1 add#(0(),X) -> c_7(activate#(X)):3
             -->_1 add#(X1,X2) -> c_6():11
          
          2:S:activate#(n__from(X)) -> c_4(from#(X))
             -->_1 from#(X) -> c_12(activate#(X),activate#(X)):5
             -->_1 from#(X) -> c_13():15
          
          3:S:add#(0(),X) -> c_7(activate#(X))
             -->_1 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_1 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_1 activate#(X) -> c_1():8
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
          
          4:S:and#(true(),X) -> c_9(activate#(X))
             -->_1 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_1 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_1 activate#(X) -> c_1():8
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
          
          5:S:from#(X) -> c_12(activate#(X),activate#(X))
             -->_2 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_1 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_2 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_1 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_2 activate#(X) -> c_1():8
             -->_1 activate#(X) -> c_1():8
             -->_2 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_2 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
             -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
          
          6:S:if#(false(),X,Y) -> c_14(activate#(Y))
             -->_1 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_1 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_1 activate#(X) -> c_1():8
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
          
          7:S:if#(true(),X,Y) -> c_15(activate#(X))
             -->_1 activate#(n__s(X)) -> c_5(s#(X)):10
             -->_1 activate#(n__first(X1,X2)) -> c_3(first#(X1,X2)):9
             -->_1 activate#(X) -> c_1():8
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
             -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
          
          8:W:activate#(X) -> c_1()
             
          
          9:W:activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
             -->_1 first#(0(),X) -> c_11():14
             -->_1 first#(X1,X2) -> c_10():13
          
          10:W:activate#(n__s(X)) -> c_5(s#(X))
             -->_1 s#(X) -> c_16():16
          
          11:W:add#(X1,X2) -> c_6()
             
          
          12:W:and#(false(),Y) -> c_8()
             
          
          13:W:first#(X1,X2) -> c_10()
             
          
          14:W:first#(0(),X) -> c_11()
             
          
          15:W:from#(X) -> c_13()
             
          
          16:W:s#(X) -> c_16()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: and#(false(),Y) -> c_8()
          11: add#(X1,X2) -> c_6()
          15: from#(X) -> c_13()
          8: activate#(X) -> c_1()
          9: activate#(n__first(X1,X2)) -> c_3(first#(X1,X2))
          13: first#(X1,X2) -> c_10()
          14: first#(0(),X) -> c_11()
          10: activate#(n__s(X)) -> c_5(s#(X))
          16: s#(X) -> c_16()
* Step 7: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            add#(0(),X) -> c_7(activate#(X))
            and#(true(),X) -> c_9(activate#(X))
            from#(X) -> c_12(activate#(X),activate#(X))
            if#(false(),X,Y) -> c_14(activate#(Y))
            if#(true(),X,Y) -> c_15(activate#(X))
        - Signature:
            {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
            ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
            ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
           -->_1 add#(0(),X) -> c_7(activate#(X)):3
        
        2:S:activate#(n__from(X)) -> c_4(from#(X))
           -->_1 from#(X) -> c_12(activate#(X),activate#(X)):5
        
        3:S:add#(0(),X) -> c_7(activate#(X))
           -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
        
        4:S:and#(true(),X) -> c_9(activate#(X))
           -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
        
        5:S:from#(X) -> c_12(activate#(X),activate#(X))
           -->_2 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_2 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
           -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
        
        6:S:if#(false(),X,Y) -> c_14(activate#(Y))
           -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
        
        7:S:if#(true(),X,Y) -> c_15(activate#(X))
           -->_1 activate#(n__from(X)) -> c_4(from#(X)):2
           -->_1 activate#(n__add(X1,X2)) -> c_2(add#(X1,X2)):1
        
        
        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).
        
        [(4,and#(true(),X) -> c_9(activate#(X)))
        ,(6,if#(false(),X,Y) -> c_14(activate#(Y)))
        ,(7,if#(true(),X,Y) -> c_15(activate#(X)))]
* Step 8: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          activate#(n__add(X1,X2)) -> c_2(add#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          add#(0(),X) -> c_7(activate#(X))
          from#(X) -> c_12(activate#(X),activate#(X))
      - Signature:
          {activate/1,add/2,and/2,first/2,from/1,if/3,s/1,activate#/1,add#/2,and#/2,first#/2,from#/1,if#/3
          ,s#/1} / {0/0,cons/2,false/0,n__add/2,n__first/2,n__from/1,n__s/1,nil/0,true/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1
          ,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2,c_13/0,c_14/1,c_15/1,c_16/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {activate#,add#,and#,first#,from#,if#
          ,s#} and constructors {0,cons,false,n__add,n__first,n__from,n__s,nil,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE