MAYBE

Problem:
 div(x,s(y)) -> d(x,s(y),0())
 d(x,s(y),z) -> cond(ge(x,z),x,y,z)
 cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
 cond(false(),x,y,z) -> 0()
 ge(u,0()) -> true()
 ge(0(),s(v)) -> false()
 ge(s(u),s(v)) -> ge(u,v)
 plus(n,0()) -> n
 plus(n,s(m)) -> s(plus(n,m))
 plus(plus(n,m),u) -> plus(n,plus(m,u))

Proof:
 DP Processor:
  DPs:
   div#(x,s(y)) -> d#(x,s(y),0())
   d#(x,s(y),z) -> ge#(x,z)
   d#(x,s(y),z) -> cond#(ge(x,z),x,y,z)
   cond#(true(),x,y,z) -> plus#(s(y),z)
   cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z))
   ge#(s(u),s(v)) -> ge#(u,v)
   plus#(n,s(m)) -> plus#(n,m)
   plus#(plus(n,m),u) -> plus#(m,u)
   plus#(plus(n,m),u) -> plus#(n,plus(m,u))
  TRS:
   div(x,s(y)) -> d(x,s(y),0())
   d(x,s(y),z) -> cond(ge(x,z),x,y,z)
   cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
   cond(false(),x,y,z) -> 0()
   ge(u,0()) -> true()
   ge(0(),s(v)) -> false()
   ge(s(u),s(v)) -> ge(u,v)
   plus(n,0()) -> n
   plus(n,s(m)) -> s(plus(n,m))
   plus(plus(n,m),u) -> plus(n,plus(m,u))
  EDG Processor:
   DPs:
    div#(x,s(y)) -> d#(x,s(y),0())
    d#(x,s(y),z) -> ge#(x,z)
    d#(x,s(y),z) -> cond#(ge(x,z),x,y,z)
    cond#(true(),x,y,z) -> plus#(s(y),z)
    cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z))
    ge#(s(u),s(v)) -> ge#(u,v)
    plus#(n,s(m)) -> plus#(n,m)
    plus#(plus(n,m),u) -> plus#(m,u)
    plus#(plus(n,m),u) -> plus#(n,plus(m,u))
   TRS:
    div(x,s(y)) -> d(x,s(y),0())
    d(x,s(y),z) -> cond(ge(x,z),x,y,z)
    cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
    cond(false(),x,y,z) -> 0()
    ge(u,0()) -> true()
    ge(0(),s(v)) -> false()
    ge(s(u),s(v)) -> ge(u,v)
    plus(n,0()) -> n
    plus(n,s(m)) -> s(plus(n,m))
    plus(plus(n,m),u) -> plus(n,plus(m,u))
   graph:
    plus#(plus(n,m),u) -> plus#(m,u) -> plus#(n,s(m)) -> plus#(n,m)
    plus#(plus(n,m),u) -> plus#(m,u) ->
    plus#(plus(n,m),u) -> plus#(m,u)
    plus#(plus(n,m),u) -> plus#(m,u) ->
    plus#(plus(n,m),u) -> plus#(n,plus(m,u))
    plus#(plus(n,m),u) -> plus#(n,plus(m,u)) ->
    plus#(n,s(m)) -> plus#(n,m)
    plus#(n,s(m)) -> plus#(n,m) -> plus#(n,s(m)) -> plus#(n,m)
    plus#(n,s(m)) -> plus#(n,m) -> plus#(plus(n,m),u) -> plus#(m,u)
    plus#(n,s(m)) -> plus#(n,m) ->
    plus#(plus(n,m),u) -> plus#(n,plus(m,u))
    cond#(true(),x,y,z) -> plus#(s(y),z) ->
    plus#(n,s(m)) -> plus#(n,m)
    cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z)) ->
    d#(x,s(y),z) -> ge#(x,z)
    cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z)) ->
    d#(x,s(y),z) -> cond#(ge(x,z),x,y,z)
    ge#(s(u),s(v)) -> ge#(u,v) -> ge#(s(u),s(v)) -> ge#(u,v)
    d#(x,s(y),z) -> cond#(ge(x,z),x,y,z) ->
    cond#(true(),x,y,z) -> plus#(s(y),z)
    d#(x,s(y),z) -> cond#(ge(x,z),x,y,z) ->
    cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z))
    d#(x,s(y),z) -> ge#(x,z) -> ge#(s(u),s(v)) -> ge#(u,v)
    div#(x,s(y)) -> d#(x,s(y),0()) -> d#(x,s(y),z) -> ge#(x,z)
    div#(x,s(y)) -> d#(x,s(y),0()) -> d#(x,s(y),z) -> cond#(ge(x,z),x,y,z)
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 16/81
    DPs:
     cond#(true(),x,y,z) -> d#(x,s(y),plus(s(y),z))
     d#(x,s(y),z) -> cond#(ge(x,z),x,y,z)
    TRS:
     div(x,s(y)) -> d(x,s(y),0())
     d(x,s(y),z) -> cond(ge(x,z),x,y,z)
     cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
     cond(false(),x,y,z) -> 0()
     ge(u,0()) -> true()
     ge(0(),s(v)) -> false()
     ge(s(u),s(v)) -> ge(u,v)
     plus(n,0()) -> n
     plus(n,s(m)) -> s(plus(n,m))
     plus(plus(n,m),u) -> plus(n,plus(m,u))
    Open
    
    DPs:
     ge#(s(u),s(v)) -> ge#(u,v)
    TRS:
     div(x,s(y)) -> d(x,s(y),0())
     d(x,s(y),z) -> cond(ge(x,z),x,y,z)
     cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
     cond(false(),x,y,z) -> 0()
     ge(u,0()) -> true()
     ge(0(),s(v)) -> false()
     ge(s(u),s(v)) -> ge(u,v)
     plus(n,0()) -> n
     plus(n,s(m)) -> s(plus(n,m))
     plus(plus(n,m),u) -> plus(n,plus(m,u))
    Open
    
    DPs:
     plus#(plus(n,m),u) -> plus#(m,u)
     plus#(plus(n,m),u) -> plus#(n,plus(m,u))
     plus#(n,s(m)) -> plus#(n,m)
    TRS:
     div(x,s(y)) -> d(x,s(y),0())
     d(x,s(y),z) -> cond(ge(x,z),x,y,z)
     cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
     cond(false(),x,y,z) -> 0()
     ge(u,0()) -> true()
     ge(0(),s(v)) -> false()
     ge(s(u),s(v)) -> ge(u,v)
     plus(n,0()) -> n
     plus(n,s(m)) -> s(plus(n,m))
     plus(plus(n,m),u) -> plus(n,plus(m,u))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [plus#](x0, x1) = x0,
      
      [false] = 1,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [true] = 0,
      
      [cond](x0, x1, x2, x3) = x0,
      
      [ge](x0, x1) = 1,
      
      [d](x0, x1, x2) = x0 + 1,
      
      [0] = 0,
      
      [div](x0, x1) = x0 + 1,
      
      [s](x0) = 0
     orientation:
      plus#(plus(n,m),u) = m + n + 1 >= m = plus#(m,u)
      
      plus#(plus(n,m),u) = m + n + 1 >= n = plus#(n,plus(m,u))
      
      plus#(n,s(m)) = n >= n = plus#(n,m)
      
      div(x,s(y)) = x + 1 >= x + 1 = d(x,s(y),0())
      
      d(x,s(y),z) = x + 1 >= 1 = cond(ge(x,z),x,y,z)
      
      cond(true(),x,y,z) = 0 >= 0 = s(d(x,s(y),plus(s(y),z)))
      
      cond(false(),x,y,z) = 1 >= 0 = 0()
      
      ge(u,0()) = 1 >= 0 = true()
      
      ge(0(),s(v)) = 1 >= 1 = false()
      
      ge(s(u),s(v)) = 1 >= 1 = ge(u,v)
      
      plus(n,0()) = n + 1 >= n = n
      
      plus(n,s(m)) = n + 1 >= 0 = s(plus(n,m))
      
      plus(plus(n,m),u) = m + n + u + 2 >= m + n + u + 2 = plus(n,plus(m,u))
     problem:
      DPs:
       plus#(n,s(m)) -> plus#(n,m)
      TRS:
       div(x,s(y)) -> d(x,s(y),0())
       d(x,s(y),z) -> cond(ge(x,z),x,y,z)
       cond(true(),x,y,z) -> s(d(x,s(y),plus(s(y),z)))
       cond(false(),x,y,z) -> 0()
       ge(u,0()) -> true()
       ge(0(),s(v)) -> false()
       ge(s(u),s(v)) -> ge(u,v)
       plus(n,0()) -> n
       plus(n,s(m)) -> s(plus(n,m))
       plus(plus(n,m),u) -> plus(n,plus(m,u))
     Open