MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            id(0()) -> 0()
            id(s(x)) -> s(id(x))
            if2(false(),b2,x,y) -> if3(b2,x,y)
            if2(true(),b2,x,y) -> 0()
            if3(false(),x,y) -> x
            if3(true(),x,y) -> mod(minus(x,y),s(y))
            if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
            if_mod(true(),b1,b2,x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id,if2,if3,if_mod,le,minus,mod,zero} and constructors {0
            ,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          id#(0()) -> c_1()
          id#(s(x)) -> c_2(id#(x))
          if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
          if2#(true(),b2,x,y) -> c_4()
          if3#(false(),x,y) -> c_5()
          if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
          if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
          if_mod#(true(),b1,b2,x,y) -> c_8()
          le#(0(),y) -> c_9()
          le#(s(x),0()) -> c_10()
          le#(s(x),s(y)) -> c_11(le#(x,y))
          minus#(x,0()) -> c_12()
          minus#(s(x),s(y)) -> c_13(minus#(x,y))
          mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
          zero#(0()) -> c_15()
          zero#(s(x)) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            id#(0()) -> c_1()
            id#(s(x)) -> c_2(id#(x))
            if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
            if2#(true(),b2,x,y) -> c_4()
            if3#(false(),x,y) -> c_5()
            if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
            if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
            if_mod#(true(),b1,b2,x,y) -> c_8()
            le#(0(),y) -> c_9()
            le#(s(x),0()) -> c_10()
            le#(s(x),s(y)) -> c_11(le#(x,y))
            minus#(x,0()) -> c_12()
            minus#(s(x),s(y)) -> c_13(minus#(x,y))
            mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
            zero#(0()) -> c_15()
            zero#(s(x)) -> c_16()
        - Weak TRS:
            id(0()) -> 0()
            id(s(x)) -> s(id(x))
            if2(false(),b2,x,y) -> if3(b2,x,y)
            if2(true(),b2,x,y) -> 0()
            if3(false(),x,y) -> x
            if3(true(),x,y) -> mod(minus(x,y),s(y))
            if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
            if_mod(true(),b1,b2,x,y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
            ,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
            ,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          id(0()) -> 0()
          id(s(x)) -> s(id(x))
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          zero(0()) -> true()
          zero(s(x)) -> false()
          id#(0()) -> c_1()
          id#(s(x)) -> c_2(id#(x))
          if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
          if2#(true(),b2,x,y) -> c_4()
          if3#(false(),x,y) -> c_5()
          if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
          if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
          if_mod#(true(),b1,b2,x,y) -> c_8()
          le#(0(),y) -> c_9()
          le#(s(x),0()) -> c_10()
          le#(s(x),s(y)) -> c_11(le#(x,y))
          minus#(x,0()) -> c_12()
          minus#(s(x),s(y)) -> c_13(minus#(x,y))
          mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
          zero#(0()) -> c_15()
          zero#(s(x)) -> c_16()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            id#(0()) -> c_1()
            id#(s(x)) -> c_2(id#(x))
            if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
            if2#(true(),b2,x,y) -> c_4()
            if3#(false(),x,y) -> c_5()
            if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
            if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
            if_mod#(true(),b1,b2,x,y) -> c_8()
            le#(0(),y) -> c_9()
            le#(s(x),0()) -> c_10()
            le#(s(x),s(y)) -> c_11(le#(x,y))
            minus#(x,0()) -> c_12()
            minus#(s(x),s(y)) -> c_13(minus#(x,y))
            mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
            zero#(0()) -> c_15()
            zero#(s(x)) -> c_16()
        - Weak TRS:
            id(0()) -> 0()
            id(s(x)) -> s(id(x))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
            ,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
            ,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,8,9,10,12,15,16}
        by application of
          Pre({1,4,5,8,9,10,12,15,16}) = {2,3,6,7,11,13,14}.
        Here rules are labelled as follows:
          1: id#(0()) -> c_1()
          2: id#(s(x)) -> c_2(id#(x))
          3: if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
          4: if2#(true(),b2,x,y) -> c_4()
          5: if3#(false(),x,y) -> c_5()
          6: if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
          7: if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
          8: if_mod#(true(),b1,b2,x,y) -> c_8()
          9: le#(0(),y) -> c_9()
          10: le#(s(x),0()) -> c_10()
          11: le#(s(x),s(y)) -> c_11(le#(x,y))
          12: minus#(x,0()) -> c_12()
          13: minus#(s(x),s(y)) -> c_13(minus#(x,y))
          14: mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
          15: zero#(0()) -> c_15()
          16: zero#(s(x)) -> c_16()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            id#(s(x)) -> c_2(id#(x))
            if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
            if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
            if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
            le#(s(x),s(y)) -> c_11(le#(x,y))
            minus#(s(x),s(y)) -> c_13(minus#(x,y))
            mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
        - Weak DPs:
            id#(0()) -> c_1()
            if2#(true(),b2,x,y) -> c_4()
            if3#(false(),x,y) -> c_5()
            if_mod#(true(),b1,b2,x,y) -> c_8()
            le#(0(),y) -> c_9()
            le#(s(x),0()) -> c_10()
            minus#(x,0()) -> c_12()
            zero#(0()) -> c_15()
            zero#(s(x)) -> c_16()
        - Weak TRS:
            id(0()) -> 0()
            id(s(x)) -> s(id(x))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
            ,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
            ,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:id#(s(x)) -> c_2(id#(x))
             -->_1 id#(0()) -> c_1():8
             -->_1 id#(s(x)) -> c_2(id#(x)):1
          
          2:S:if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
             -->_1 if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y)):3
             -->_1 if3#(false(),x,y) -> c_5():10
          
          3:S:if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
                                    ,zero#(x)
                                    ,zero#(y)
                                    ,le#(y,x)
                                    ,id#(x)
                                    ,id#(y)):7
             -->_2 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
             -->_2 minus#(x,0()) -> c_12():14
          
          4:S:if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
             -->_1 if2#(true(),b2,x,y) -> c_4():9
             -->_1 if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y)):2
          
          5:S:le#(s(x),s(y)) -> c_11(le#(x,y))
             -->_1 le#(s(x),0()) -> c_10():13
             -->_1 le#(0(),y) -> c_9():12
             -->_1 le#(s(x),s(y)) -> c_11(le#(x,y)):5
          
          6:S:minus#(s(x),s(y)) -> c_13(minus#(x,y))
             -->_1 minus#(x,0()) -> c_12():14
             -->_1 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
          
          7:S:mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
             -->_3 zero#(s(x)) -> c_16():16
             -->_2 zero#(s(x)) -> c_16():16
             -->_3 zero#(0()) -> c_15():15
             -->_2 zero#(0()) -> c_15():15
             -->_4 le#(s(x),0()) -> c_10():13
             -->_4 le#(0(),y) -> c_9():12
             -->_1 if_mod#(true(),b1,b2,x,y) -> c_8():11
             -->_6 id#(0()) -> c_1():8
             -->_5 id#(0()) -> c_1():8
             -->_4 le#(s(x),s(y)) -> c_11(le#(x,y)):5
             -->_1 if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y)):4
             -->_6 id#(s(x)) -> c_2(id#(x)):1
             -->_5 id#(s(x)) -> c_2(id#(x)):1
          
          8:W:id#(0()) -> c_1()
             
          
          9:W:if2#(true(),b2,x,y) -> c_4()
             
          
          10:W:if3#(false(),x,y) -> c_5()
             
          
          11:W:if_mod#(true(),b1,b2,x,y) -> c_8()
             
          
          12:W:le#(0(),y) -> c_9()
             
          
          13:W:le#(s(x),0()) -> c_10()
             
          
          14:W:minus#(x,0()) -> c_12()
             
          
          15:W:zero#(0()) -> c_15()
             
          
          16:W:zero#(s(x)) -> c_16()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: if3#(false(),x,y) -> c_5()
          14: minus#(x,0()) -> c_12()
          9: if2#(true(),b2,x,y) -> c_4()
          11: if_mod#(true(),b1,b2,x,y) -> c_8()
          12: le#(0(),y) -> c_9()
          13: le#(s(x),0()) -> c_10()
          15: zero#(0()) -> c_15()
          16: zero#(s(x)) -> c_16()
          8: id#(0()) -> c_1()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            id#(s(x)) -> c_2(id#(x))
            if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
            if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
            if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
            le#(s(x),s(y)) -> c_11(le#(x,y))
            minus#(s(x),s(y)) -> c_13(minus#(x,y))
            mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
        - Weak TRS:
            id(0()) -> 0()
            id(s(x)) -> s(id(x))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
            ,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
            ,c_12/0,c_13/1,c_14/6,c_15/0,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:id#(s(x)) -> c_2(id#(x))
             -->_1 id#(s(x)) -> c_2(id#(x)):1
          
          2:S:if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
             -->_1 if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y)):3
          
          3:S:if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
                                    ,zero#(x)
                                    ,zero#(y)
                                    ,le#(y,x)
                                    ,id#(x)
                                    ,id#(y)):7
             -->_2 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
          
          4:S:if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
             -->_1 if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y)):2
          
          5:S:le#(s(x),s(y)) -> c_11(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_11(le#(x,y)):5
          
          6:S:minus#(s(x),s(y)) -> c_13(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_13(minus#(x,y)):6
          
          7:S:mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),zero#(x),zero#(y),le#(y,x),id#(x),id#(y))
             -->_4 le#(s(x),s(y)) -> c_11(le#(x,y)):5
             -->_1 if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y)):4
             -->_6 id#(s(x)) -> c_2(id#(x)):1
             -->_5 id#(s(x)) -> c_2(id#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),le#(y,x),id#(x),id#(y))
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          id#(s(x)) -> c_2(id#(x))
          if2#(false(),b2,x,y) -> c_3(if3#(b2,x,y))
          if3#(true(),x,y) -> c_6(mod#(minus(x,y),s(y)),minus#(x,y))
          if_mod#(false(),b1,b2,x,y) -> c_7(if2#(b1,b2,x,y))
          le#(s(x),s(y)) -> c_11(le#(x,y))
          minus#(s(x),s(y)) -> c_13(minus#(x,y))
          mod#(x,y) -> c_14(if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)),le#(y,x),id#(x),id#(y))
      - Weak TRS:
          id(0()) -> 0()
          id(s(x)) -> s(id(x))
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          zero(0()) -> true()
          zero(s(x)) -> false()
      - Signature:
          {id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2,zero/1,id#/1,if2#/4,if3#/3,if_mod#/5,le#/2,minus#/2,mod#/2
          ,zero#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1
          ,c_12/0,c_13/1,c_14/4,c_15/0,c_16/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {id#,if2#,if3#,if_mod#,le#,minus#,mod#
          ,zero#} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE