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