YES

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 minus(minus(x,y),z) -> minus(x,plus(y,z))
 app(nil(),k) -> k
 app(l,nil()) -> l
 app(cons(x,l),k) -> cons(x,app(l,k))
 sum(cons(x,nil())) -> cons(x,nil())
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   plus#(s(x),y) -> plus#(x,y)
   minus#(minus(x,y),z) -> plus#(y,z)
   minus#(minus(x,y),z) -> minus#(x,plus(y,z))
   app#(cons(x,l),k) -> app#(l,k)
   sum#(cons(x,cons(y,l))) -> plus#(x,y)
   sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
   sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
  TRS:
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   minus(minus(x,y),z) -> minus(x,plus(y,z))
   app(nil(),k) -> k
   app(l,nil()) -> l
   app(cons(x,l),k) -> cons(x,app(l,k))
   sum(cons(x,nil())) -> cons(x,nil())
   sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
   sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
  EDG Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    plus#(s(x),y) -> plus#(x,y)
    minus#(minus(x,y),z) -> plus#(y,z)
    minus#(minus(x,y),z) -> minus#(x,plus(y,z))
    app#(cons(x,l),k) -> app#(l,k)
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
   TRS:
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    minus(minus(x,y),z) -> minus(x,plus(y,z))
    app(nil(),k) -> k
    app(l,nil()) -> l
    app(cons(x,l),k) -> cons(x,app(l,k))
    sum(cons(x,nil())) -> cons(x,nil())
    sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
    sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
   graph:
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),y) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k)))) ->
    app#(cons(x,l),k) -> app#(l,k)
    app#(cons(x,l),k) -> app#(l,k) -> app#(cons(x,l),k) -> app#(l,k)
    plus#(s(x),y) -> plus#(x,y) ->
    plus#(s(x),y) -> plus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(minus(x,y),z) -> plus#(y,z)
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(minus(x,y),z) -> minus#(x,plus(y,z))
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(minus(x,y),z) -> plus#(y,z)
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(minus(x,y),z) -> minus#(x,plus(y,z))
    minus#(minus(x,y),z) -> plus#(y,z) ->
    plus#(s(x),y) -> plus#(x,y)
    minus#(minus(x,y),z) -> minus#(x,plus(y,z)) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(minus(x,y),z) -> minus#(x,plus(y,z)) ->
    minus#(minus(x,y),z) -> plus#(y,z)
    minus#(minus(x,y),z) -> minus#(x,plus(y,z)) -> minus#(minus(x,y),z) -> minus#(x,plus(y,z))
   SCC Processor:
    #sccs: 6
    #rules: 7
    #arcs: 25/144
    DPs:
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [quot#](x0, x1) = x0,
      
      [sum](x0) = 1,
      
      [cons](x0, x1) = 1,
      
      [app](x0, x1) = x0 + x1 + 1,
      
      [nil] = 0,
      
      [plus](x0, x1) = x0 + x1,
      
      [quot](x0, x1) = x0 + x1 + 1,
      
      [s](x0) = x0 + 1,
      
      [minus](x0, x1) = x0,
      
      [0] = 0
     orientation:
      quot#(s(x),s(y)) = x + 1 >= x = quot#(minus(x,y),s(y))
      
      minus(x,0()) = x >= x = x
      
      minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
      
      quot(0(),s(y)) = y + 2 >= 0 = 0()
      
      quot(s(x),s(y)) = x + y + 3 >= x + y + 3 = s(quot(minus(x,y),s(y)))
      
      plus(0(),y) = y >= y = y
      
      plus(s(x),y) = x + y + 1 >= x + y + 1 = s(plus(x,y))
      
      minus(minus(x,y),z) = x >= x = minus(x,plus(y,z))
      
      app(nil(),k) = k + 1 >= k = k
      
      app(l,nil()) = l + 1 >= l = l
      
      app(cons(x,l),k) = k + 2 >= 1 = cons(x,app(l,k))
      
      sum(cons(x,nil())) = 1 >= 1 = cons(x,nil())
      
      sum(cons(x,cons(y,l))) = 1 >= 1 = sum(cons(plus(x,y),l))
      
      sum(app(l,cons(x,cons(y,k)))) = 1 >= 1 = sum(app(l,sum(cons(x,cons(y,k)))))
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed
    
    DPs:
     minus#(minus(x,y),z) -> minus#(x,plus(y,z))
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed
    
    DPs:
     sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [sum#](x0) = x0,
      
      [sum](x0) = 1,
      
      [cons](x0, x1) = x1 + 1,
      
      [app](x0, x1) = x0 + x1 + 1,
      
      [nil] = 0,
      
      [plus](x0, x1) = x1,
      
      [quot](x0, x1) = 0,
      
      [s](x0) = x0,
      
      [minus](x0, x1) = x0,
      
      [0] = 0
     orientation:
      sum#(app(l,cons(x,cons(y,k)))) = k + l + 3 >= l + 2 = sum#(app(l,sum(cons(x,cons(y,k)))))
      
      minus(x,0()) = x >= x = x
      
      minus(s(x),s(y)) = x >= x = minus(x,y)
      
      quot(0(),s(y)) = 0 >= 0 = 0()
      
      quot(s(x),s(y)) = 0 >= 0 = s(quot(minus(x,y),s(y)))
      
      plus(0(),y) = y >= y = y
      
      plus(s(x),y) = y >= y = s(plus(x,y))
      
      minus(minus(x,y),z) = x >= x = minus(x,plus(y,z))
      
      app(nil(),k) = k + 1 >= k = k
      
      app(l,nil()) = l + 1 >= l = l
      
      app(cons(x,l),k) = k + l + 2 >= k + l + 2 = cons(x,app(l,k))
      
      sum(cons(x,nil())) = 1 >= 1 = cons(x,nil())
      
      sum(cons(x,cons(y,l))) = 1 >= 1 = sum(cons(plus(x,y),l))
      
      sum(app(l,cons(x,cons(y,k)))) = 1 >= 1 = sum(app(l,sum(cons(x,cons(y,k)))))
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed
    
    DPs:
     app#(cons(x,l),k) -> app#(l,k)
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed
    
    DPs:
     sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [sum#](x0) = x0,
      
      [sum](x0) = 1,
      
      [cons](x0, x1) = x1 + 1,
      
      [app](x0, x1) = x0 + x1,
      
      [nil] = 0,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [quot](x0, x1) = x0,
      
      [s](x0) = x0 + 1,
      
      [minus](x0, x1) = x0,
      
      [0] = 0
     orientation:
      sum#(cons(x,cons(y,l))) = l + 2 >= l + 1 = sum#(cons(plus(x,y),l))
      
      minus(x,0()) = x >= x = x
      
      minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
      
      quot(0(),s(y)) = 0 >= 0 = 0()
      
      quot(s(x),s(y)) = x + 1 >= x + 1 = s(quot(minus(x,y),s(y)))
      
      plus(0(),y) = y + 1 >= y = y
      
      plus(s(x),y) = x + y + 2 >= x + y + 2 = s(plus(x,y))
      
      minus(minus(x,y),z) = x >= x = minus(x,plus(y,z))
      
      app(nil(),k) = k >= k = k
      
      app(l,nil()) = l >= l = l
      
      app(cons(x,l),k) = k + l + 1 >= k + l + 1 = cons(x,app(l,k))
      
      sum(cons(x,nil())) = 1 >= 1 = cons(x,nil())
      
      sum(cons(x,cons(y,l))) = 1 >= 1 = sum(cons(plus(x,y),l))
      
      sum(app(l,cons(x,cons(y,k)))) = 1 >= 1 = sum(app(l,sum(cons(x,cons(y,k)))))
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     minus(minus(x,y),z) -> minus(x,plus(y,z))
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       minus(minus(x,y),z) -> minus(x,plus(y,z))
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Qed