MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            fac(x) -> help(x,0())
            help(x,c) -> if(lt(c,x),x,c)
            if(false(),x,c) -> s(0())
            if(true(),x,c) -> times(s(c),help(x,s(c)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2} / {0/0,false/0,s/1,times/2,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,help,if,lt} and constructors {0,false,s,times,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fac#(x) -> c_1(help#(x,0()))
          help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
          if#(false(),x,c) -> c_3()
          if#(true(),x,c) -> c_4(help#(x,s(c)))
          lt#(x,0()) -> c_5()
          lt#(0(),s(x)) -> c_6()
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(x) -> c_1(help#(x,0()))
            help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
            if#(false(),x,c) -> c_3()
            if#(true(),x,c) -> c_4(help#(x,s(c)))
            lt#(x,0()) -> c_5()
            lt#(0(),s(x)) -> c_6()
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
        - Weak TRS:
            fac(x) -> help(x,0())
            help(x,c) -> if(lt(c,x),x,c)
            if(false(),x,c) -> s(0())
            if(true(),x,c) -> times(s(c),help(x,s(c)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
            ,c_4/1,c_5/0,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
            ,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)
          fac#(x) -> c_1(help#(x,0()))
          help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
          if#(false(),x,c) -> c_3()
          if#(true(),x,c) -> c_4(help#(x,s(c)))
          lt#(x,0()) -> c_5()
          lt#(0(),s(x)) -> c_6()
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(x) -> c_1(help#(x,0()))
            help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
            if#(false(),x,c) -> c_3()
            if#(true(),x,c) -> c_4(help#(x,s(c)))
            lt#(x,0()) -> c_5()
            lt#(0(),s(x)) -> c_6()
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
            ,c_4/1,c_5/0,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,5,6}
        by application of
          Pre({3,5,6}) = {2,7}.
        Here rules are labelled as follows:
          1: fac#(x) -> c_1(help#(x,0()))
          2: help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
          3: if#(false(),x,c) -> c_3()
          4: if#(true(),x,c) -> c_4(help#(x,s(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))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(x) -> c_1(help#(x,0()))
            help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
            if#(true(),x,c) -> c_4(help#(x,s(c)))
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
        - Weak DPs:
            if#(false(),x,c) -> c_3()
            lt#(x,0()) -> c_5()
            lt#(0(),s(x)) -> c_6()
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
            ,c_4/1,c_5/0,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fac#(x) -> c_1(help#(x,0()))
             -->_1 help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x)):2
          
          2:S:help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
             -->_2 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
             -->_1 if#(true(),x,c) -> c_4(help#(x,s(c))):3
             -->_2 lt#(0(),s(x)) -> c_6():7
             -->_2 lt#(x,0()) -> c_5():6
             -->_1 if#(false(),x,c) -> c_3():5
          
          3:S:if#(true(),x,c) -> c_4(help#(x,s(c)))
             -->_1 help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x)):2
          
          4:S:lt#(s(x),s(y)) -> c_7(lt#(x,y))
             -->_1 lt#(0(),s(x)) -> c_6():7
             -->_1 lt#(x,0()) -> c_5():6
             -->_1 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
          
          5:W:if#(false(),x,c) -> c_3()
             
          
          6:W:lt#(x,0()) -> c_5()
             
          
          7:W:lt#(0(),s(x)) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: if#(false(),x,c) -> c_3()
          6: lt#(x,0()) -> c_5()
          7: lt#(0(),s(x)) -> c_6()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(x) -> c_1(help#(x,0()))
            help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
            if#(true(),x,c) -> c_4(help#(x,s(c)))
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
            ,c_4/1,c_5/0,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
            ,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:fac#(x) -> c_1(help#(x,0()))
           -->_1 help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x)):2
        
        2:S:help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
           -->_2 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
           -->_1 if#(true(),x,c) -> c_4(help#(x,s(c))):3
        
        3:S:if#(true(),x,c) -> c_4(help#(x,s(c)))
           -->_1 help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x)):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
        
        
        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(help#(x,0())))]
* Step 6: WeightGap MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
            if#(true(),x,c) -> c_4(help#(x,s(c)))
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
            ,c_4/1,c_5/0,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
            ,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(if#) = {1},
            uargs(c_2) = {1,2},
            uargs(c_4) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]                  
              p(fac) = [0]                  
            p(false) = [1]                  
             p(help) = [0]                  
               p(if) = [0]                  
               p(lt) = [2]                  
                p(s) = [0]                  
            p(times) = [0]                  
             p(true) = [2]                  
             p(fac#) = [0]                  
            p(help#) = [0]                  
              p(if#) = [1] x1 + [1]         
              p(lt#) = [1]                  
              p(c_1) = [0]                  
              p(c_2) = [1] x1 + [1] x2 + [0]
              p(c_3) = [0]                  
              p(c_4) = [1] x1 + [0]         
              p(c_5) = [0]                  
              p(c_6) = [2]                  
              p(c_7) = [1] x1 + [8]         
          
          Following rules are strictly oriented:
          if#(true(),x,c) = [3]               
                          > [0]               
                          = c_4(help#(x,s(c)))
          
          
          Following rules are (at-least) weakly oriented:
              help#(x,c) =  [0]                           
                         >= [4]                           
                         =  c_2(if#(lt(c,x),x,c),lt#(c,x))
          
          lt#(s(x),s(y)) =  [1]                           
                         >= [9]                           
                         =  c_7(lt#(x,y))                 
          
               lt(x,0()) =  [2]                           
                         >= [1]                           
                         =  false()                       
          
            lt(0(),s(x)) =  [2]                           
                         >= [2]                           
                         =  true()                        
          
           lt(s(x),s(y)) =  [2]                           
                         >= [2]                           
                         =  lt(x,y)                       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          help#(x,c) -> c_2(if#(lt(c,x),x,c),lt#(c,x))
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
      - Weak DPs:
          if#(true(),x,c) -> c_4(help#(x,s(c)))
      - Weak TRS:
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
      - Signature:
          {fac/1,help/2,if/3,lt/2,fac#/1,help#/2,if#/3,lt#/2} / {0/0,false/0,s/1,times/2,true/0,c_1/1,c_2/2,c_3/0
          ,c_4/1,c_5/0,c_6/0,c_7/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {fac#,help#,if#,lt#} and constructors {0,false,s,times
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE