MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            if(false(),b,x,y,z) -> div_by_zero()
            if(true(),false(),x,y,z) -> z
            if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
            quot(x,y) -> div(x,y,0())
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2} / {0/0,div_by_zero/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,ge,id_inc,if,minus,quot} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
          ge#(x,0()) -> c_2()
          ge#(0(),s(y)) -> c_3()
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          id_inc#(x) -> c_5()
          id_inc#(x) -> c_6()
          if#(false(),b,x,y,z) -> c_7()
          if#(true(),false(),x,y,z) -> c_8()
          if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
          minus#(x,0()) -> c_10()
          minus#(0(),y) -> c_11()
          minus#(s(x),s(y)) -> c_12(minus#(x,y))
          quot#(x,y) -> c_13(div#(x,y,0()))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
            ge#(x,0()) -> c_2()
            ge#(0(),s(y)) -> c_3()
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            id_inc#(x) -> c_5()
            id_inc#(x) -> c_6()
            if#(false(),b,x,y,z) -> c_7()
            if#(true(),false(),x,y,z) -> c_8()
            if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
            minus#(x,0()) -> c_10()
            minus#(0(),y) -> c_11()
            minus#(s(x),s(y)) -> c_12(minus#(x,y))
            quot#(x,y) -> c_13(div#(x,y,0()))
        - Weak TRS:
            div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            if(false(),b,x,y,z) -> div_by_zero()
            if(true(),false(),x,y,z) -> z
            if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
            quot(x,y) -> div(x,y,0())
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
            ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/3,c_10/0,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          id_inc(x) -> x
          id_inc(x) -> s(x)
          minus(x,0()) -> x
          minus(0(),y) -> 0()
          minus(s(x),s(y)) -> minus(x,y)
          div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
          ge#(x,0()) -> c_2()
          ge#(0(),s(y)) -> c_3()
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          id_inc#(x) -> c_5()
          id_inc#(x) -> c_6()
          if#(false(),b,x,y,z) -> c_7()
          if#(true(),false(),x,y,z) -> c_8()
          if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
          minus#(x,0()) -> c_10()
          minus#(0(),y) -> c_11()
          minus#(s(x),s(y)) -> c_12(minus#(x,y))
          quot#(x,y) -> c_13(div#(x,y,0()))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
            ge#(x,0()) -> c_2()
            ge#(0(),s(y)) -> c_3()
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            id_inc#(x) -> c_5()
            id_inc#(x) -> c_6()
            if#(false(),b,x,y,z) -> c_7()
            if#(true(),false(),x,y,z) -> c_8()
            if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
            minus#(x,0()) -> c_10()
            minus#(0(),y) -> c_11()
            minus#(s(x),s(y)) -> c_12(minus#(x,y))
            quot#(x,y) -> c_13(div#(x,y,0()))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
            ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/3,c_10/0,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,5,6,7,8,10,11}
        by application of
          Pre({2,3,5,6,7,8,10,11}) = {1,4,9,12}.
        Here rules are labelled as follows:
          1: div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
          2: ge#(x,0()) -> c_2()
          3: ge#(0(),s(y)) -> c_3()
          4: ge#(s(x),s(y)) -> c_4(ge#(x,y))
          5: id_inc#(x) -> c_5()
          6: id_inc#(x) -> c_6()
          7: if#(false(),b,x,y,z) -> c_7()
          8: if#(true(),false(),x,y,z) -> c_8()
          9: if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
          10: minus#(x,0()) -> c_10()
          11: minus#(0(),y) -> c_11()
          12: minus#(s(x),s(y)) -> c_12(minus#(x,y))
          13: quot#(x,y) -> c_13(div#(x,y,0()))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
            minus#(s(x),s(y)) -> c_12(minus#(x,y))
            quot#(x,y) -> c_13(div#(x,y,0()))
        - Weak DPs:
            ge#(x,0()) -> c_2()
            ge#(0(),s(y)) -> c_3()
            id_inc#(x) -> c_5()
            id_inc#(x) -> c_6()
            if#(false(),b,x,y,z) -> c_7()
            if#(true(),false(),x,y,z) -> c_8()
            minus#(x,0()) -> c_10()
            minus#(0(),y) -> c_11()
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
            ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/3,c_10/0,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
             -->_1 if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z)):3
             -->_3 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
             -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
             -->_1 if#(true(),false(),x,y,z) -> c_8():11
             -->_1 if#(false(),b,x,y,z) -> c_7():10
             -->_3 ge#(0(),s(y)) -> c_3():7
             -->_2 ge#(0(),s(y)) -> c_3():7
             -->_3 ge#(x,0()) -> c_2():6
          
          2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y))
             -->_1 ge#(0(),s(y)) -> c_3():7
             -->_1 ge#(x,0()) -> c_2():6
             -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
          
          3:S:if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
             -->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
             -->_2 minus#(0(),y) -> c_11():13
             -->_2 minus#(x,0()) -> c_10():12
             -->_3 id_inc#(x) -> c_6():9
             -->_3 id_inc#(x) -> c_5():8
             -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):1
          
          4:S:minus#(s(x),s(y)) -> c_12(minus#(x,y))
             -->_1 minus#(0(),y) -> c_11():13
             -->_1 minus#(x,0()) -> c_10():12
             -->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
          
          5:S:quot#(x,y) -> c_13(div#(x,y,0()))
             -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):1
          
          6:W:ge#(x,0()) -> c_2()
             
          
          7:W:ge#(0(),s(y)) -> c_3()
             
          
          8:W:id_inc#(x) -> c_5()
             
          
          9:W:id_inc#(x) -> c_6()
             
          
          10:W:if#(false(),b,x,y,z) -> c_7()
             
          
          11:W:if#(true(),false(),x,y,z) -> c_8()
             
          
          12:W:minus#(x,0()) -> c_10()
             
          
          13:W:minus#(0(),y) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: if#(false(),b,x,y,z) -> c_7()
          11: if#(true(),false(),x,y,z) -> c_8()
          6: ge#(x,0()) -> c_2()
          7: ge#(0(),s(y)) -> c_3()
          8: id_inc#(x) -> c_5()
          9: id_inc#(x) -> c_6()
          12: minus#(x,0()) -> c_10()
          13: minus#(0(),y) -> c_11()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
            minus#(s(x),s(y)) -> c_12(minus#(x,y))
            quot#(x,y) -> c_13(div#(x,y,0()))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
            ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/3,c_10/0,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
             -->_1 if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z)):3
             -->_3 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
             -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
          
          2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
          
          3:S:if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y),id_inc#(z))
             -->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
             -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):1
          
          4:S:minus#(s(x),s(y)) -> c_12(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
          
          5:S:quot#(x,y) -> c_13(div#(x,y,0()))
             -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y))
* Step 6: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y))
            minus#(s(x),s(y)) -> c_12(minus#(x,y))
            quot#(x,y) -> c_13(div#(x,y,0()))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(y)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            id_inc(x) -> x
            id_inc(x) -> s(x)
            minus(x,0()) -> x
            minus(0(),y) -> 0()
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
            ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
            ,div_by_zero,false,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
           -->_1 if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y)):3
           -->_3 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
           -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
        
        2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y))
           -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
        
        3:S:if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y))
           -->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
           -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):1
        
        4:S:minus#(s(x),s(y)) -> c_12(minus#(x,y))
           -->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):4
        
        5:S:quot#(x,y) -> c_13(div#(x,y,0()))
           -->_1 div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y)):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,y) -> c_13(div#(x,y,0())))]
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          div#(x,y,z) -> c_1(if#(ge(y,s(0())),ge(x,y),x,y,z),ge#(y,s(0())),ge#(x,y))
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          if#(true(),true(),x,y,z) -> c_9(div#(minus(x,y),y,id_inc(z)),minus#(x,y))
          minus#(s(x),s(y)) -> c_12(minus#(x,y))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(y)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          id_inc(x) -> x
          id_inc(x) -> s(x)
          minus(x,0()) -> x
          minus(0(),y) -> 0()
          minus(s(x),s(y)) -> minus(x,y)
      - Signature:
          {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2,div#/3,ge#/2,id_inc#/1,if#/5,minus#/2,quot#/2} / {0/0,div_by_zero/0
          ,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1,c_13/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {div#,ge#,id_inc#,if#,minus#,quot#} and constructors {0
          ,div_by_zero,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE