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))

Instead we prove termination of the uncurried system:

  app(branch_0(),y) -> branch_1(y)
  app(branch_1(x1),y) -> branch_2(x1,y)
  app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
  app(leaf_0(),y) -> leaf_1(y)
  app(mapbt_0(),y) -> mapbt_1(y)
  app(mapbt_1(x1),y) -> mapbt_2(x1,y)
  mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
  mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))


-- SCC decomposition.

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

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,leaf_1(x)) -> app#(f,x)
p3: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)
p4: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p5: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)
p3: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p4: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)
p5: mapbt_2#(f,leaf_1(x)) -> app#(f,x)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The set of usable rules consists of

  r1, r2, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{6, x1 - 7}
        mapbt_1_A(x1) = x1 + 15
        mapbt_2#_A(x1,x2) = x1 + 7
        branch_3_A(x1,x2,x3) = max{4, x1 - 18, x2 + 1, x3 - 18}
        leaf_1_A(x1) = max{2, x1 - 18}
        app_A(x1,x2) = max{17, x1 - 16, x2 - 7}
        branch_0_A = 1
        branch_1_A(x1) = max{2, x1 - 8}
        branch_2_A(x1,x2) = max{3, x1 - 25, x2 - 8}
        leaf_0_A = 1
  
    base order 2:
  
      precedence:
      
        mapbt_1 = mapbt_2# > app# = app = leaf_0 > branch_3 = leaf_1 > branch_2 > branch_1 > branch_0
      
  
    partial status:
  
      pi(app#) = []
      pi(mapbt_1) = [1]
      pi(mapbt_2#) = [1]
      pi(branch_3) = [2]
      pi(leaf_1) = []
      pi(app) = []
      pi(branch_0) = []
      pi(branch_1) = []
      pi(branch_2) = []
      pi(leaf_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)
p3: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p4: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,branch_3(x,l,r)) -> app#(f,x)
p3: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p4: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{1, x1 - 2}
        mapbt_1_A(x1) = x1 + 4
        mapbt_2#_A(x1,x2) = max{2, x1 - 1}
        branch_3_A(x1,x2,x3) = max{0, x1 - 23, x2 - 23, x3 - 23}
        app_A(x1,x2) = max{5, x1 - 5, x2 - 10}
        branch_0_A = 1
        branch_1_A(x1) = max{2, x1 - 11}
        branch_2_A(x1,x2) = max{4, x1 - 17, x2 - 17}
  
    base order 2:
  
      precedence:
      
        app# = mapbt_1 > mapbt_2# = app > branch_3 > branch_2 > branch_1 > branch_0
      
  
    partial status:
  
      pi(app#) = []
      pi(mapbt_1) = [1]
      pi(mapbt_2#) = []
      pi(branch_3) = []
      pi(app) = []
      pi(branch_0) = []
      pi(branch_1) = []
      pi(branch_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(mapbt_1(x1),y) -> mapbt_2#(x1,y)
p2: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p3: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The estimated dependency graph contains the following SCCs:

  {p2, p3}


-- Reduction pair.

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

p1: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)
p2: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,r)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The set of usable rules consists of

  r1, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mapbt_2#_A(x1,x2) = x2
        branch_3_A(x1,x2,x3) = max{x1, x2, x3}
        app_A(x1,x2) = max{0, x1 - 3, x2 - 1}
        branch_0_A = 2
        branch_1_A(x1) = max{0, x1 - 2}
        leaf_0_A = 2
        leaf_1_A(x1) = max{0, x1 - 2}
  
    base order 2:
  
      precedence:
      
        branch_3 = app = branch_0 > mapbt_2# = leaf_1 > branch_1 = leaf_0
      
  
    partial status:
  
      pi(mapbt_2#) = [2]
      pi(branch_3) = [2, 3]
      pi(app) = []
      pi(branch_0) = []
      pi(branch_1) = []
      pi(leaf_0) = []
      pi(leaf_1) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(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: mapbt_2#(f,branch_3(x,l,r)) -> mapbt_2#(f,l)

and R consists of:

r1: app(branch_0(),y) -> branch_1(y)
r2: app(branch_1(x1),y) -> branch_2(x1,y)
r3: app(branch_2(x1,x2),y) -> branch_3(x1,x2,y)
r4: app(leaf_0(),y) -> leaf_1(y)
r5: app(mapbt_0(),y) -> mapbt_1(y)
r6: app(mapbt_1(x1),y) -> mapbt_2(x1,y)
r7: mapbt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r8: mapbt_2(f,branch_3(x,l,r)) -> branch_3(app(f,x),mapbt_2(f,l),mapbt_2(f,r))

The set of usable rules consists of

  r5

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        mapbt_2#_A(x1,x2) = x2 + 9
        branch_3_A(x1,x2,x3) = max{x1 - 3, x2 + 10, x3 - 7}
        app_A(x1,x2) = max{2, x1 - 12, x2 - 19}
        mapbt_0_A = 26
        mapbt_1_A(x1) = max{13, x1 - 20}
  
    base order 2:
  
      precedence:
      
        branch_3 = app > mapbt_2# = mapbt_1 > mapbt_0
      
  
    partial status:
  
      pi(mapbt_2#) = [2]
      pi(branch_3) = [2]
      pi(app) = []
      pi(mapbt_0) = []
      pi(mapbt_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.