YES
We show the termination of the TRS R:

  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(treemap(),f),app(app(node(),x),xs)) -> app(app(node(),app(f,x)),app(app(map(),app(treemap(),f)),xs))

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(node_0(),y) -> node_1(y)
  app(node_1(x1),y) -> node_2(x1,y)
  app(treemap_0(),y) -> treemap_1(y)
  app(treemap_1(x1),y) -> treemap_2(x1,y)
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  treemap_2(f,node_2(x,xs)) -> node_2(app(f,x),map_2(treemap_1(f),xs))


-- SCC decomposition.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: treemap_2#(f,node_2(x,xs)) -> app#(f,x)
p6: treemap_2#(f,node_2(x,xs)) -> map_2#(treemap_1(f),xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(map_0(),y) -> map_1(y)
r4: app(map_1(x1),y) -> map_2(x1,y)
r5: app(node_0(),y) -> node_1(y)
r6: app(node_1(x1),y) -> node_2(x1,y)
r7: app(treemap_0(),y) -> treemap_1(y)
r8: app(treemap_1(x1),y) -> treemap_2(x1,y)
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r11: treemap_2(f,node_2(x,xs)) -> node_2(app(f,x),map_2(treemap_1(f),xs))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p3: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p5: treemap_2#(f,node_2(x,xs)) -> map_2#(treemap_1(f),xs)
p6: treemap_2#(f,node_2(x,xs)) -> app#(f,x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(map_0(),y) -> map_1(y)
r4: app(map_1(x1),y) -> map_2(x1,y)
r5: app(node_0(),y) -> node_1(y)
r6: app(node_1(x1),y) -> node_2(x1,y)
r7: app(treemap_0(),y) -> treemap_1(y)
r8: app(treemap_1(x1),y) -> treemap_2(x1,y)
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r11: treemap_2(f,node_2(x,xs)) -> node_2(app(f,x),map_2(treemap_1(f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{0, x2 - 2}
        map_1_A(x1) = max{2, x1 - 5}
        map_2#_A(x1,x2) = max{0, x2 - 2}
        cons_2_A(x1,x2) = max{10, x1 + 2, x2 + 2}
        treemap_1_A(x1) = max{11, x1 + 4}
        treemap_2#_A(x1,x2) = max{0, x2 - 2}
        node_2_A(x1,x2) = max{6, x1 + 2, x2 + 2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app#_A(x1,x2) = x1
            map_1_A(x1) = x1 + 5
            map_2#_A(x1,x2) = x1
            cons_2_A(x1,x2) = 10
            treemap_1_A(x1) = x1 + 5
            treemap_2#_A(x1,x2) = x1 + 5
            node_2_A(x1,x2) = 10
            app#!_A(x1,x2) = x2 + 3
            map_1!_A(x1) = 16
            map_2#!_A(x1,x2) = 2
            cons_2!_A(x1,x2) = x1 + 2
            treemap_1!_A(x1) = 14
            treemap_2#!_A(x1,x2) = x2 + 3
            node_2!_A(x1,x2) = x1 + 14
  
    partial status:
  
      pi(app#) = []
      pi(map_1) = []
      pi(map_2#) = []
      pi(cons_2) = [1, 2]
      pi(treemap_1) = []
      pi(treemap_2#) = []
      pi(node_2) = [1, 2]

The following rules are removed:

  p1, p2, p3, p5, p6



-- SCC decomposition.

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

p1: app#(treemap_1(x1),y) -> treemap_2#(x1,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(map_0(),y) -> map_1(y)
r4: app(map_1(x1),y) -> map_2(x1,y)
r5: app(node_0(),y) -> node_1(y)
r6: app(node_1(x1),y) -> node_2(x1,y)
r7: app(treemap_0(),y) -> treemap_1(y)
r8: app(treemap_1(x1),y) -> treemap_2(x1,y)
r9: map_2(f,nil_0()) -> nil_0()
r10: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r11: treemap_2(f,node_2(x,xs)) -> node_2(app(f,x),map_2(treemap_1(f),xs))

The estimated dependency graph contains the following SCCs:

  (no SCCs)