MAYBE

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(app(comp(),f),g),x) -> app(f,app(g,x))
 app(twice(),f) -> app(app(comp(),f),f)

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(app(comp(),f),g),x) -> app#(g,x)
   app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
   app#(twice(),f) -> app#(comp(),f)
   app#(twice(),f) -> app#(app(comp(),f),f)
  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(app(comp(),f),g),x) -> app(f,app(g,x))
   app(twice(),f) -> app(app(comp(),f),f)
  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(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(twice(),f) -> app#(comp(),f)
    app#(twice(),f) -> app#(app(comp(),f),f)
   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(app(comp(),f),g),x) -> app(f,app(g,x))
    app(twice(),f) -> app(app(comp(),f),f)
   graph:
    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(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(twice(),f) -> app#(comp(),f)
    app#(app(app(comp(),f),g),x) -> app#(g,x) ->
    app#(twice(),f) -> app#(app(comp(),f),f)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(app(comp(),f),g),x) -> app#(g,x)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x))
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(twice(),f) -> app#(comp(),f)
    app#(app(app(comp(),f),g),x) -> app#(f,app(g,x)) ->
    app#(twice(),f) -> app#(app(comp(),f),f)
    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: 32/121
    DPs:
     app#(app(app(comp(),f),g),x) -> app#(g,x)
     app#(app(app(comp(),f),g),x) -> app#(f,app(g,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(app(comp(),f),g),x) -> app(f,app(g,x))
     app(twice(),f) -> app(app(comp(),f),f)
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     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(app(comp(),f),g),x) -> app(f,app(g,x))
       app(twice(),f) -> app(app(comp(),f),f)
     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(app(comp(),f),g),x) -> app(f,app(g,x))
     app(twice(),f) -> app(app(comp(),f),f)
    Open
    
    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(app(comp(),f),g),x) -> app(f,app(g,x))
     app(twice(),f) -> app(app(comp(),f),f)
    Open