MAYBE

Problem:
 p(0()) -> s(s(0()))
 p(s(x)) -> x
 p(p(s(x))) -> p(x)
 le(p(s(x)),x) -> le(x,x)
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,y) -> if(le(x,y),x,y)
 if(true(),x,y) -> 0()
 if(false(),x,y) -> s(minus(p(x),y))

Proof:
 DP Processor:
  DPs:
   p#(p(s(x))) -> p#(x)
   le#(p(s(x)),x) -> le#(x,x)
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,y) -> le#(x,y)
   minus#(x,y) -> if#(le(x,y),x,y)
   if#(false(),x,y) -> p#(x)
   if#(false(),x,y) -> minus#(p(x),y)
  TRS:
   p(0()) -> s(s(0()))
   p(s(x)) -> x
   p(p(s(x))) -> p(x)
   le(p(s(x)),x) -> le(x,x)
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(x,y) -> if(le(x,y),x,y)
   if(true(),x,y) -> 0()
   if(false(),x,y) -> s(minus(p(x),y))
  TDG Processor:
   DPs:
    p#(p(s(x))) -> p#(x)
    le#(p(s(x)),x) -> le#(x,x)
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,y) -> le#(x,y)
    minus#(x,y) -> if#(le(x,y),x,y)
    if#(false(),x,y) -> p#(x)
    if#(false(),x,y) -> minus#(p(x),y)
   TRS:
    p(0()) -> s(s(0()))
    p(s(x)) -> x
    p(p(s(x))) -> p(x)
    le(p(s(x)),x) -> le(x,x)
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(x,y) -> if(le(x,y),x,y)
    if(true(),x,y) -> 0()
    if(false(),x,y) -> s(minus(p(x),y))
   graph:
    if#(false(),x,y) -> minus#(p(x),y) ->
    minus#(x,y) -> if#(le(x,y),x,y)
    if#(false(),x,y) -> minus#(p(x),y) -> minus#(x,y) -> le#(x,y)
    if#(false(),x,y) -> p#(x) -> p#(p(s(x))) -> p#(x)
    minus#(x,y) -> if#(le(x,y),x,y) ->
    if#(false(),x,y) -> minus#(p(x),y)
    minus#(x,y) -> if#(le(x,y),x,y) -> if#(false(),x,y) -> p#(x)
    minus#(x,y) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    minus#(x,y) -> le#(x,y) -> le#(p(s(x)),x) -> le#(x,x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(p(s(x)),x) -> le#(x,x)
    le#(p(s(x)),x) -> le#(x,x) -> le#(s(x),s(y)) -> le#(x,y)
    le#(p(s(x)),x) -> le#(x,x) -> le#(p(s(x)),x) -> le#(x,x)
    p#(p(s(x))) -> p#(x) -> p#(p(s(x))) -> p#(x)
   SCC Processor:
    #sccs: 3
    #rules: 5
    #arcs: 12/49
    DPs:
     if#(false(),x,y) -> minus#(p(x),y)
     minus#(x,y) -> if#(le(x,y),x,y)
    TRS:
     p(0()) -> s(s(0()))
     p(s(x)) -> x
     p(p(s(x))) -> p(x)
     le(p(s(x)),x) -> le(x,x)
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,y) -> if(le(x,y),x,y)
     if(true(),x,y) -> 0()
     if(false(),x,y) -> s(minus(p(x),y))
    Open
    
    DPs:
     p#(p(s(x))) -> p#(x)
    TRS:
     p(0()) -> s(s(0()))
     p(s(x)) -> x
     p(p(s(x))) -> p(x)
     le(p(s(x)),x) -> le(x,x)
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,y) -> if(le(x,y),x,y)
     if(true(),x,y) -> 0()
     if(false(),x,y) -> s(minus(p(x),y))
    Subterm Criterion Processor:
     simple projection:
      pi(p#) = 0
     problem:
      DPs:
       
      TRS:
       p(0()) -> s(s(0()))
       p(s(x)) -> x
       p(p(s(x))) -> p(x)
       le(p(s(x)),x) -> le(x,x)
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,y) -> if(le(x,y),x,y)
       if(true(),x,y) -> 0()
       if(false(),x,y) -> s(minus(p(x),y))
     Qed
    
    DPs:
     le#(p(s(x)),x) -> le#(x,x)
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     p(0()) -> s(s(0()))
     p(s(x)) -> x
     p(p(s(x))) -> p(x)
     le(p(s(x)),x) -> le(x,x)
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,y) -> if(le(x,y),x,y)
     if(true(),x,y) -> 0()
     if(false(),x,y) -> s(minus(p(x),y))
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       le#(p(s(x)),x) -> le#(x,x)
      TRS:
       p(0()) -> s(s(0()))
       p(s(x)) -> x
       p(p(s(x))) -> p(x)
       le(p(s(x)),x) -> le(x,x)
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,y) -> if(le(x,y),x,y)
       if(true(),x,y) -> 0()
       if(false(),x,y) -> s(minus(p(x),y))
     Subterm Criterion Processor:
      simple projection:
       pi(le#) = 0
      problem:
       DPs:
        
       TRS:
        p(0()) -> s(s(0()))
        p(s(x)) -> x
        p(p(s(x))) -> p(x)
        le(p(s(x)),x) -> le(x,x)
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        minus(x,y) -> if(le(x,y),x,y)
        if(true(),x,y) -> 0()
        if(false(),x,y) -> s(minus(p(x),y))
      Qed