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(le(),|0|()),y) -> true()
  app(app(le(),app(s(),x)),|0|()) -> false()
  app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
  app(app(maxlist(),x),app(app(cons(),y),ys)) -> app(app(if(),app(app(le(),x),y)),app(app(maxlist(),y),ys))
  app(app(maxlist(),x),nil()) -> x
  app(height(),app(app(node(),x),xs)) -> app(s(),app(app(maxlist(),|0|()),app(app(map(),height()),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(height_0(),y) -> height_1(y)
  app(if_0(),y) -> if_1(y)
  app(if_1(x1),y) -> if_2(x1,y)
  app(le_0(),y) -> le_1(y)
  app(le_1(x1),y) -> le_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(maxlist_0(),y) -> maxlist_1(y)
  app(maxlist_1(x1),y) -> maxlist_2(x1,y)
  app(node_0(),y) -> node_1(y)
  app(node_1(x1),y) -> node_2(x1,y)
  app(s_0(),y) -> s_1(y)
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  le_2(|0|_0(),y) -> true_0()
  le_2(s_1(x),|0|_0()) -> false_0()
  le_2(s_1(x),s_1(y)) -> le_2(x,y)
  maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
  maxlist_2(x,nil_0()) -> x
  height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),xs)))


-- SCC decomposition.

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

p1: app#(height_0(),y) -> height_1#(y)
p2: app#(le_1(x1),y) -> le_2#(x1,y)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: app#(maxlist_1(x1),y) -> maxlist_2#(x1,y)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p7: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p8: maxlist_2#(x,cons_2(y,ys)) -> le_2#(x,y)
p9: maxlist_2#(x,cons_2(y,ys)) -> maxlist_2#(y,ys)
p10: height_1#(node_2(x,xs)) -> maxlist_2#(|0|_0(),map_2(height_0(),xs))
p11: height_1#(node_2(x,xs)) -> map_2#(height_0(),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(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),xs)))

The estimated dependency graph contains the following SCCs:

  {p1, p3, p5, p6, p11}
  {p9}
  {p7}


-- Reduction pair.

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

p1: app#(height_0(),y) -> height_1#(y)
p2: height_1#(node_2(x,xs)) -> map_2#(height_0(),xs)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(map_1(x1),y) -> map_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(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),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) = 1
        height_0_A = 2
        height_1#_A(x1) = 1
        node_2_A(x1,x2) = max{18, x1 - 3, x2 - 3}
        map_2#_A(x1,x2) = 1
        cons_2_A(x1,x2) = max{2, x1 - 21, x2 - 21}
        map_1_A(x1) = max{2, x1 - 30}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 8, x2 - 1}
          height_0_A = 7
          height_1#_A(x1) = max{0, x1 - 7}
          node_2_A(x1,x2) = max{x1 + 5, x2 + 5}
          map_2#_A(x1,x2) = max{0, x1 - 8, x2 - 12}
          cons_2_A(x1,x2) = max{x1 + 11, x2 + 3}
          map_1_A(x1) = x1 + 6
          app#!_A(x1,x2) = max{5, x1 - 21, x2 - 5}
          height_0!_A = 5
          height_1#!_A(x1) = max{4, x1 - 6}
          node_2!_A(x1,x2) = max{6, x1 - 1, x2 - 1}
          map_2#!_A(x1,x2) = max{3, x1 - 21, x2 - 6}
          cons_2!_A(x1,x2) = max{0, x1 - 39, x2 - 42}
          map_1!_A(x1) = max{2, x1 - 20}
  
    partial status:
  
      pi(app#) = []
      pi(height_0) = []
      pi(height_1#) = []
      pi(node_2) = []
      pi(map_2#) = []
      pi(cons_2) = []
      pi(map_1) = []

The following rules are removed:

  p1, p2, 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)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),xs)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- 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(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),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:
      
        map_2#_A(x1,x2) = max{13, x2 - 1}
        cons_2_A(x1,x2) = max{x1 + 15, x2 + 5}
  
    base order 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{4, x1 - 1, x2 - 14}
          map_2#!_A(x1,x2) = 0
          cons_2!_A(x1,x2) = max{0, x1 - 54, x2 - 104}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_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: maxlist_2#(x,cons_2(y,ys)) -> maxlist_2#(y,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),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:
      
        maxlist_2#_A(x1,x2) = max{11, x1 + 4, x2}
        cons_2_A(x1,x2) = max{11, x1 + 4, x2 + 9}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          maxlist_2#_A(x1,x2) = 0
          cons_2_A(x1,x2) = max{15, x1 - 1, x2 - 6}
          maxlist_2#!_A(x1,x2) = 0
          cons_2!_A(x1,x2) = max{6, x1 - 26, x2 - 82}
  
    partial status:
  
      pi(maxlist_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: le_2#(s_1(x),s_1(y)) -> le_2#(x,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(height_0(),y) -> height_1(y)
r4: app(if_0(),y) -> if_1(y)
r5: app(if_1(x1),y) -> if_2(x1,y)
r6: app(le_0(),y) -> le_1(y)
r7: app(le_1(x1),y) -> le_2(x1,y)
r8: app(map_0(),y) -> map_1(y)
r9: app(map_1(x1),y) -> map_2(x1,y)
r10: app(maxlist_0(),y) -> maxlist_1(y)
r11: app(maxlist_1(x1),y) -> maxlist_2(x1,y)
r12: app(node_0(),y) -> node_1(y)
r13: app(node_1(x1),y) -> node_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: map_2(f,nil_0()) -> nil_0()
r16: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r17: le_2(|0|_0(),y) -> true_0()
r18: le_2(s_1(x),|0|_0()) -> false_0()
r19: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r20: maxlist_2(x,cons_2(y,ys)) -> if_2(le_2(x,y),maxlist_2(y,ys))
r21: maxlist_2(x,nil_0()) -> x
r22: height_1(node_2(x,xs)) -> s_1(maxlist_2(|0|_0(),map_2(height_0(),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:
      
        le_2#_A(x1,x2) = max{1, x1 - 1, x2 - 4}
        s_1_A(x1) = x1 + 30
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le_2#_A(x1,x2) = 0
          s_1_A(x1) = max{12, x1 - 3}
          le_2#!_A(x1,x2) = max{10, x2 - 1}
          s_1!_A(x1) = max{9, x1 - 53}
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.