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