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:

  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) = []

The next rules are strictly ordered:

  p1

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)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- 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#(f,x)
p3: 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 reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        app#_A(x1,x2) = max{0, x2 - 14}
        app_A(x1,x2) = max{x1 + 5, x2 + 8}
        mapbt_A = 7
        branch_A = 8
  
    precedence:
    
      app# = app = mapbt = branch
    
    partial status:
  
      pi(app#) = []
      pi(app) = [2]
      pi(mapbt) = []
      pi(branch) = []

The next rules are strictly ordered:

  p3

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#(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 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#(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:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        app#_A(x1,x2) = x1 + 11
        app_A(x1,x2) = max{9, x1 - 2, x2 + 1}
        mapbt_A = 11
        branch_A = 18
  
    precedence:
    
      app# = app = mapbt = branch
    
    partial status:
  
      pi(app#) = []
      pi(app) = [2]
      pi(mapbt) = []
      pi(branch) = []

The next rules are strictly ordered:

  p2

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)

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}


-- 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)

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:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        app#_A(x1,x2) = max{x1 + 6, x2 + 1}
        app_A(x1,x2) = max{x1 + 2, x2 + 2}
        mapbt_A = 0
        branch_A = 1
  
    precedence:
    
      app# = app = mapbt = branch
    
    partial status:
  
      pi(app#) = [2]
      pi(app) = [1, 2]
      pi(mapbt) = []
      pi(branch) = []

The next rules are strictly ordered:

  p1

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