MAYBE
* Step 1: InnermostRuleRemoval MAYBE
    + Considered Problem:
        - Strict TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
            plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
            plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
            plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
            plus(zero(),y) -> y
            times(x,0()) -> 0()
            times(x,1()) -> x
            times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2} / {0/0,1/0,false/0,s/1,true/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt,id,if,not,plus,times} and constructors {0,1,false,s
            ,true,zero}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
            plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
            plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
            plus(zero(),y) -> y
            times(x,0()) -> 0()
            times(x,1()) -> x
            times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2} / {0/0,1/0,false/0,s/1,true/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt,id,if,not,plus,times} and constructors {0,1,false,s
            ,true,zero}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          gt#(s(x),s(y)) -> c_1(gt#(x,y))
          gt#(s(x),zero()) -> c_2()
          gt#(zero(),y) -> c_3()
          id#(x) -> c_4()
          if#(false(),x,y) -> c_5()
          if#(true(),x,y) -> c_6()
          not#(x) -> c_7(if#(x,false(),true()))
          plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
          plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                 ,if#(gt(x,y),x,y)
                                 ,gt#(x,y)
                                 ,if#(not(gt(x,y)),id(x),id(y))
                                 ,not#(gt(x,y))
                                 ,gt#(x,y)
                                 ,id#(x)
                                 ,id#(y))
          plus#(zero(),y) -> c_10()
          times#(x,0()) -> c_11()
          times#(x,1()) -> c_12()
          times#(x,plus(y,1())) -> c_13(plus#(times(x,plus(y,times(1(),0()))),x)
                                       ,times#(x,plus(y,times(1(),0())))
                                       ,plus#(y,times(1(),0()))
                                       ,times#(1(),0()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            gt#(s(x),zero()) -> c_2()
            gt#(zero(),y) -> c_3()
            id#(x) -> c_4()
            if#(false(),x,y) -> c_5()
            if#(true(),x,y) -> c_6()
            not#(x) -> c_7(if#(x,false(),true()))
            plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
            plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                   ,if#(gt(x,y),x,y)
                                   ,gt#(x,y)
                                   ,if#(not(gt(x,y)),id(x),id(y))
                                   ,not#(gt(x,y))
                                   ,gt#(x,y)
                                   ,id#(x)
                                   ,id#(y))
            plus#(zero(),y) -> c_10()
            times#(x,0()) -> c_11()
            times#(x,1()) -> c_12()
            times#(x,plus(y,1())) -> c_13(plus#(times(x,plus(y,times(1(),0()))),x)
                                         ,times#(x,plus(y,times(1(),0())))
                                         ,plus#(y,times(1(),0()))
                                         ,times#(1(),0()))
        - Weak TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
            plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
            plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
            plus(zero(),y) -> y
            times(x,0()) -> 0()
            times(x,1()) -> x
            times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
            ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/5,c_9/8,c_10/0,c_11/0,c_12/0,c_13/4}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
            ,false,s,true,zero}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          gt(s(x),s(y)) -> gt(x,y)
          gt(s(x),zero()) -> true()
          gt(zero(),y) -> false()
          id(x) -> x
          if(false(),x,y) -> y
          if(true(),x,y) -> x
          not(x) -> if(x,false(),true())
          gt#(s(x),s(y)) -> c_1(gt#(x,y))
          gt#(s(x),zero()) -> c_2()
          gt#(zero(),y) -> c_3()
          id#(x) -> c_4()
          if#(false(),x,y) -> c_5()
          if#(true(),x,y) -> c_6()
          not#(x) -> c_7(if#(x,false(),true()))
          plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
          plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                 ,if#(gt(x,y),x,y)
                                 ,gt#(x,y)
                                 ,if#(not(gt(x,y)),id(x),id(y))
                                 ,not#(gt(x,y))
                                 ,gt#(x,y)
                                 ,id#(x)
                                 ,id#(y))
          plus#(zero(),y) -> c_10()
          times#(x,0()) -> c_11()
          times#(x,1()) -> c_12()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            gt#(s(x),zero()) -> c_2()
            gt#(zero(),y) -> c_3()
            id#(x) -> c_4()
            if#(false(),x,y) -> c_5()
            if#(true(),x,y) -> c_6()
            not#(x) -> c_7(if#(x,false(),true()))
            plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
            plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                   ,if#(gt(x,y),x,y)
                                   ,gt#(x,y)
                                   ,if#(not(gt(x,y)),id(x),id(y))
                                   ,not#(gt(x,y))
                                   ,gt#(x,y)
                                   ,id#(x)
                                   ,id#(y))
            plus#(zero(),y) -> c_10()
            times#(x,0()) -> c_11()
            times#(x,1()) -> c_12()
        - Weak TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
            ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/5,c_9/8,c_10/0,c_11/0,c_12/0,c_13/4}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
            ,false,s,true,zero}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,5,6,10,11,12}
        by application of
          Pre({2,3,4,5,6,10,11,12}) = {1,7,8,9}.
        Here rules are labelled as follows:
          1: gt#(s(x),s(y)) -> c_1(gt#(x,y))
          2: gt#(s(x),zero()) -> c_2()
          3: gt#(zero(),y) -> c_3()
          4: id#(x) -> c_4()
          5: if#(false(),x,y) -> c_5()
          6: if#(true(),x,y) -> c_6()
          7: not#(x) -> c_7(if#(x,false(),true()))
          8: plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
          9: plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                    ,if#(gt(x,y),x,y)
                                    ,gt#(x,y)
                                    ,if#(not(gt(x,y)),id(x),id(y))
                                    ,not#(gt(x,y))
                                    ,gt#(x,y)
                                    ,id#(x)
                                    ,id#(y))
          10: plus#(zero(),y) -> c_10()
          11: times#(x,0()) -> c_11()
          12: times#(x,1()) -> c_12()
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            not#(x) -> c_7(if#(x,false(),true()))
            plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
            plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                   ,if#(gt(x,y),x,y)
                                   ,gt#(x,y)
                                   ,if#(not(gt(x,y)),id(x),id(y))
                                   ,not#(gt(x,y))
                                   ,gt#(x,y)
                                   ,id#(x)
                                   ,id#(y))
        - Weak DPs:
            gt#(s(x),zero()) -> c_2()
            gt#(zero(),y) -> c_3()
            id#(x) -> c_4()
            if#(false(),x,y) -> c_5()
            if#(true(),x,y) -> c_6()
            plus#(zero(),y) -> c_10()
            times#(x,0()) -> c_11()
            times#(x,1()) -> c_12()
        - Weak TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
            ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/5,c_9/8,c_10/0,c_11/0,c_12/0,c_13/4}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
            ,false,s,true,zero}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {4}.
        Here rules are labelled as follows:
          1: gt#(s(x),s(y)) -> c_1(gt#(x,y))
          2: not#(x) -> c_7(if#(x,false(),true()))
          3: plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
          4: plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                    ,if#(gt(x,y),x,y)
                                    ,gt#(x,y)
                                    ,if#(not(gt(x,y)),id(x),id(y))
                                    ,not#(gt(x,y))
                                    ,gt#(x,y)
                                    ,id#(x)
                                    ,id#(y))
          5: gt#(s(x),zero()) -> c_2()
          6: gt#(zero(),y) -> c_3()
          7: id#(x) -> c_4()
          8: if#(false(),x,y) -> c_5()
          9: if#(true(),x,y) -> c_6()
          10: plus#(zero(),y) -> c_10()
          11: times#(x,0()) -> c_11()
          12: times#(x,1()) -> c_12()
* Step 6: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
            plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                   ,if#(gt(x,y),x,y)
                                   ,gt#(x,y)
                                   ,if#(not(gt(x,y)),id(x),id(y))
                                   ,not#(gt(x,y))
                                   ,gt#(x,y)
                                   ,id#(x)
                                   ,id#(y))
        - Weak DPs:
            gt#(s(x),zero()) -> c_2()
            gt#(zero(),y) -> c_3()
            id#(x) -> c_4()
            if#(false(),x,y) -> c_5()
            if#(true(),x,y) -> c_6()
            not#(x) -> c_7(if#(x,false(),true()))
            plus#(zero(),y) -> c_10()
            times#(x,0()) -> c_11()
            times#(x,1()) -> c_12()
        - Weak TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
            ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/5,c_9/8,c_10/0,c_11/0,c_12/0,c_13/4}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
            ,false,s,true,zero}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:gt#(s(x),s(y)) -> c_1(gt#(x,y))
             -->_1 gt#(zero(),y) -> c_3():5
             -->_1 gt#(s(x),zero()) -> c_2():4
             -->_1 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          2:S:plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                  ,if#(gt(x,x),id(x),id(x))
                                  ,gt#(x,x)
                                  ,id#(x)
                                  ,id#(x))
             -->_1 plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                          ,if#(gt(x,y),x,y)
                                          ,gt#(x,y)
                                          ,if#(not(gt(x,y)),id(x),id(y))
                                          ,not#(gt(x,y))
                                          ,gt#(x,y)
                                          ,id#(x)
                                          ,id#(y)):3
             -->_1 plus#(zero(),y) -> c_10():10
             -->_2 if#(true(),x,y) -> c_6():8
             -->_2 if#(false(),x,y) -> c_5():7
             -->_5 id#(x) -> c_4():6
             -->_4 id#(x) -> c_4():6
             -->_3 gt#(zero(),y) -> c_3():5
             -->_1 plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                       ,if#(gt(x,x),id(x),id(x))
                                       ,gt#(x,x)
                                       ,id#(x)
                                       ,id#(x)):2
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          3:S:plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                     ,if#(gt(x,y),x,y)
                                     ,gt#(x,y)
                                     ,if#(not(gt(x,y)),id(x),id(y))
                                     ,not#(gt(x,y))
                                     ,gt#(x,y)
                                     ,id#(x)
                                     ,id#(y))
             -->_5 not#(x) -> c_7(if#(x,false(),true())):9
             -->_1 plus#(zero(),y) -> c_10():10
             -->_4 if#(true(),x,y) -> c_6():8
             -->_2 if#(true(),x,y) -> c_6():8
             -->_4 if#(false(),x,y) -> c_5():7
             -->_2 if#(false(),x,y) -> c_5():7
             -->_8 id#(x) -> c_4():6
             -->_7 id#(x) -> c_4():6
             -->_6 gt#(zero(),y) -> c_3():5
             -->_3 gt#(zero(),y) -> c_3():5
             -->_6 gt#(s(x),zero()) -> c_2():4
             -->_3 gt#(s(x),zero()) -> c_2():4
             -->_1 plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                          ,if#(gt(x,y),x,y)
                                          ,gt#(x,y)
                                          ,if#(not(gt(x,y)),id(x),id(y))
                                          ,not#(gt(x,y))
                                          ,gt#(x,y)
                                          ,id#(x)
                                          ,id#(y)):3
             -->_1 plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                       ,if#(gt(x,x),id(x),id(x))
                                       ,gt#(x,x)
                                       ,id#(x)
                                       ,id#(x)):2
             -->_6 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          4:W:gt#(s(x),zero()) -> c_2()
             
          
          5:W:gt#(zero(),y) -> c_3()
             
          
          6:W:id#(x) -> c_4()
             
          
          7:W:if#(false(),x,y) -> c_5()
             
          
          8:W:if#(true(),x,y) -> c_6()
             
          
          9:W:not#(x) -> c_7(if#(x,false(),true()))
             -->_1 if#(true(),x,y) -> c_6():8
             -->_1 if#(false(),x,y) -> c_5():7
          
          10:W:plus#(zero(),y) -> c_10()
             
          
          11:W:times#(x,0()) -> c_11()
             
          
          12:W:times#(x,1()) -> c_12()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: times#(x,1()) -> c_12()
          11: times#(x,0()) -> c_11()
          6: id#(x) -> c_4()
          10: plus#(zero(),y) -> c_10()
          9: not#(x) -> c_7(if#(x,false(),true()))
          7: if#(false(),x,y) -> c_5()
          8: if#(true(),x,y) -> c_6()
          4: gt#(s(x),zero()) -> c_2()
          5: gt#(zero(),y) -> c_3()
* Step 7: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),if#(gt(x,x),id(x),id(x)),gt#(x,x),id#(x),id#(x))
            plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                   ,if#(gt(x,y),x,y)
                                   ,gt#(x,y)
                                   ,if#(not(gt(x,y)),id(x),id(y))
                                   ,not#(gt(x,y))
                                   ,gt#(x,y)
                                   ,id#(x)
                                   ,id#(y))
        - Weak TRS:
            gt(s(x),s(y)) -> gt(x,y)
            gt(s(x),zero()) -> true()
            gt(zero(),y) -> false()
            id(x) -> x
            if(false(),x,y) -> y
            if(true(),x,y) -> x
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
            ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/5,c_9/8,c_10/0,c_11/0,c_12/0,c_13/4}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
            ,false,s,true,zero}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gt#(s(x),s(y)) -> c_1(gt#(x,y))
             -->_1 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          2:S:plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                  ,if#(gt(x,x),id(x),id(x))
                                  ,gt#(x,x)
                                  ,id#(x)
                                  ,id#(x))
             -->_1 plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                          ,if#(gt(x,y),x,y)
                                          ,gt#(x,y)
                                          ,if#(not(gt(x,y)),id(x),id(y))
                                          ,not#(gt(x,y))
                                          ,gt#(x,y)
                                          ,id#(x)
                                          ,id#(y)):3
             -->_1 plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                       ,if#(gt(x,x),id(x),id(x))
                                       ,gt#(x,x)
                                       ,id#(x)
                                       ,id#(x)):2
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          3:S:plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                     ,if#(gt(x,y),x,y)
                                     ,gt#(x,y)
                                     ,if#(not(gt(x,y)),id(x),id(y))
                                     ,not#(gt(x,y))
                                     ,gt#(x,y)
                                     ,id#(x)
                                     ,id#(y))
             -->_1 plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
                                          ,if#(gt(x,y),x,y)
                                          ,gt#(x,y)
                                          ,if#(not(gt(x,y)),id(x),id(y))
                                          ,not#(gt(x,y))
                                          ,gt#(x,y)
                                          ,id#(x)
                                          ,id#(y)):3
             -->_1 plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x))
                                       ,if#(gt(x,x),id(x),id(x))
                                       ,gt#(x,x)
                                       ,id#(x)
                                       ,id#(x)):2
             -->_6 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),gt#(x,x))
          plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))),gt#(x,y),gt#(x,y))
* Step 8: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          gt#(s(x),s(y)) -> c_1(gt#(x,y))
          plus#(s(x),x) -> c_8(plus#(if(gt(x,x),id(x),id(x)),s(x)),gt#(x,x))
          plus#(s(x),s(y)) -> c_9(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))),gt#(x,y),gt#(x,y))
      - Weak TRS:
          gt(s(x),s(y)) -> gt(x,y)
          gt(s(x),zero()) -> true()
          gt(zero(),y) -> false()
          id(x) -> x
          if(false(),x,y) -> y
          if(true(),x,y) -> x
          not(x) -> if(x,false(),true())
      - Signature:
          {gt/2,id/1,if/3,not/1,plus/2,times/2,gt#/2,id#/1,if#/3,not#/1,plus#/2,times#/2} / {0/0,1/0,false/0,s/1
          ,true/0,zero/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/3,c_10/0,c_11/0,c_12/0,c_13/4}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {gt#,id#,if#,not#,plus#,times#} and constructors {0,1
          ,false,s,true,zero}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE