YES
We show the termination of the TRS R:

  app(app(mapt(),f),app(leaf(),x)) -> app(leaf(),app(f,x))
  app(app(mapt(),f),app(node(),xs)) -> app(node(),app(app(maptlist(),f),xs))
  app(app(maptlist(),f),nil()) -> nil()
  app(app(maptlist(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(app(mapt(),f),x)),app(app(maptlist(),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(leaf_0(),y) -> leaf_1(y)
  app(mapt_0(),y) -> mapt_1(y)
  app(mapt_1(x1),y) -> mapt_2(x1,y)
  app(maptlist_0(),y) -> maptlist_1(y)
  app(maptlist_1(x1),y) -> maptlist_2(x1,y)
  app(node_0(),y) -> node_1(y)
  mapt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
  mapt_2(f,node_1(xs)) -> node_1(maptlist_2(f,xs))
  maptlist_2(f,nil_0()) -> nil_0()
  maptlist_2(f,cons_2(x,xs)) -> cons_2(mapt_2(f,x),maptlist_2(f,xs))


-- SCC decomposition.

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

p1: app#(mapt_1(x1),y) -> mapt_2#(x1,y)
p2: app#(maptlist_1(x1),y) -> maptlist_2#(x1,y)
p3: mapt_2#(f,leaf_1(x)) -> app#(f,x)
p4: mapt_2#(f,node_1(xs)) -> maptlist_2#(f,xs)
p5: maptlist_2#(f,cons_2(x,xs)) -> mapt_2#(f,x)
p6: maptlist_2#(f,cons_2(x,xs)) -> maptlist_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(leaf_0(),y) -> leaf_1(y)
r4: app(mapt_0(),y) -> mapt_1(y)
r5: app(mapt_1(x1),y) -> mapt_2(x1,y)
r6: app(maptlist_0(),y) -> maptlist_1(y)
r7: app(maptlist_1(x1),y) -> maptlist_2(x1,y)
r8: app(node_0(),y) -> node_1(y)
r9: mapt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r10: mapt_2(f,node_1(xs)) -> node_1(maptlist_2(f,xs))
r11: maptlist_2(f,nil_0()) -> nil_0()
r12: maptlist_2(f,cons_2(x,xs)) -> cons_2(mapt_2(f,x),maptlist_2(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#(mapt_1(x1),y) -> mapt_2#(x1,y)
p2: mapt_2#(f,node_1(xs)) -> maptlist_2#(f,xs)
p3: maptlist_2#(f,cons_2(x,xs)) -> maptlist_2#(f,xs)
p4: maptlist_2#(f,cons_2(x,xs)) -> mapt_2#(f,x)
p5: mapt_2#(f,leaf_1(x)) -> app#(f,x)
p6: app#(maptlist_1(x1),y) -> maptlist_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(leaf_0(),y) -> leaf_1(y)
r4: app(mapt_0(),y) -> mapt_1(y)
r5: app(mapt_1(x1),y) -> mapt_2(x1,y)
r6: app(maptlist_0(),y) -> maptlist_1(y)
r7: app(maptlist_1(x1),y) -> maptlist_2(x1,y)
r8: app(node_0(),y) -> node_1(y)
r9: mapt_2(f,leaf_1(x)) -> leaf_1(app(f,x))
r10: mapt_2(f,node_1(xs)) -> node_1(maptlist_2(f,xs))
r11: maptlist_2(f,nil_0()) -> nil_0()
r12: maptlist_2(f,cons_2(x,xs)) -> cons_2(mapt_2(f,x),maptlist_2(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!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              app#_A(x1,x2) = x1 + x2
              mapt_1_A(x1) = x1 + 4
              mapt_2#_A(x1,x2) = x1 + x2
              node_1_A(x1) = x1 + 7
              maptlist_2#_A(x1,x2) = x1 + x2
              cons_2_A(x1,x2) = x1 + x2 + 7
              leaf_1_A(x1) = x1 + 8
              maptlist_1_A(x1) = x1 + 4
              app#!_A(x1,x2) = x2 + 8
              mapt_1!_A(x1) = 2
              mapt_2#!_A(x1,x2) = x2 + 1
              node_1!_A(x1) = 6
              maptlist_2#!_A(x1,x2) = x2 + 7
              cons_2!_A(x1,x2) = x1 + 3
              leaf_1!_A(x1) = 4
              maptlist_1!_A(x1) = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 3, x2 - 8}
            mapt_1_A(x1) = max{2, x1 + 1}
            mapt_2#_A(x1,x2) = max{0, x1 - 2, x2 - 24}
            node_1_A(x1) = x1 + 16
            maptlist_2#_A(x1,x2) = max{0, x1 - 2, x2 - 12}
            cons_2_A(x1,x2) = max{11, x1 - 12, x2 + 10}
            leaf_1_A(x1) = max{18, x1 + 16}
            maptlist_1_A(x1) = max{2, x1 + 1}
            app#!_A(x1,x2) = 2
            mapt_1!_A(x1) = max{0, x1 - 15}
            mapt_2#!_A(x1,x2) = max{3, x2 - 17}
            node_1!_A(x1) = 6
            maptlist_2#!_A(x1,x2) = 1
            cons_2!_A(x1,x2) = max{4, x1 - 16, x2 - 13}
            leaf_1!_A(x1) = max{0, x1 - 2}
            maptlist_1!_A(x1) = max{0, x1 - 15}
        
  
    partial status:
  
      pi(app#) = []
      pi(mapt_1) = []
      pi(mapt_2#) = []
      pi(node_1) = [1]
      pi(maptlist_2#) = []
      pi(cons_2) = [1]
      pi(leaf_1) = [1]
      pi(maptlist_1) = [1]

The following rules are removed:

  p1, p2, p3, p4, p5, p6

Then no dependency pair remains.