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:
 DP Processor:
  DPs:
   app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
   app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
   app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   app#(app(times(),app(s(),x)),y) -> app#(times(),x)
   app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
   app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
   app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
   inc#() -> app#(s(),0())
   inc#() -> app#(plus(),app(s(),0()))
   inc#() -> app#(map(),app(plus(),app(s(),0())))
   double#() -> app#(s(),0())
   double#() -> app#(s(),app(s(),0()))
   double#() -> app#(times(),app(s(),app(s(),0())))
   double#() -> app#(map(),app(times(),app(s(),app(s(),0()))))
  TRS:
   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()))))
  EDG Processor:
   DPs:
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    inc#() -> app#(s(),0())
    inc#() -> app#(plus(),app(s(),0()))
    inc#() -> app#(map(),app(plus(),app(s(),0())))
    double#() -> app#(s(),0())
    double#() -> app#(s(),app(s(),0()))
    double#() -> app#(times(),app(s(),app(s(),0())))
    double#() -> app#(map(),app(times(),app(s(),app(s(),0()))))
   TRS:
    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()))))
   graph:
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 25/324
    DPs:
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    TRS:
     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()))))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
      TRS:
       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()))))
     Subterm Criterion Processor:
      simple projection:
       pi(app#) = 1
      problem:
       DPs:
        
       TRS:
        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()))))
      Qed
    
    DPs:
     app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    TRS:
     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()))))
    Usable Rule Processor:
     DPs:
      app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
     TRS:
      
     Bounds Processor:
      bound: 1
      enrichment: match
      automaton:
       final states: {7}
       transitions:
        s0() -> 6*
        app{#,1}(11,6) -> 7*
        app{#,1}(11,5) -> 7*
        app1(10,5) -> 11*
        app1(10,6) -> 11*
        times1() -> 10*
        app{#,0}(5,5) -> 7*
        app{#,0}(6,6) -> 7*
        app{#,0}(5,6) -> 7*
        app{#,0}(6,5) -> 7*
        app0(5,5) -> 5*
        app0(6,6) -> 5*
        app0(5,6) -> 5*
        app0(6,5) -> 5*
        times0() -> 5*
      problem:
       DPs:
        
       TRS:
        
      Qed
    
    DPs:
     app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    TRS:
     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()))))
    Usable Rule Processor:
     DPs:
      app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
     TRS:
      
     Bounds Processor:
      bound: 1
      enrichment: match
      automaton:
       final states: {7}
       transitions:
        s0() -> 6*
        app{#,1}(11,6) -> 7*
        app{#,1}(11,5) -> 7*
        app1(10,5) -> 11*
        app1(10,6) -> 11*
        plus0() -> 5*
        plus1() -> 10*
        app{#,0}(5,5) -> 7*
        app{#,0}(6,6) -> 7*
        app{#,0}(5,6) -> 7*
        app{#,0}(6,5) -> 7*
        app0(5,5) -> 5*
        app0(6,6) -> 5*
        app0(5,6) -> 5*
        app0(6,5) -> 5*
      problem:
       DPs:
        
       TRS:
        
      Qed