MAYBE

Problem:
 minus(0(),y) -> 0()
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 plus(0(),y) -> y
 plus(s(x),y) -> plus(x,s(y))
 zero(s(x)) -> false()
 zero(0()) -> true()
 p(s(x)) -> x
 div(x,y) -> quot(x,y,0())
 quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
 if(true(),x,y,z) -> p(z)
 if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   plus#(s(x),y) -> plus#(x,s(y))
   div#(x,y) -> quot#(x,y,0())
   quot#(x,y,z) -> plus#(z,s(0()))
   quot#(x,y,z) -> zero#(x)
   quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0())))
   if#(true(),x,y,z) -> p#(z)
   if#(false(),x,s(y),z) -> minus#(x,s(y))
   if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z)
  TRS:
   minus(0(),y) -> 0()
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   plus(0(),y) -> y
   plus(s(x),y) -> plus(x,s(y))
   zero(s(x)) -> false()
   zero(0()) -> true()
   p(s(x)) -> x
   div(x,y) -> quot(x,y,0())
   quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
   if(true(),x,y,z) -> p(z)
   if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
  TDG Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    plus#(s(x),y) -> plus#(x,s(y))
    div#(x,y) -> quot#(x,y,0())
    quot#(x,y,z) -> plus#(z,s(0()))
    quot#(x,y,z) -> zero#(x)
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0())))
    if#(true(),x,y,z) -> p#(z)
    if#(false(),x,s(y),z) -> minus#(x,s(y))
    if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z)
   TRS:
    minus(0(),y) -> 0()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    plus(0(),y) -> y
    plus(s(x),y) -> plus(x,s(y))
    zero(s(x)) -> false()
    zero(0()) -> true()
    p(s(x)) -> x
    div(x,y) -> quot(x,y,0())
    quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
    if(true(),x,y,z) -> p(z)
    if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
   graph:
    if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z) ->
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0())))
    if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z) ->
    quot#(x,y,z) -> zero#(x)
    if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z) ->
    quot#(x,y,z) -> plus#(z,s(0()))
    if#(false(),x,s(y),z) -> minus#(x,s(y)) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0()))) ->
    if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z)
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0()))) ->
    if#(false(),x,s(y),z) -> minus#(x,s(y))
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0()))) ->
    if#(true(),x,y,z) -> p#(z)
    quot#(x,y,z) -> plus#(z,s(0())) -> plus#(s(x),y) -> plus#(x,s(y))
    div#(x,y) -> quot#(x,y,0()) ->
    quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0())))
    div#(x,y) -> quot#(x,y,0()) -> quot#(x,y,z) -> zero#(x)
    div#(x,y) -> quot#(x,y,0()) -> quot#(x,y,z) -> plus#(z,s(0()))
    plus#(s(x),y) -> plus#(x,s(y)) ->
    plus#(s(x),y) -> plus#(x,s(y))
    minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 13/81
    DPs:
     if#(false(),x,s(y),z) -> quot#(minus(x,s(y)),s(y),z)
     quot#(x,y,z) -> if#(zero(x),x,y,plus(z,s(0())))
    TRS:
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> plus(x,s(y))
     zero(s(x)) -> false()
     zero(0()) -> true()
     p(s(x)) -> x
     div(x,y) -> quot(x,y,0())
     quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
     if(true(),x,y,z) -> p(z)
     if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
    Open
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> plus(x,s(y))
     zero(s(x)) -> false()
     zero(0()) -> true()
     p(s(x)) -> x
     div(x,y) -> quot(x,y,0())
     quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
     if(true(),x,y,z) -> p(z)
     if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       plus(0(),y) -> y
       plus(s(x),y) -> plus(x,s(y))
       zero(s(x)) -> false()
       zero(0()) -> true()
       p(s(x)) -> x
       div(x,y) -> quot(x,y,0())
       quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
       if(true(),x,y,z) -> p(z)
       if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,s(y))
    TRS:
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> plus(x,s(y))
     zero(s(x)) -> false()
     zero(0()) -> true()
     p(s(x)) -> x
     div(x,y) -> quot(x,y,0())
     quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
     if(true(),x,y,z) -> p(z)
     if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       plus(0(),y) -> y
       plus(s(x),y) -> plus(x,s(y))
       zero(s(x)) -> false()
       zero(0()) -> true()
       p(s(x)) -> x
       div(x,y) -> quot(x,y,0())
       quot(x,y,z) -> if(zero(x),x,y,plus(z,s(0())))
       if(true(),x,y,z) -> p(z)
       if(false(),x,s(y),z) -> quot(minus(x,s(y)),s(y),z)
     Qed