YES

Problem:
 app(app(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(times(),0()),y) -> 0()
 app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 inc() -> app(map(),app(plus(),app(s(),0())))
 double() -> app(map(),app(times(),app(s(),app(s(),0()))))

Proof:
 Uncurry Processor:
  plus2(0(),y) -> y
  plus2(s1(x),y) -> s1(plus2(x,y))
  times2(0(),y) -> 0()
  times2(s1(x),y) -> plus2(times2(x,y),y)
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  inc() -> map1(plus1(s1(0())))
  double() -> map1(times1(s1(s1(0()))))
  app(plus1(x4),x5) -> plus2(x4,x5)
  app(plus(),x5) -> plus1(x5)
  app(s(),x5) -> s1(x5)
  app(times1(x4),x5) -> times2(x4,x5)
  app(times(),x5) -> times1(x5)
  app(map1(x4),x5) -> map2(x4,x5)
  app(map(),x5) -> map1(x5)
  app(cons1(x4),x5) -> cons2(x4,x5)
  app(cons(),x5) -> cons1(x5)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    times{2,#}(s1(x),y) -> times{2,#}(x,y)
    times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(times1(x4),x5) -> times{2,#}(x4,x5)
    app#(map1(x4),x5) -> map{2,#}(x4,x5)
   TRS:
    plus2(0(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    times2(0(),y) -> 0()
    times2(s1(x),y) -> plus2(times2(x,y),y)
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    inc() -> map1(plus1(s1(0())))
    double() -> map1(times1(s1(s1(0()))))
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus(),x5) -> plus1(x5)
    app(s(),x5) -> s1(x5)
    app(times1(x4),x5) -> times2(x4,x5)
    app(times(),x5) -> times1(x5)
    app(map1(x4),x5) -> map2(x4,x5)
    app(map(),x5) -> map1(x5)
    app(cons1(x4),x5) -> cons2(x4,x5)
    app(cons(),x5) -> cons1(x5)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(times1(x4),x5) -> times{2,#}(x4,x5)
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
    TRS:
     plus2(0(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     times2(0(),y) -> 0()
     times2(s1(x),y) -> plus2(times2(x,y),y)
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     inc() -> map1(plus1(s1(0())))
     double() -> map1(times1(s1(s1(0()))))
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus(),x5) -> plus1(x5)
     app(s(),x5) -> s1(x5)
     app(times1(x4),x5) -> times2(x4,x5)
     app(times(),x5) -> times1(x5)
     app(map1(x4),x5) -> map2(x4,x5)
     app(map(),x5) -> map1(x5)
     app(cons1(x4),x5) -> cons2(x4,x5)
     app(cons(),x5) -> cons1(x5)
    graph:
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(times1(x4),x5) -> times{2,#}(x4,x5) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     app#(times1(x4),x5) -> times{2,#}(x4,x5) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x4),x5) -> times{2,#}(x4,x5)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) -> plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 5
     #arcs: 14/64
     DPs:
      app#(map1(x4),x5) -> map{2,#}(x4,x5)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     TRS:
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(0(),y) -> 0()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc() -> map1(plus1(s1(0())))
      double() -> map1(times1(s1(s1(0()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(s(),x5) -> s1(x5)
      app(times1(x4),x5) -> times2(x4,x5)
      app(times(),x5) -> times1(x5)
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
      problem:
       DPs:
        app#(map1(x4),x5) -> map{2,#}(x4,x5)
       TRS:
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(0(),y) -> 0()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc() -> map1(plus1(s1(0())))
        double() -> map1(times1(s1(s1(0()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(s(),x5) -> s1(x5)
        app(times1(x4),x5) -> times2(x4,x5)
        app(times(),x5) -> times1(x5)
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 5/1
       
     
     DPs:
      times{2,#}(s1(x),y) -> times{2,#}(x,y)
     TRS:
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(0(),y) -> 0()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc() -> map1(plus1(s1(0())))
      double() -> map1(times1(s1(s1(0()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(s(),x5) -> s1(x5)
      app(times1(x4),x5) -> times2(x4,x5)
      app(times(),x5) -> times1(x5)
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(times{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(0(),y) -> 0()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc() -> map1(plus1(s1(0())))
        double() -> map1(times1(s1(s1(0()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(s(),x5) -> s1(x5)
        app(times1(x4),x5) -> times2(x4,x5)
        app(times(),x5) -> times1(x5)
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
      Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(0(),y) -> 0()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc() -> map1(plus1(s1(0())))
      double() -> map1(times1(s1(s1(0()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(s(),x5) -> s1(x5)
      app(times1(x4),x5) -> times2(x4,x5)
      app(times(),x5) -> times1(x5)
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(0(),y) -> 0()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc() -> map1(plus1(s1(0())))
        double() -> map1(times1(s1(s1(0()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(s(),x5) -> s1(x5)
        app(times1(x4),x5) -> times2(x4,x5)
        app(times(),x5) -> times1(x5)
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
      Qed