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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            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
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2} / {0/0,false/0,s/1,true/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt,id,if,minus,not,plus,quot} and constructors {0,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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            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
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2} / {0/0,false/0,s/1,true/0,zero/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt,id,if,minus,not,plus,quot} and constructors {0,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()
          minus#(x,0()) -> c_7()
          minus#(s(x),s(y)) -> c_8(minus#(x,y))
          not#(x) -> c_9(if#(x,false(),true()))
          plus#(s(x),x) -> c_10(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_11(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_12()
          quot#(0(),s(y)) -> c_13()
          quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
        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()
            minus#(x,0()) -> c_7()
            minus#(s(x),s(y)) -> c_8(minus#(x,y))
            not#(x) -> c_9(if#(x,false(),true()))
            plus#(s(x),x) -> c_10(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_11(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_12()
            quot#(0(),s(y)) -> c_13()
            quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            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
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
            ,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_7/0,c_8/1,c_9/1,c_10/5,c_11/8,c_12/0,c_13/0
            ,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
            ,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
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          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()
          minus#(x,0()) -> c_7()
          minus#(s(x),s(y)) -> c_8(minus#(x,y))
          not#(x) -> c_9(if#(x,false(),true()))
          plus#(s(x),x) -> c_10(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_11(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_12()
          quot#(0(),s(y)) -> c_13()
          quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
* 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()
            minus#(x,0()) -> c_7()
            minus#(s(x),s(y)) -> c_8(minus#(x,y))
            not#(x) -> c_9(if#(x,false(),true()))
            plus#(s(x),x) -> c_10(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_11(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_12()
            quot#(0(),s(y)) -> c_13()
            quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
            ,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_7/0,c_8/1,c_9/1,c_10/5,c_11/8,c_12/0,c_13/0
            ,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
            ,false,s,true,zero}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,5,6,7,12,13}
        by application of
          Pre({2,3,4,5,6,7,12,13}) = {1,8,9,10,11,14}.
        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: minus#(x,0()) -> c_7()
          8: minus#(s(x),s(y)) -> c_8(minus#(x,y))
          9: not#(x) -> c_9(if#(x,false(),true()))
          10: plus#(s(x),x) -> c_10(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))
          11: plus#(s(x),s(y)) -> c_11(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))
          12: plus#(zero(),y) -> c_12()
          13: quot#(0(),s(y)) -> c_13()
          14: quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 5: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            minus#(s(x),s(y)) -> c_8(minus#(x,y))
            not#(x) -> c_9(if#(x,false(),true()))
            plus#(s(x),x) -> c_10(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_11(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))
            quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,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()
            minus#(x,0()) -> c_7()
            plus#(zero(),y) -> c_12()
            quot#(0(),s(y)) -> c_13()
        - 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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
            ,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_7/0,c_8/1,c_9/1,c_10/5,c_11/8,c_12/0,c_13/0
            ,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
            ,false,s,true,zero}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3}
        by application of
          Pre({3}) = {5}.
        Here rules are labelled as follows:
          1: gt#(s(x),s(y)) -> c_1(gt#(x,y))
          2: minus#(s(x),s(y)) -> c_8(minus#(x,y))
          3: not#(x) -> c_9(if#(x,false(),true()))
          4: plus#(s(x),x) -> c_10(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))
          5: plus#(s(x),s(y)) -> c_11(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))
          6: quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
          7: gt#(s(x),zero()) -> c_2()
          8: gt#(zero(),y) -> c_3()
          9: id#(x) -> c_4()
          10: if#(false(),x,y) -> c_5()
          11: if#(true(),x,y) -> c_6()
          12: minus#(x,0()) -> c_7()
          13: plus#(zero(),y) -> c_12()
          14: quot#(0(),s(y)) -> c_13()
* Step 6: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            minus#(s(x),s(y)) -> c_8(minus#(x,y))
            plus#(s(x),x) -> c_10(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_11(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))
            quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,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()
            minus#(x,0()) -> c_7()
            not#(x) -> c_9(if#(x,false(),true()))
            plus#(zero(),y) -> c_12()
            quot#(0(),s(y)) -> c_13()
        - 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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
            ,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_7/0,c_8/1,c_9/1,c_10/5,c_11/8,c_12/0,c_13/0
            ,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
            ,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():7
             -->_1 gt#(s(x),zero()) -> c_2():6
             -->_1 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          2:S:minus#(s(x),s(y)) -> c_8(minus#(x,y))
             -->_1 minus#(x,0()) -> c_7():11
             -->_1 minus#(s(x),s(y)) -> c_8(minus#(x,y)):2
          
          3:S:plus#(s(x),x) -> c_10(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_11(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)):4
             -->_1 plus#(zero(),y) -> c_12():13
             -->_2 if#(true(),x,y) -> c_6():10
             -->_2 if#(false(),x,y) -> c_5():9
             -->_5 id#(x) -> c_4():8
             -->_4 id#(x) -> c_4():8
             -->_3 gt#(zero(),y) -> c_3():7
             -->_1 plus#(s(x),x) -> c_10(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)):3
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          4:S:plus#(s(x),s(y)) -> c_11(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_9(if#(x,false(),true())):12
             -->_1 plus#(zero(),y) -> c_12():13
             -->_4 if#(true(),x,y) -> c_6():10
             -->_2 if#(true(),x,y) -> c_6():10
             -->_4 if#(false(),x,y) -> c_5():9
             -->_2 if#(false(),x,y) -> c_5():9
             -->_8 id#(x) -> c_4():8
             -->_7 id#(x) -> c_4():8
             -->_6 gt#(zero(),y) -> c_3():7
             -->_3 gt#(zero(),y) -> c_3():7
             -->_6 gt#(s(x),zero()) -> c_2():6
             -->_3 gt#(s(x),zero()) -> c_2():6
             -->_1 plus#(s(x),s(y)) -> c_11(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)):4
             -->_1 plus#(s(x),x) -> c_10(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)):3
             -->_6 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          5:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(0(),s(y)) -> c_13():14
             -->_2 minus#(x,0()) -> c_7():11
             -->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):5
             -->_2 minus#(s(x),s(y)) -> c_8(minus#(x,y)):2
          
          6:W:gt#(s(x),zero()) -> c_2()
             
          
          7:W:gt#(zero(),y) -> c_3()
             
          
          8:W:id#(x) -> c_4()
             
          
          9:W:if#(false(),x,y) -> c_5()
             
          
          10:W:if#(true(),x,y) -> c_6()
             
          
          11:W:minus#(x,0()) -> c_7()
             
          
          12:W:not#(x) -> c_9(if#(x,false(),true()))
             -->_1 if#(true(),x,y) -> c_6():10
             -->_1 if#(false(),x,y) -> c_5():9
          
          13:W:plus#(zero(),y) -> c_12()
             
          
          14:W:quot#(0(),s(y)) -> c_13()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          14: quot#(0(),s(y)) -> c_13()
          8: id#(x) -> c_4()
          13: plus#(zero(),y) -> c_12()
          12: not#(x) -> c_9(if#(x,false(),true()))
          9: if#(false(),x,y) -> c_5()
          10: if#(true(),x,y) -> c_6()
          11: minus#(x,0()) -> c_7()
          6: gt#(s(x),zero()) -> c_2()
          7: gt#(zero(),y) -> c_3()
* Step 7: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            gt#(s(x),s(y)) -> c_1(gt#(x,y))
            minus#(s(x),s(y)) -> c_8(minus#(x,y))
            plus#(s(x),x) -> c_10(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_11(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))
            quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(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
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            not(x) -> if(x,false(),true())
        - Signature:
            {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
            ,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_7/0,c_8/1,c_9/1,c_10/5,c_11/8,c_12/0,c_13/0
            ,c_14/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
            ,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:minus#(s(x),s(y)) -> c_8(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_8(minus#(x,y)):2
          
          3:S:plus#(s(x),x) -> c_10(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_11(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)):4
             -->_1 plus#(s(x),x) -> c_10(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)):3
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          4:S:plus#(s(x),s(y)) -> c_11(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_11(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)):4
             -->_1 plus#(s(x),x) -> c_10(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)):3
             -->_6 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
             -->_3 gt#(s(x),s(y)) -> c_1(gt#(x,y)):1
          
          5:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):5
             -->_2 minus#(s(x),s(y)) -> c_8(minus#(x,y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          plus#(s(x),x) -> c_10(plus#(if(gt(x,x),id(x),id(x)),s(x)),gt#(x,x))
          plus#(s(x),s(y)) -> c_11(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))
          minus#(s(x),s(y)) -> c_8(minus#(x,y))
          plus#(s(x),x) -> c_10(plus#(if(gt(x,x),id(x),id(x)),s(x)),gt#(x,x))
          plus#(s(x),s(y)) -> c_11(plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))),gt#(x,y),gt#(x,y))
          quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(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
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          not(x) -> if(x,false(),true())
      - Signature:
          {gt/2,id/1,if/3,minus/2,not/1,plus/2,quot/2,gt#/2,id#/1,if#/3,minus#/2,not#/1,plus#/2,quot#/2} / {0/0
          ,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_7/0,c_8/1,c_9/1,c_10/2,c_11/3,c_12/0,c_13/0
          ,c_14/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {gt#,id#,if#,minus#,not#,plus#,quot#} and constructors {0
          ,false,s,true,zero}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE