MAYBE

Problem:
 app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
 app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
 app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))

Proof:
 DP Processor:
  DPs:
   app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
   app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
   app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
   app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
   app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
  TRS:
   app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
   app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
   app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
  CDG Processor:
   DPs:
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
   TRS:
    app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
    app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
   graph:
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x) ->
    app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
    app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
   Restore Modifier:
    DPs:
     app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
     app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
     app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
     app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
     app#(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
    TRS:
     app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
     app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
     app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 32/64
     DPs:
      app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
      app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
      app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
      app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
     TRS:
      app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
      app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
      app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [app#](x0, x1) = x0,
       
       [branch] = 0,
       
       [leaf] = 0,
       
       [app](x0, x1) = x1 + 1,
       
       [mapbt] = 0
      orientation:
       app#(app(mapbt(),f),app(leaf(),x)) = f + 1 >= f = app#(f,x)
       
       app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) = f + 1 >= f = app#(f,x)
       
       app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) = f + 1 >= f + 1 = app#(app(mapbt(),f),l)
       
       app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) = f + 1 >= f + 1 = app#(app(mapbt(),f),r)
       
       app(app(mapbt(),f),app(leaf(),x)) = x + 2 >= x + 2 = app(leaf(),app(f,x))
       
       app(app(mapbt(),f),app(app(app(branch(),x),l),r)) = r + 2 >= r + 2 = app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
      problem:
       DPs:
        app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
        app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
       TRS:
        app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
        app(app(mapbt(),f),app(app(app(branch(),x),l),r)) ->
        app(app(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))
      Open