MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            div(x,y,z) -> if(lt(x,y),x,y,inc(z))
            division(x,y) -> div(x,y,0())
            if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
            if(true(),x,y,z) -> z
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,division/2,if/4,inc/1,lt/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,division,if,inc,lt,minus} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
          division#(x,y) -> c_2(div#(x,y,0()))
          if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
          if#(true(),x,y,z) -> c_4()
          inc#(0()) -> c_5()
          inc#(s(x)) -> c_6(inc#(x))
          lt#(x,0()) -> c_7()
          lt#(0(),s(y)) -> c_8()
          lt#(s(x),s(y)) -> c_9(lt#(x,y))
          minus#(x,0()) -> c_10()
          minus#(s(x),s(y)) -> c_11(minus#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
            division#(x,y) -> c_2(div#(x,y,0()))
            if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
            if#(true(),x,y,z) -> c_4()
            inc#(0()) -> c_5()
            inc#(s(x)) -> c_6(inc#(x))
            lt#(x,0()) -> c_7()
            lt#(0(),s(y)) -> c_8()
            lt#(s(x),s(y)) -> c_9(lt#(x,y))
            minus#(x,0()) -> c_10()
            minus#(s(x),s(y)) -> c_11(minus#(x,y))
        - Weak TRS:
            div(x,y,z) -> if(lt(x,y),x,y,inc(z))
            division(x,y) -> div(x,y,0())
            if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
            if(true(),x,y,z) -> z
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,division/2,if/4,inc/1,lt/2,minus/2,div#/3,division#/2,if#/4,inc#/1,lt#/2,minus#/2} / {0/0,false/0,s/1
            ,true/0,c_1/3,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,division#,if#,inc#,lt#,minus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          inc(0()) -> s(0())
          inc(s(x)) -> s(inc(x))
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
          division#(x,y) -> c_2(div#(x,y,0()))
          if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
          if#(true(),x,y,z) -> c_4()
          inc#(0()) -> c_5()
          inc#(s(x)) -> c_6(inc#(x))
          lt#(x,0()) -> c_7()
          lt#(0(),s(y)) -> c_8()
          lt#(s(x),s(y)) -> c_9(lt#(x,y))
          minus#(x,0()) -> c_10()
          minus#(s(x),s(y)) -> c_11(minus#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
            division#(x,y) -> c_2(div#(x,y,0()))
            if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
            if#(true(),x,y,z) -> c_4()
            inc#(0()) -> c_5()
            inc#(s(x)) -> c_6(inc#(x))
            lt#(x,0()) -> c_7()
            lt#(0(),s(y)) -> c_8()
            lt#(s(x),s(y)) -> c_9(lt#(x,y))
            minus#(x,0()) -> c_10()
            minus#(s(x),s(y)) -> c_11(minus#(x,y))
        - Weak TRS:
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,division/2,if/4,inc/1,lt/2,minus/2,div#/3,division#/2,if#/4,inc#/1,lt#/2,minus#/2} / {0/0,false/0,s/1
            ,true/0,c_1/3,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,division#,if#,inc#,lt#,minus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5,7,8,10}
        by application of
          Pre({4,5,7,8,10}) = {1,6,9,11}.
        Here rules are labelled as follows:
          1: div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
          2: division#(x,y) -> c_2(div#(x,y,0()))
          3: if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
          4: if#(true(),x,y,z) -> c_4()
          5: inc#(0()) -> c_5()
          6: inc#(s(x)) -> c_6(inc#(x))
          7: lt#(x,0()) -> c_7()
          8: lt#(0(),s(y)) -> c_8()
          9: lt#(s(x),s(y)) -> c_9(lt#(x,y))
          10: minus#(x,0()) -> c_10()
          11: minus#(s(x),s(y)) -> c_11(minus#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
            division#(x,y) -> c_2(div#(x,y,0()))
            if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
            inc#(s(x)) -> c_6(inc#(x))
            lt#(s(x),s(y)) -> c_9(lt#(x,y))
            minus#(s(x),s(y)) -> c_11(minus#(x,y))
        - Weak DPs:
            if#(true(),x,y,z) -> c_4()
            inc#(0()) -> c_5()
            lt#(x,0()) -> c_7()
            lt#(0(),s(y)) -> c_8()
            minus#(x,0()) -> c_10()
        - Weak TRS:
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,division/2,if/4,inc/1,lt/2,minus/2,div#/3,division#/2,if#/4,inc#/1,lt#/2,minus#/2} / {0/0,false/0,s/1
            ,true/0,c_1/3,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,division#,if#,inc#,lt#,minus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
             -->_2 lt#(s(x),s(y)) -> c_9(lt#(x,y)):5
             -->_3 inc#(s(x)) -> c_6(inc#(x)):4
             -->_1 if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y))):3
             -->_2 lt#(0(),s(y)) -> c_8():10
             -->_2 lt#(x,0()) -> c_7():9
             -->_3 inc#(0()) -> c_5():8
             -->_1 if#(true(),x,y,z) -> c_4():7
          
          2:S:division#(x,y) -> c_2(div#(x,y,0()))
             -->_1 div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z)):1
          
          3:S:if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
             -->_2 minus#(s(x),s(y)) -> c_11(minus#(x,y)):6
             -->_1 div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z)):1
          
          4:S:inc#(s(x)) -> c_6(inc#(x))
             -->_1 inc#(0()) -> c_5():8
             -->_1 inc#(s(x)) -> c_6(inc#(x)):4
          
          5:S:lt#(s(x),s(y)) -> c_9(lt#(x,y))
             -->_1 lt#(0(),s(y)) -> c_8():10
             -->_1 lt#(x,0()) -> c_7():9
             -->_1 lt#(s(x),s(y)) -> c_9(lt#(x,y)):5
          
          6:S:minus#(s(x),s(y)) -> c_11(minus#(x,y))
             -->_1 minus#(x,0()) -> c_10():11
             -->_1 minus#(s(x),s(y)) -> c_11(minus#(x,y)):6
          
          7:W:if#(true(),x,y,z) -> c_4()
             
          
          8:W:inc#(0()) -> c_5()
             
          
          9:W:lt#(x,0()) -> c_7()
             
          
          10:W:lt#(0(),s(y)) -> c_8()
             
          
          11:W:minus#(x,0()) -> c_10()
             
          
        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,z) -> c_4()
          11: minus#(x,0()) -> c_10()
          8: inc#(0()) -> c_5()
          9: lt#(x,0()) -> c_7()
          10: lt#(0(),s(y)) -> c_8()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
            division#(x,y) -> c_2(div#(x,y,0()))
            if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
            inc#(s(x)) -> c_6(inc#(x))
            lt#(s(x),s(y)) -> c_9(lt#(x,y))
            minus#(s(x),s(y)) -> c_11(minus#(x,y))
        - Weak TRS:
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,division/2,if/4,inc/1,lt/2,minus/2,div#/3,division#/2,if#/4,inc#/1,lt#/2,minus#/2} / {0/0,false/0,s/1
            ,true/0,c_1/3,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,division#,if#,inc#,lt#,minus#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
           -->_2 lt#(s(x),s(y)) -> c_9(lt#(x,y)):5
           -->_3 inc#(s(x)) -> c_6(inc#(x)):4
           -->_1 if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y))):3
        
        2:S:division#(x,y) -> c_2(div#(x,y,0()))
           -->_1 div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z)):1
        
        3:S:if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
           -->_2 minus#(s(x),s(y)) -> c_11(minus#(x,y)):6
           -->_1 div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z)):1
        
        4:S:inc#(s(x)) -> c_6(inc#(x))
           -->_1 inc#(s(x)) -> c_6(inc#(x)):4
        
        5:S:lt#(s(x),s(y)) -> c_9(lt#(x,y))
           -->_1 lt#(s(x),s(y)) -> c_9(lt#(x,y)):5
        
        6:S:minus#(s(x),s(y)) -> c_11(minus#(x,y))
           -->_1 minus#(s(x),s(y)) -> c_11(minus#(x,y)):6
        
        
        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).
        
        [(2,division#(x,y) -> c_2(div#(x,y,0())))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          div#(x,y,z) -> c_1(if#(lt(x,y),x,y,inc(z)),lt#(x,y),inc#(z))
          if#(false(),x,s(y),z) -> c_3(div#(minus(x,s(y)),s(y),z),minus#(x,s(y)))
          inc#(s(x)) -> c_6(inc#(x))
          lt#(s(x),s(y)) -> c_9(lt#(x,y))
          minus#(s(x),s(y)) -> c_11(minus#(x,y))
      - Weak TRS:
          inc(0()) -> s(0())
          inc(s(x)) -> s(inc(x))
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
      - Signature:
          {div/3,division/2,if/4,inc/1,lt/2,minus/2,div#/3,division#/2,if#/4,inc#/1,lt#/2,minus#/2} / {0/0,false/0,s/1
          ,true/0,c_1/3,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {div#,division#,if#,inc#,lt#,minus#} and constructors {0
          ,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE