MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            div(x,y) -> if1(ge(x,y),x,y)
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),x,y) -> 0()
            if(true(),x,y) -> s(minus(p(x),y))
            if1(false(),x,y) -> 0()
            if1(true(),x,y) -> if2(gt(y,0()),x,y)
            if2(false(),x,y) -> 0()
            if2(true(),x,y) -> s(div(minus(x,y),y))
            minus(x,y) -> if(gt(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,ge,gt,if,if1,if2,minus,p} 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(if1#(ge(x,y),x,y),ge#(x,y))
          ge#(x,0()) -> c_2()
          ge#(0(),s(x)) -> c_3()
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          gt#(0(),y) -> c_5()
          gt#(s(x),0()) -> c_6()
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(false(),x,y) -> c_8()
          if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
          if1#(false(),x,y) -> c_10()
          if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
          if2#(false(),x,y) -> c_12()
          if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
          minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
          p#(0()) -> c_15()
          p#(s(x)) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
            ge#(x,0()) -> c_2()
            ge#(0(),s(x)) -> c_3()
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(false(),x,y) -> c_8()
            if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
            if1#(false(),x,y) -> c_10()
            if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
            if2#(false(),x,y) -> c_12()
            if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
            minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
            p#(0()) -> c_15()
            p#(s(x)) -> c_16()
        - Weak TRS:
            div(x,y) -> if1(ge(x,y),x,y)
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),x,y) -> 0()
            if(true(),x,y) -> s(minus(p(x),y))
            if1(false(),x,y) -> 0()
            if1(true(),x,y) -> if2(gt(y,0()),x,y)
            if2(false(),x,y) -> 0()
            if2(true(),x,y) -> s(div(minus(x,y),y))
            minus(x,y) -> if(gt(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1,div#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2,p#/1} / {0/0
            ,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2,c_12/0,c_13/2,c_14/2
            ,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,gt#,if#,if1#,if2#,minus#,p#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ge(x,0()) -> true()
          ge(0(),s(x)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          gt(0(),y) -> false()
          gt(s(x),0()) -> true()
          gt(s(x),s(y)) -> gt(x,y)
          if(false(),x,y) -> 0()
          if(true(),x,y) -> s(minus(p(x),y))
          minus(x,y) -> if(gt(x,y),x,y)
          p(0()) -> 0()
          p(s(x)) -> x
          div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
          ge#(x,0()) -> c_2()
          ge#(0(),s(x)) -> c_3()
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          gt#(0(),y) -> c_5()
          gt#(s(x),0()) -> c_6()
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(false(),x,y) -> c_8()
          if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
          if1#(false(),x,y) -> c_10()
          if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
          if2#(false(),x,y) -> c_12()
          if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
          minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
          p#(0()) -> c_15()
          p#(s(x)) -> c_16()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
            ge#(x,0()) -> c_2()
            ge#(0(),s(x)) -> c_3()
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(false(),x,y) -> c_8()
            if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
            if1#(false(),x,y) -> c_10()
            if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
            if2#(false(),x,y) -> c_12()
            if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
            minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
            p#(0()) -> c_15()
            p#(s(x)) -> c_16()
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),x,y) -> 0()
            if(true(),x,y) -> s(minus(p(x),y))
            minus(x,y) -> if(gt(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1,div#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2,p#/1} / {0/0
            ,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2,c_12/0,c_13/2,c_14/2
            ,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,gt#,if#,if1#,if2#,minus#,p#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,5,6,8,10,12,15,16}
        by application of
          Pre({2,3,5,6,8,10,12,15,16}) = {1,4,7,9,11,14}.
        Here rules are labelled as follows:
          1: div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
          2: ge#(x,0()) -> c_2()
          3: ge#(0(),s(x)) -> c_3()
          4: ge#(s(x),s(y)) -> c_4(ge#(x,y))
          5: gt#(0(),y) -> c_5()
          6: gt#(s(x),0()) -> c_6()
          7: gt#(s(x),s(y)) -> c_7(gt#(x,y))
          8: if#(false(),x,y) -> c_8()
          9: if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
          10: if1#(false(),x,y) -> c_10()
          11: if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
          12: if2#(false(),x,y) -> c_12()
          13: if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
          14: minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
          15: p#(0()) -> c_15()
          16: p#(s(x)) -> c_16()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
            if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
            if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
            minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
        - Weak DPs:
            ge#(x,0()) -> c_2()
            ge#(0(),s(x)) -> c_3()
            gt#(0(),y) -> c_5()
            gt#(s(x),0()) -> c_6()
            if#(false(),x,y) -> c_8()
            if1#(false(),x,y) -> c_10()
            if2#(false(),x,y) -> c_12()
            p#(0()) -> c_15()
            p#(s(x)) -> c_16()
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),x,y) -> 0()
            if(true(),x,y) -> s(minus(p(x),y))
            minus(x,y) -> if(gt(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1,div#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2,p#/1} / {0/0
            ,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2,c_12/0,c_13/2,c_14/2
            ,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,gt#,if#,if1#,if2#,minus#,p#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
             -->_1 if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())):5
             -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
             -->_1 if1#(false(),x,y) -> c_10():13
             -->_2 ge#(0(),s(x)) -> c_3():9
             -->_2 ge#(x,0()) -> c_2():8
          
          2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y))
             -->_1 ge#(0(),s(x)) -> c_3():9
             -->_1 ge#(x,0()) -> c_2():8
             -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2
          
          3:S:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),0()) -> c_6():11
             -->_1 gt#(0(),y) -> c_5():10
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3
          
          4:S:if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
             -->_1 minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y)):7
             -->_2 p#(s(x)) -> c_16():16
             -->_2 p#(0()) -> c_15():15
          
          5:S:if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
             -->_1 if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y)):6
             -->_1 if2#(false(),x,y) -> c_12():14
             -->_2 gt#(s(x),0()) -> c_6():11
             -->_2 gt#(0(),y) -> c_5():10
          
          6:S:if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
             -->_2 minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y)):7
             -->_1 div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1
          
          7:S:minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
             -->_1 if#(false(),x,y) -> c_8():12
             -->_2 gt#(s(x),0()) -> c_6():11
             -->_2 gt#(0(),y) -> c_5():10
             -->_1 if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x)):4
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3
          
          8:W:ge#(x,0()) -> c_2()
             
          
          9:W:ge#(0(),s(x)) -> c_3()
             
          
          10:W:gt#(0(),y) -> c_5()
             
          
          11:W:gt#(s(x),0()) -> c_6()
             
          
          12:W:if#(false(),x,y) -> c_8()
             
          
          13:W:if1#(false(),x,y) -> c_10()
             
          
          14:W:if2#(false(),x,y) -> c_12()
             
          
          15:W:p#(0()) -> c_15()
             
          
          16:W:p#(s(x)) -> c_16()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: if1#(false(),x,y) -> c_10()
          8: ge#(x,0()) -> c_2()
          9: ge#(0(),s(x)) -> c_3()
          14: if2#(false(),x,y) -> c_12()
          15: p#(0()) -> c_15()
          16: p#(s(x)) -> c_16()
          10: gt#(0(),y) -> c_5()
          11: gt#(s(x),0()) -> c_6()
          12: if#(false(),x,y) -> c_8()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
            ge#(s(x),s(y)) -> c_4(ge#(x,y))
            gt#(s(x),s(y)) -> c_7(gt#(x,y))
            if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
            if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
            if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
            minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
        - Weak TRS:
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            gt(0(),y) -> false()
            gt(s(x),0()) -> true()
            gt(s(x),s(y)) -> gt(x,y)
            if(false(),x,y) -> 0()
            if(true(),x,y) -> s(minus(p(x),y))
            minus(x,y) -> if(gt(x,y),x,y)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1,div#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2,p#/1} / {0/0
            ,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2,c_12/0,c_13/2,c_14/2
            ,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,ge#,gt#,if#,if1#,if2#,minus#,p#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
             -->_1 if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())):5
             -->_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:gt#(s(x),s(y)) -> c_7(gt#(x,y))
             -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3
          
          4:S:if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x))
             -->_1 minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y)):7
          
          5:S:if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0()))
             -->_1 if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y)):6
          
          6:S:if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
             -->_2 minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y)):7
             -->_1 div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1
          
          7:S:minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
             -->_1 if#(true(),x,y) -> c_9(minus#(p(x),y),p#(x)):4
             -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if#(true(),x,y) -> c_9(minus#(p(x),y))
          if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y))
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          div#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y))
          ge#(s(x),s(y)) -> c_4(ge#(x,y))
          gt#(s(x),s(y)) -> c_7(gt#(x,y))
          if#(true(),x,y) -> c_9(minus#(p(x),y))
          if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y))
          if2#(true(),x,y) -> c_13(div#(minus(x,y),y),minus#(x,y))
          minus#(x,y) -> c_14(if#(gt(x,y),x,y),gt#(x,y))
      - Weak TRS:
          ge(x,0()) -> true()
          ge(0(),s(x)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          gt(0(),y) -> false()
          gt(s(x),0()) -> true()
          gt(s(x),s(y)) -> gt(x,y)
          if(false(),x,y) -> 0()
          if(true(),x,y) -> s(minus(p(x),y))
          minus(x,y) -> if(gt(x,y),x,y)
          p(0()) -> 0()
          p(s(x)) -> x
      - Signature:
          {div/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,p/1,div#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2,p#/1} / {0/0
          ,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/1,c_12/0,c_13/2,c_14/2
          ,c_15/0,c_16/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {div#,ge#,gt#,if#,if1#,if2#,minus#,p#} and constructors {0
          ,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE