MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            help(x,s(y),c) -> if(lt(c,x),x,s(y),c)
            if(false(),x,s(y),c) -> 0()
            if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y))))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            quot(x,s(y)) -> help(x,s(y),0())
        - Signature:
            {help/3,if/4,lt/2,plus/2,quot/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help,if,lt,plus,quot} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
          if#(false(),x,s(y),c) -> c_2()
          if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
          lt#(x,0()) -> c_4()
          lt#(0(),s(y)) -> c_5()
          lt#(s(x),s(y)) -> c_6(lt#(x,y))
          plus#(x,0()) -> c_7()
          plus#(x,s(y)) -> c_8(plus#(x,y))
          quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
            if#(false(),x,s(y),c) -> c_2()
            if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
            lt#(x,0()) -> c_4()
            lt#(0(),s(y)) -> c_5()
            lt#(s(x),s(y)) -> c_6(lt#(x,y))
            plus#(x,0()) -> c_7()
            plus#(x,s(y)) -> c_8(plus#(x,y))
            quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
        - Weak TRS:
            help(x,s(y),c) -> if(lt(c,x),x,s(y),c)
            if(false(),x,s(y),c) -> 0()
            if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y))))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            quot(x,s(y)) -> help(x,s(y),0())
        - Signature:
            {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
          if#(false(),x,s(y),c) -> c_2()
          if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
          lt#(x,0()) -> c_4()
          lt#(0(),s(y)) -> c_5()
          lt#(s(x),s(y)) -> c_6(lt#(x,y))
          plus#(x,0()) -> c_7()
          plus#(x,s(y)) -> c_8(plus#(x,y))
          quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
            if#(false(),x,s(y),c) -> c_2()
            if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
            lt#(x,0()) -> c_4()
            lt#(0(),s(y)) -> c_5()
            lt#(s(x),s(y)) -> c_6(lt#(x,y))
            plus#(x,0()) -> c_7()
            plus#(x,s(y)) -> c_8(plus#(x,y))
            quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
        - Signature:
            {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5,7}
        by application of
          Pre({2,4,5,7}) = {1,6,8}.
        Here rules are labelled as follows:
          1: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
          2: if#(false(),x,s(y),c) -> c_2()
          3: if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
          4: lt#(x,0()) -> c_4()
          5: lt#(0(),s(y)) -> c_5()
          6: lt#(s(x),s(y)) -> c_6(lt#(x,y))
          7: plus#(x,0()) -> c_7()
          8: plus#(x,s(y)) -> c_8(plus#(x,y))
          9: quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
            if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
            lt#(s(x),s(y)) -> c_6(lt#(x,y))
            plus#(x,s(y)) -> c_8(plus#(x,y))
            quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
        - Weak DPs:
            if#(false(),x,s(y),c) -> c_2()
            lt#(x,0()) -> c_4()
            lt#(0(),s(y)) -> c_5()
            plus#(x,0()) -> c_7()
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
        - Signature:
            {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
             -->_2 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3
             -->_1 if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))):2
             -->_2 lt#(0(),s(y)) -> c_5():8
             -->_2 lt#(x,0()) -> c_4():7
             -->_1 if#(false(),x,s(y),c) -> c_2():6
          
          2:S:if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
             -->_2 plus#(x,s(y)) -> c_8(plus#(x,y)):4
             -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1
          
          3:S:lt#(s(x),s(y)) -> c_6(lt#(x,y))
             -->_1 lt#(0(),s(y)) -> c_5():8
             -->_1 lt#(x,0()) -> c_4():7
             -->_1 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3
          
          4:S:plus#(x,s(y)) -> c_8(plus#(x,y))
             -->_1 plus#(x,0()) -> c_7():9
             -->_1 plus#(x,s(y)) -> c_8(plus#(x,y)):4
          
          5:S:quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
             -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1
          
          6:W:if#(false(),x,s(y),c) -> c_2()
             
          
          7:W:lt#(x,0()) -> c_4()
             
          
          8:W:lt#(0(),s(y)) -> c_5()
             
          
          9:W:plus#(x,0()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: if#(false(),x,s(y),c) -> c_2()
          9: plus#(x,0()) -> c_7()
          7: lt#(x,0()) -> c_4()
          8: lt#(0(),s(y)) -> c_5()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
            if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
            lt#(s(x),s(y)) -> c_6(lt#(x,y))
            plus#(x,s(y)) -> c_8(plus#(x,y))
            quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
        - Signature:
            {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0
            ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
           -->_2 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3
           -->_1 if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))):2
        
        2:S:if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
           -->_2 plus#(x,s(y)) -> c_8(plus#(x,y)):4
           -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1
        
        3:S:lt#(s(x),s(y)) -> c_6(lt#(x,y))
           -->_1 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3
        
        4:S:plus#(x,s(y)) -> c_8(plus#(x,y))
           -->_1 plus#(x,s(y)) -> c_8(plus#(x,y)):4
        
        5:S:quot#(x,s(y)) -> c_9(help#(x,s(y),0()))
           -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):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).
        
        [(5,quot#(x,s(y)) -> c_9(help#(x,s(y),0())))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x))
          if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y)))
          lt#(s(x),s(y)) -> c_6(lt#(x,y))
          plus#(x,s(y)) -> c_8(plus#(x,y))
      - Weak TRS:
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
      - Signature:
          {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0
          ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE