MAYBE

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

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,s(y)) -> p#(s(y))
   minus#(x,s(y)) -> minus#(x,p(s(y)))
   minus#(x,s(y)) -> p#(minus(x,p(s(y))))
   minus#(x,s(y)) -> le#(x,s(y))
   minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
  TRS:
   p(0()) -> 0()
   p(s(x)) -> x
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(x,0()) -> x
   minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
   if(true(),x,y) -> x
   if(false(),x,y) -> y
  CDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> p#(s(y))
    minus#(x,s(y)) -> minus#(x,p(s(y)))
    minus#(x,s(y)) -> p#(minus(x,p(s(y))))
    minus#(x,s(y)) -> le#(x,s(y))
    minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
   TRS:
    p(0()) -> 0()
    p(s(x)) -> x
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(x,0()) -> x
    minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
    if(true(),x,y) -> x
    if(false(),x,y) -> y
   graph:
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> p#(s(y))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> minus#(x,p(s(y)))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> p#(minus(x,p(s(y))))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> le#(x,s(y))
    minus#(x,s(y)) -> minus#(x,p(s(y))) ->
    minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
    minus#(x,s(y)) -> le#(x,s(y)) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   Restore Modifier:
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
     minus#(x,s(y)) -> p#(s(y))
     minus#(x,s(y)) -> minus#(x,p(s(y)))
     minus#(x,s(y)) -> p#(minus(x,p(s(y))))
     minus#(x,s(y)) -> le#(x,s(y))
     minus#(x,s(y)) -> if#(le(x,s(y)),0(),p(minus(x,p(s(y)))))
    TRS:
     p(0()) -> 0()
     p(s(x)) -> x
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
     if(true(),x,y) -> x
     if(false(),x,y) -> y
    SCC Processor:
     #sccs: 2
     #rules: 2
     #arcs: 7/36
     DPs:
      minus#(x,s(y)) -> minus#(x,p(s(y)))
     TRS:
      p(0()) -> 0()
      p(s(x)) -> x
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      minus(x,0()) -> x
      minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
      if(true(),x,y) -> x
      if(false(),x,y) -> y
     Open
     
     DPs:
      le#(s(x),s(y)) -> le#(x,y)
     TRS:
      p(0()) -> 0()
      p(s(x)) -> x
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      minus(x,0()) -> x
      minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
      if(true(),x,y) -> x
      if(false(),x,y) -> y
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [le#](x0, x1) = x1 + 1,
       
       [if](x0, x1, x2) = x1 + x2,
       
       [minus](x0, x1) = x0,
       
       [false] = 0,
       
       [true] = 0,
       
       [le](x0, x1) = 0,
       
       [s](x0) = x0 + 1,
       
       [p](x0) = x0,
       
       [0] = 0
      orientation:
       le#(s(x),s(y)) = y + 2 >= y + 1 = le#(x,y)
       
       p(0()) = 0 >= 0 = 0()
       
       p(s(x)) = x + 1 >= x = x
       
       le(0(),y) = 0 >= 0 = true()
       
       le(s(x),0()) = 0 >= 0 = false()
       
       le(s(x),s(y)) = 0 >= 0 = le(x,y)
       
       minus(x,0()) = x >= x = x
       
       minus(x,s(y)) = x >= x = if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
       
       if(true(),x,y) = x + y >= x = x
       
       if(false(),x,y) = x + y >= y = y
      problem:
       DPs:
        
       TRS:
        p(0()) -> 0()
        p(s(x)) -> x
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        minus(x,0()) -> x
        minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y)))))
        if(true(),x,y) -> x
        if(false(),x,y) -> y
      Qed