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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 35, x2 - 37}
            map_1_A(x1) = max{34, x1 + 28}
            map_2#_A(x1,x2) = max{0, x1 - 35, x2 - 41}
            cons_2_A(x1,x2) = max{31, x1 + 30, x2 + 1}
            treemap_1_A(x1) = max{3, x1 + 2}
            treemap_2#_A(x1,x2) = max{0, x1 - 33, x2 - 37}
            node_2_A(x1,x2) = max{x1 + 4, x2 - 4}
            app#!_A(x1,x2) = max{22, x1 - 35, x2 - 5}
            map_1!_A(x1) = max{20, x1 - 36}
            map_2#!_A(x1,x2) = max{21, x1 - 35, x2 - 9}
            cons_2!_A(x1,x2) = max{3, x1 - 6, x2 - 37}
            treemap_1!_A(x1) = max{16, x1 - 29}
            treemap_2#!_A(x1,x2) = max{22, x1 - 33, x2 - 5}
            node_2!_A(x1,x2) = max{x1 + 18, x2 + 17}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 63, x2 - 58}
            map_1_A(x1) = max{62, x1 + 22}
            map_2#_A(x1,x2) = max{0, x1 - 63, x2 - 82}
            cons_2_A(x1,x2) = max{x1 + 24, x2 + 81}
            treemap_1_A(x1) = max{61, x1 + 56}
            treemap_2#_A(x1,x2) = max{0, x1 - 7, x2 - 61}
            node_2_A(x1,x2) = max{x1 + 58, x2 - 19}
            app#!_A(x1,x2) = max{8, x1 - 63, x2 - 51}
            map_1!_A(x1) = max{5, x1 - 14}
            map_2#!_A(x1,x2) = max{6, x1 - 63, x2 - 70}
            cons_2!_A(x1,x2) = max{2, x1 - 106, x2 - 12}
            treemap_1!_A(x1) = max{9, x1 - 6}
            treemap_2#!_A(x1,x2) = max{8, x1 - 7, x2 - 51}
            node_2!_A(x1,x2) = max{3, x1 - 14, x2 - 24}
        
  
    partial status:
  
      pi(app#) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(treemap_1) = [1]
      pi(treemap_2#) = []
      pi(node_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p4: treemap_2#(f,node_2(x,xs)) -> map_2#(treemap_1(f),xs)
p5: 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 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: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p4: treemap_2#(f,node_2(x,xs)) -> app#(f,x)
p5: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{0, x1 - 8, x2 - 42}
            cons_2_A(x1,x2) = max{x1 + 17, x2 + 1}
            app#_A(x1,x2) = max{0, x1 - 8, x2 - 33}
            treemap_1_A(x1) = max{7, x1}
            treemap_2#_A(x1,x2) = max{0, x1 - 8, x2 - 33}
            node_2_A(x1,x2) = max{31, x1 + 13, x2 - 9}
            map_2#!_A(x1,x2) = max{5, x1 - 53, x2 - 42}
            cons_2!_A(x1,x2) = max{0, x1 - 30, x2 - 10}
            app#!_A(x1,x2) = max{5, x1 - 53, x2 - 31}
            treemap_1!_A(x1) = max{4, x1 - 52}
            treemap_2#!_A(x1,x2) = max{5, x1 - 53, x2 - 32}
            node_2!_A(x1,x2) = max{0, x1 - 2, x2 - 42}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{0, x1 - 8, x2 - 32}
            cons_2_A(x1,x2) = max{x1 + 11, x2 + 28}
            app#_A(x1,x2) = max{0, x1 - 8, x2 - 21}
            treemap_1_A(x1) = max{7, x1}
            treemap_2#_A(x1,x2) = max{0, x1 - 8, x2 - 24}
            node_2_A(x1,x2) = max{22, x1 + 20, x2 - 8}
            map_2#!_A(x1,x2) = max{34, x1 - 8, x2 + 3}
            cons_2!_A(x1,x2) = max{30, x1 - 18, x2 - 40}
            app#!_A(x1,x2) = max{34, x1 - 8, x2 + 13}
            treemap_1!_A(x1) = 32
            treemap_2#!_A(x1,x2) = max{34, x1 - 8, x2 + 11}
            node_2!_A(x1,x2) = max{x1 - 14, x2 + 30}
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app#) = [2]
      pi(treemap_1) = []
      pi(treemap_2#) = [2]
      pi(node_2) = [1, 2]

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p4: 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 estimated dependency graph contains the following SCCs:

  {p1}
  {p3, p4}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = 0
            cons_2_A(x1,x2) = max{5, x1 - 4, x2 - 6}
            map_2#!_A(x1,x2) = max{3, x2 - 6}
            cons_2!_A(x1,x2) = max{x1 + 4, x2 - 5}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = 0
            cons_2_A(x1,x2) = max{9, x1 - 4, x2 - 6}
            map_2#!_A(x1,x2) = max{4, x1 - 1, x2 - 10}
            cons_2!_A(x1,x2) = max{3, x1 - 22, x2 - 5}
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: app#(treemap_1(x1),y) -> treemap_2#(x1,y)
p2: 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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 7, x2 - 3}
            treemap_1_A(x1) = x1 + 6
            treemap_2#_A(x1,x2) = max{0, x1 - 1, x2 - 7}
            node_2_A(x1,x2) = max{6, x1 + 4, x2 - 15}
            app#!_A(x1,x2) = max{3, x1 - 7, x2 - 23}
            treemap_1!_A(x1) = max{2, x1 - 31}
            treemap_2#!_A(x1,x2) = max{3, x1 - 7, x2 - 27}
            node_2!_A(x1,x2) = max{1, x1 - 23, x2 - 20}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 31, x2 - 34}
            treemap_1_A(x1) = x1 + 30
            treemap_2#_A(x1,x2) = max{0, x1 - 1, x2 - 34}
            node_2_A(x1,x2) = max{x1 + 33, x2 - 3}
            app#!_A(x1,x2) = max{1, x1 - 53, x2 - 53}
            treemap_1!_A(x1) = max{3, x1 - 51}
            treemap_2#!_A(x1,x2) = max{1, x1 - 52, x2 - 53}
            node_2!_A(x1,x2) = max{2, x1 - 54, x2 - 19}
        
  
    partial status:
  
      pi(app#) = [2]
      pi(treemap_1) = []
      pi(treemap_2#) = [2]
      pi(node_2) = [1, 2]

The following rules are removed:

  p2



-- 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)