MAYBE

Problem:
 lt(0(),s(x)) -> true()
 lt(x,0()) -> false()
 lt(s(x),s(y)) -> lt(x,y)
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(x,y))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 fac(x) -> loop(x,s(0()),s(0()))
 loop(x,c,y) -> if(lt(x,c),x,c,y)
 if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
 if(true(),x,c,y) -> y

Proof:
 DP Processor:
  DPs:
   lt#(s(x),s(y)) -> lt#(x,y)
   times#(s(x),y) -> times#(x,y)
   times#(s(x),y) -> plus#(y,times(x,y))
   plus#(s(x),y) -> plus#(x,y)
   fac#(x) -> loop#(x,s(0()),s(0()))
   loop#(x,c,y) -> lt#(x,c)
   loop#(x,c,y) -> if#(lt(x,c),x,c,y)
   if#(false(),x,c,y) -> times#(y,s(c))
   if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c)))
  TRS:
   lt(0(),s(x)) -> true()
   lt(x,0()) -> false()
   lt(s(x),s(y)) -> lt(x,y)
   times(0(),y) -> 0()
   times(s(x),y) -> plus(y,times(x,y))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   fac(x) -> loop(x,s(0()),s(0()))
   loop(x,c,y) -> if(lt(x,c),x,c,y)
   if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
   if(true(),x,c,y) -> y
  TDG Processor:
   DPs:
    lt#(s(x),s(y)) -> lt#(x,y)
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y))
    plus#(s(x),y) -> plus#(x,y)
    fac#(x) -> loop#(x,s(0()),s(0()))
    loop#(x,c,y) -> lt#(x,c)
    loop#(x,c,y) -> if#(lt(x,c),x,c,y)
    if#(false(),x,c,y) -> times#(y,s(c))
    if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c)))
   TRS:
    lt(0(),s(x)) -> true()
    lt(x,0()) -> false()
    lt(s(x),s(y)) -> lt(x,y)
    times(0(),y) -> 0()
    times(s(x),y) -> plus(y,times(x,y))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    fac(x) -> loop(x,s(0()),s(0()))
    loop(x,c,y) -> if(lt(x,c),x,c,y)
    if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
    if(true(),x,c,y) -> y
   graph:
    if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c))) ->
    loop#(x,c,y) -> if#(lt(x,c),x,c,y)
    if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c))) ->
    loop#(x,c,y) -> lt#(x,c)
    if#(false(),x,c,y) -> times#(y,s(c)) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    if#(false(),x,c,y) -> times#(y,s(c)) ->
    times#(s(x),y) -> times#(x,y)
    loop#(x,c,y) -> if#(lt(x,c),x,c,y) ->
    if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c)))
    loop#(x,c,y) -> if#(lt(x,c),x,c,y) ->
    if#(false(),x,c,y) -> times#(y,s(c))
    loop#(x,c,y) -> lt#(x,c) -> lt#(s(x),s(y)) -> lt#(x,y)
    fac#(x) -> loop#(x,s(0()),s(0())) ->
    loop#(x,c,y) -> if#(lt(x,c),x,c,y)
    fac#(x) -> loop#(x,s(0()),s(0())) -> loop#(x,c,y) -> lt#(x,c)
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y)) ->
    plus#(s(x),y) -> plus#(x,y)
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    times#(s(x),y) -> times#(x,y) -> times#(s(x),y) -> times#(x,y)
    lt#(s(x),s(y)) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 14/81
    DPs:
     if#(false(),x,c,y) -> loop#(x,s(c),times(y,s(c)))
     loop#(x,c,y) -> if#(lt(x,c),x,c,y)
    TRS:
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     fac(x) -> loop(x,s(0()),s(0()))
     loop(x,c,y) -> if(lt(x,c),x,c,y)
     if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
     if(true(),x,c,y) -> y
    Open
    
    DPs:
     times#(s(x),y) -> times#(x,y)
    TRS:
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     fac(x) -> loop(x,s(0()),s(0()))
     loop(x,c,y) -> if(lt(x,c),x,c,y)
     if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
     if(true(),x,c,y) -> y
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       lt(0(),s(x)) -> true()
       lt(x,0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       fac(x) -> loop(x,s(0()),s(0()))
       loop(x,c,y) -> if(lt(x,c),x,c,y)
       if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
       if(true(),x,c,y) -> y
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
    TRS:
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     fac(x) -> loop(x,s(0()),s(0()))
     loop(x,c,y) -> if(lt(x,c),x,c,y)
     if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
     if(true(),x,c,y) -> y
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       lt(0(),s(x)) -> true()
       lt(x,0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       fac(x) -> loop(x,s(0()),s(0()))
       loop(x,c,y) -> if(lt(x,c),x,c,y)
       if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
       if(true(),x,c,y) -> y
     Qed
    
    DPs:
     lt#(s(x),s(y)) -> lt#(x,y)
    TRS:
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     fac(x) -> loop(x,s(0()),s(0()))
     loop(x,c,y) -> if(lt(x,c),x,c,y)
     if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
     if(true(),x,c,y) -> y
    Subterm Criterion Processor:
     simple projection:
      pi(lt#) = 1
     problem:
      DPs:
       
      TRS:
       lt(0(),s(x)) -> true()
       lt(x,0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       fac(x) -> loop(x,s(0()),s(0()))
       loop(x,c,y) -> if(lt(x,c),x,c,y)
       if(false(),x,c,y) -> loop(x,s(c),times(y,s(c)))
       if(true(),x,c,y) -> y
     Qed