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