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. lexicographic path order with precedence:
    
      precedence:
      
        app# > mapbt > app > branch > leaf
      
      argument filter:
    
        pi(app#) = 1
        pi(app) = [1, 2]
        pi(mapbt) = []
        pi(leaf) = []
        pi(branch) = []
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        mapbt > app > app# > branch > leaf
      
      argument filter:
    
        pi(app#) = []
        pi(app) = 1
        pi(mapbt) = []
        pi(leaf) = []
        pi(branch) = []
    

The next rules are strictly ordered:

  p1, p4

We remove them from the problem.

-- SCC decomposition.

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

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

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:

  {p1, p2}


-- Reduction pair.

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

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

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 monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. lexicographic path order with precedence:
    
      precedence:
      
        mapbt > app > app# > branch
      
      argument filter:
    
        pi(app#) = [1, 2]
        pi(app) = [1, 2]
        pi(mapbt) = []
        pi(branch) = []
    
    2. lexicographic path order with precedence:
    
      precedence:
      
        branch > mapbt > app > app#
      
      argument filter:
    
        pi(app#) = [1, 2]
        pi(app) = [1, 2]
        pi(mapbt) = []
        pi(branch) = []
    

The next rules are strictly ordered:

  p1, p2
  r1, r2

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