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

  r10, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{x1 + 5, x2}
        height_0_A = 18
        height_1#_A(x1) = max{23, x1 - 41}
        node_2_A(x1,x2) = max{40, x1 + 24, x2 + 24}
        map_2#_A(x1,x2) = max{x1 + 5, x2 - 17}
        cons_2_A(x1,x2) = max{x1 + 26, x2 + 21}
        map_1_A(x1) = max{13, x1 + 4}
        app_A(x1,x2) = max{27, x1 + 11}
        maxlist_0_A = 1
        maxlist_1_A(x1) = 13
        le_2_A(x1,x2) = max{x1 + 41, x2 - 28}
        |0|_0_A = 25
        true_0_A = 24
        s_1_A(x1) = max{27, x1 - 42}
        false_0_A = 0
  
    base order 2:
  
      precedence:
      
        node_2 = app > le_2 > |0|_0 = s_1 > cons_2 = map_1 = true_0 = false_0 > app# = height_1# = map_2# > height_0 = maxlist_1 > maxlist_0
      
  
    partial status:
  
      pi(app#) = []
      pi(height_0) = []
      pi(height_1#) = []
      pi(node_2) = []
      pi(map_2#) = []
      pi(cons_2) = []
      pi(map_1) = [1]
      pi(app) = [1]
      pi(maxlist_0) = []
      pi(maxlist_1) = []
      pi(le_2) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = []
      pi(false_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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)

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, p2, p3, p4}


-- 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)) -> app#(f,x)
p4: 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

  r1, r4, r5, r6, r8, r10, r12, r13, r14, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x2 + 12
        height_0_A = 29
        height_1#_A(x1) = max{11, x1 + 5}
        node_2_A(x1,x2) = max{x1 + 28, x2 - 4}
        map_2#_A(x1,x2) = max{9, x2 - 12}
        cons_2_A(x1,x2) = max{146, x1 + 25, x2 + 11}
        app_A(x1,x2) = max{62, x1 - 30, x2 + 60}
        cons_0_A = 28
        cons_1_A(x1) = x1 + 59
        if_0_A = 28
        if_1_A(x1) = max{29, x1 - 1}
        if_2_A(x1,x2) = max{27, x1 - 32, x2 - 77}
        le_0_A = 24
        le_1_A(x1) = max{25, x1 - 21}
        map_0_A = 29
        map_1_A(x1) = max{61, x1 - 1}
        maxlist_0_A = 29
        maxlist_1_A(x1) = max{61, x1 - 1}
        node_0_A = 1
        node_1_A(x1) = x1 + 59
        s_0_A = 29
        s_1_A(x1) = max{25, x1 - 30}
        le_2_A(x1,x2) = max{26, x1 - 52, x2 - 129}
        |0|_0_A = 27
        true_0_A = 0
        false_0_A = 0
  
    base order 2:
  
      precedence:
      
        node_2 > |0|_0 > app > height_1# = map_1 = node_1 = le_2 > height_0 = map_2# = cons_2 = le_1 = map_0 = node_0 = s_1 = true_0 > app# = cons_1 = le_0 = s_0 = false_0 > cons_0 = if_2 = maxlist_1 > if_1 = maxlist_0 > if_0
      
  
    partial status:
  
      pi(app#) = []
      pi(height_0) = []
      pi(height_1#) = [1]
      pi(node_2) = []
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(maxlist_0) = []
      pi(maxlist_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(height_0(),y) -> height_1#(y)
p2: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: 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 estimated dependency graph contains the following SCCs:

  {p3}


-- 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

  r4, r5, r8, r10, r12, r13, r14, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x2}
        cons_2_A(x1,x2) = max{x1 + 2, x2 + 4}
        app_A(x1,x2) = max{7, x1 - 9, x2 - 3}
        if_0_A = 6
        if_1_A(x1) = max{7, x1 - 3}
        if_2_A(x1,x2) = max{7, x1 - 12, x2 - 19}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 4}
        maxlist_0_A = 6
        maxlist_1_A(x1) = max{7, x1 - 26}
        node_0_A = 6
        node_1_A(x1) = max{7, x1 - 3}
        node_2_A(x1,x2) = max{6, x1 - 36, x2 - 36}
        s_0_A = 1
        s_1_A(x1) = max{2, x1 - 7}
        le_2_A(x1,x2) = max{5, x1 - 6, x2 - 4}
        |0|_0_A = 1
        true_0_A = 0
        false_0_A = 0
  
    base order 2:
  
      precedence:
      
        node_0 = node_2 > |0|_0 > app > if_2 = map_1 = node_1 = le_2 > cons_2 = if_1 = map_0 = s_1 = true_0 > map_2# = if_0 = s_0 = false_0 > maxlist_1 > maxlist_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1]
      pi(app) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(maxlist_0) = []
      pi(maxlist_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

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

  r1, r4, r5, r6, r8, r10, r12, r13, r14, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        maxlist_2#_A(x1,x2) = x2 + 8
        cons_2_A(x1,x2) = max{12, x1 - 6, x2 + 9}
        app_A(x1,x2) = max{5, x1 - 6, x2 - 15}
        cons_0_A = 4
        cons_1_A(x1) = max{5, x1 - 15}
        if_0_A = 5
        if_1_A(x1) = max{4, x1 - 16}
        if_2_A(x1,x2) = max{3, x1 - 37, x2 - 15}
        le_0_A = 3
        le_1_A(x1) = max{4, x1 - 16}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 15}
        maxlist_0_A = 7
        maxlist_1_A(x1) = max{4, x1 - 52}
        node_0_A = 1
        node_1_A(x1) = max{2, x1 - 16}
        node_2_A(x1,x2) = max{5, x1 - 67, x2 - 49}
        s_0_A = 1
        s_1_A(x1) = max{2, x1 - 16}
        le_2_A(x1,x2) = max{36, x1 - 15, x2 - 3}
        |0|_0_A = 1
        true_0_A = 0
        false_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > if_2 = node_2 > cons_2 = |0|_0 > maxlist_2# = cons_0 = le_2 > cons_1 = if_1 = le_1 = map_1 = maxlist_1 = node_1 = s_1 = true_0 > if_0 = le_0 = map_0 = maxlist_0 = node_0 = s_0 = false_0
      
  
    partial status:
  
      pi(maxlist_2#) = [2]
      pi(cons_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(maxlist_0) = []
      pi(maxlist_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

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

  r1, r2, r4, r5, r6, r8, r10, r12, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = max{x1 - 1, x2 + 24}
        s_1_A(x1) = max{25, x1 + 23}
        app_A(x1,x2) = max{78, x1 - 17, x2 - 57}
        cons_0_A = 15
        cons_1_A(x1) = max{16, x1 - 58}
        cons_2_A(x1,x2) = max{0, x1 - 133}
        if_0_A = 15
        if_1_A(x1) = max{16, x1 - 64}
        if_2_A(x1,x2) = max{28, x1 - 82, x2 - 58}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 58}
        map_0_A = 15
        map_1_A(x1) = max{16, x1 - 58}
        maxlist_0_A = 15
        maxlist_1_A(x1) = max{16, x1 - 57}
        node_0_A = 15
        node_1_A(x1) = max{16, x1 - 58}
        le_2_A(x1,x2) = max{26, x1 - 76, x2 - 191}
        |0|_0_A = 50
        true_0_A = 0
        false_0_A = 24
  
    base order 2:
  
      precedence:
      
        le_2# > s_1 > |0|_0 > app > if_2 = node_1 > cons_0 = maxlist_1 > cons_1 = if_1 = map_1 = maxlist_0 = node_0 = le_2 > cons_2 = if_0 = le_1 = map_0 = true_0 = false_0 > le_0
      
  
    partial status:
  
      pi(le_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(maxlist_0) = []
      pi(maxlist_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.