YES

We show the termination of the TRS 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(app(branch(),app(f,x)),app(app(mapbt(),f),l)),app(app(mapbt(),f),r))

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(app(mapbt(),f),app(leaf(),x)) -> app#(leaf(),app(f,x))
p2: app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
p3: 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))
p4: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(branch(),app(f,x)),app(app(mapbt(),f),l))
p5: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(branch(),app(f,x))
p6: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)
p7: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
p8: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)

and R consists of:

r1: app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
r2: 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))

The estimated dependency graph contains the following SCCs:

  {p2, p6, p7, p8}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(app(mapbt(),f),app(leaf(),x)) -> app#(f,x)
p2: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),r)
p3: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(app(mapbt(),f),l)
p4: app#(app(mapbt(),f),app(app(app(branch(),x),l),r)) -> app#(f,x)

and R consists of:

r1: app(app(mapbt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
r2: 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))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          app#_A(x1,x2) = max{x1 + 4, x2 + 8}
          app_A(x1,x2) = max{x1 + 2, x2 + 5}
          mapbt_A = 2
          leaf_A = 0
          branch_A = 3
    
      precedence:
      
        app# = app = mapbt = leaf = branch
      
      partial status:
    
        pi(app#) = [2]
        pi(app) = [1, 2]
        pi(mapbt) = []
        pi(leaf) = []
        pi(branch) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          app#_A(x1,x2) = max{0, x2 - 9}
          app_A(x1,x2) = max{20, x1 - 3, x2 + 11}
          mapbt_A = 4
          leaf_A = 13
          branch_A = 3
    
      precedence:
      
        app = mapbt > app# = leaf = branch
      
      partial status:
    
        pi(app#) = []
        pi(app) = [2]
        pi(mapbt) = []
        pi(leaf) = []
        pi(branch) = []
    

The next rules are strictly ordered:

  p1, p2, p3, p4

We remove them from the problem.  Then no dependency pair remains.