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