MAYBE
* Step 1: InnermostRuleRemoval MAYBE
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__s(X)) -> s(X)
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            leq(0(),Y) -> true()
            leq(s(X),0()) -> false()
            leq(s(X),s(Y)) -> leq(X,Y)
            p(0()) -> 0()
            p(s(X)) -> X
            s(X) -> n__s(X)
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1} / {false/0,n__0/0,n__s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,activate,diff,if,leq,p,s} and constructors {false,n__0
            ,n__s,true}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          leq(0(),Y) -> true()
          leq(s(X),0()) -> false()
          leq(s(X),s(Y)) -> leq(X,Y)
          p(0()) -> 0()
          p(s(X)) -> X
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            0() -> n__0()
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__s(X)) -> s(X)
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            s(X) -> n__s(X)
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1} / {false/0,n__0/0,n__s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0,activate,diff,if,leq,p,s} and constructors {false,n__0
            ,n__s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          0#() -> c_1()
          activate#(X) -> c_2()
          activate#(n__0()) -> c_3(0#())
          activate#(n__s(X)) -> c_4(s#(X))
          diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
          if#(false(),X,Y) -> c_6(activate#(Y))
          if#(true(),X,Y) -> c_7(activate#(X))
          s#(X) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__0()) -> c_3(0#())
            activate#(n__s(X)) -> c_4(s#(X))
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            s#(X) -> c_8()
        - Weak TRS:
            0() -> n__0()
            activate(X) -> X
            activate(n__0()) -> 0()
            activate(n__s(X)) -> s(X)
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
            if(false(),X,Y) -> activate(Y)
            if(true(),X,Y) -> activate(X)
            s(X) -> n__s(X)
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,n__s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
          0#() -> c_1()
          activate#(X) -> c_2()
          activate#(n__0()) -> c_3(0#())
          activate#(n__s(X)) -> c_4(s#(X))
          diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
          if#(false(),X,Y) -> c_6(activate#(Y))
          if#(true(),X,Y) -> c_7(activate#(X))
          s#(X) -> c_8()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            0#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__0()) -> c_3(0#())
            activate#(n__s(X)) -> c_4(s#(X))
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            s#(X) -> c_8()
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,n__s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,8}
        by application of
          Pre({1,2,8}) = {3,4,6,7}.
        Here rules are labelled as follows:
          1: 0#() -> c_1()
          2: activate#(X) -> c_2()
          3: activate#(n__0()) -> c_3(0#())
          4: activate#(n__s(X)) -> c_4(s#(X))
          5: diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
          6: if#(false(),X,Y) -> c_6(activate#(Y))
          7: if#(true(),X,Y) -> c_7(activate#(X))
          8: s#(X) -> c_8()
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            activate#(n__0()) -> c_3(0#())
            activate#(n__s(X)) -> c_4(s#(X))
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
        - Weak DPs:
            0#() -> c_1()
            activate#(X) -> c_2()
            s#(X) -> c_8()
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,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__0()) -> c_3(0#())
          2: activate#(n__s(X)) -> c_4(s#(X))
          3: diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
          4: if#(false(),X,Y) -> c_6(activate#(Y))
          5: if#(true(),X,Y) -> c_7(activate#(X))
          6: 0#() -> c_1()
          7: activate#(X) -> c_2()
          8: s#(X) -> c_8()
* Step 6: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
        - Weak DPs:
            0#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__0()) -> c_3(0#())
            activate#(n__s(X)) -> c_4(s#(X))
            s#(X) -> c_8()
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,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}) = {}.
        Here rules are labelled as follows:
          1: diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
          2: if#(false(),X,Y) -> c_6(activate#(Y))
          3: if#(true(),X,Y) -> c_7(activate#(X))
          4: 0#() -> c_1()
          5: activate#(X) -> c_2()
          6: activate#(n__0()) -> c_3(0#())
          7: activate#(n__s(X)) -> c_4(s#(X))
          8: s#(X) -> c_8()
* Step 7: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
        - Weak DPs:
            0#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__0()) -> c_3(0#())
            activate#(n__s(X)) -> c_4(s#(X))
            if#(false(),X,Y) -> c_6(activate#(Y))
            if#(true(),X,Y) -> c_7(activate#(X))
            s#(X) -> c_8()
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,n__s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
             -->_3 diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)):1
          
          2:W:0#() -> c_1()
             
          
          3:W:activate#(X) -> c_2()
             
          
          4:W:activate#(n__0()) -> c_3(0#())
             -->_1 0#() -> c_1():2
          
          5:W:activate#(n__s(X)) -> c_4(s#(X))
             -->_1 s#(X) -> c_8():8
          
          6:W:if#(false(),X,Y) -> c_6(activate#(Y))
             -->_1 activate#(n__s(X)) -> c_4(s#(X)):5
             -->_1 activate#(n__0()) -> c_3(0#()):4
             -->_1 activate#(X) -> c_2():3
          
          7:W:if#(true(),X,Y) -> c_7(activate#(X))
             -->_1 activate#(n__s(X)) -> c_4(s#(X)):5
             -->_1 activate#(n__0()) -> c_3(0#()):4
             -->_1 activate#(X) -> c_2():3
          
          8:W:s#(X) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: if#(true(),X,Y) -> c_7(activate#(X))
          6: if#(false(),X,Y) -> c_6(activate#(Y))
          5: activate#(n__s(X)) -> c_4(s#(X))
          8: s#(X) -> c_8()
          4: activate#(n__0()) -> c_3(0#())
          3: activate#(X) -> c_2()
          2: 0#() -> c_1()
* Step 8: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/4,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,n__s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X))
             -->_3 diff#(X,Y) -> c_5(if#(leq(X,Y),n__0(),n__s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          diff#(X,Y) -> c_5(diff#(p(X),Y))
* Step 9: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            diff#(X,Y) -> c_5(diff#(p(X),Y))
        - Weak TRS:
            diff(X,Y) -> if(leq(X,Y),n__0(),n__s(diff(p(X),Y)))
        - Signature:
            {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
            ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
            ,s#} and constructors {false,n__0,n__s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          diff#(X,Y) -> c_5(diff#(p(X),Y))
* Step 10: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          diff#(X,Y) -> c_5(diff#(p(X),Y))
      - Signature:
          {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0
          ,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p#
          ,s#} and constructors {false,n__0,n__s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE