MAYBE

Problem:
 app(app(app(if(),true()),x),y) -> x
 app(app(app(if(),false()),x),y) -> y
 app(app(app(until(),p),f),x) -> app(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))

Proof:
 DP Processor:
  DPs:
   app#(app(app(until(),p),f),x) -> app#(f,x)
   app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
   app#(app(app(until(),p),f),x) -> app#(p,x)
   app#(app(app(until(),p),f),x) -> app#(if(),app(p,x))
   app#(app(app(until(),p),f),x) -> app#(app(if(),app(p,x)),x)
   app#(app(app(until(),p),f),x) ->
   app#(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
  TRS:
   app(app(app(if(),true()),x),y) -> x
   app(app(app(if(),false()),x),y) -> y
   app(app(app(until(),p),f),x) ->
   app(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
  EDG Processor:
   DPs:
    app#(app(app(until(),p),f),x) -> app#(f,x)
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
    app#(app(app(until(),p),f),x) -> app#(p,x)
    app#(app(app(until(),p),f),x) -> app#(if(),app(p,x))
    app#(app(app(until(),p),f),x) -> app#(app(if(),app(p,x)),x)
    app#(app(app(until(),p),f),x) ->
    app#(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
   TRS:
    app(app(app(if(),true()),x),y) -> x
    app(app(app(if(),false()),x),y) -> y
    app(app(app(until(),p),f),x) ->
    app(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
   graph:
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) -> app#(f,x)
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) -> app#(p,x)
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) -> app#(if(),app(p,x))
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) -> app#(app(if(),app(p,x)),x)
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x)) ->
    app#(app(app(until(),p),f),x) ->
    app#(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) -> app#(f,x)
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) -> app#(p,x)
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) -> app#(if(),app(p,x))
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) -> app#(app(if(),app(p,x)),x)
    app#(app(app(until(),p),f),x) -> app#(p,x) ->
    app#(app(app(until(),p),f),x) ->
    app#(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) -> app#(f,x)
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) -> app#(p,x)
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) -> app#(if(),app(p,x))
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) -> app#(app(if(),app(p,x)),x)
    app#(app(app(until(),p),f),x) -> app#(f,x) ->
    app#(app(app(until(),p),f),x) ->
    app#(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 18/36
    DPs:
     app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
     app#(app(app(until(),p),f),x) -> app#(p,x)
     app#(app(app(until(),p),f),x) -> app#(f,x)
    TRS:
     app(app(app(if(),true()),x),y) -> x
     app(app(app(if(),false()),x),y) -> y
     app(app(app(until(),p),f),x) ->
     app(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       app#(app(app(until(),p),f),x) -> app#(app(app(until(),p),f),app(f,x))
      TRS:
       app(app(app(if(),true()),x),y) -> x
       app(app(app(if(),false()),x),y) -> y
       app(app(app(until(),p),f),x) ->
       app(app(app(if(),app(p,x)),x),app(app(app(until(),p),f),app(f,x)))
     Open