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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
  app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
  app(app(plus(),|0|()),x) -> |0|()
  app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))

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(plus_0(),y) -> plus_1(y)
  app(plus_1(x1),y) -> plus_2(x1,y)
  app(s_0(),y) -> s_1(y)
  app(size_0(),y) -> size_1(y)
  app(sum_0(),y) -> sum_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))
  sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
  size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
  plus_2(|0|_0(),x) -> |0|_0()
  plus_2(s_1(x),y) -> s_1(plus_2(x,y))


-- 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#(plus_1(x1),y) -> plus_2#(x1,y)
p3: app#(size_0(),y) -> size_1#(y)
p4: app#(sum_0(),y) -> sum_1#(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: sum_1#(cons_2(x,xs)) -> plus_2#(x,sum_1(xs))
p8: sum_1#(cons_2(x,xs)) -> sum_1#(xs)
p9: size_1#(node_2(x,xs)) -> sum_1#(map_2(size_0(),xs))
p10: size_1#(node_2(x,xs)) -> map_2#(size_0(),xs)
p11: plus_2#(s_1(x),y) -> plus_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The estimated dependency graph contains the following SCCs:

  {p1, p3, p5, p6, p10}
  {p8}
  {p11}


-- 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#(size_0(),y) -> size_1#(y)
p5: size_1#(node_2(x,xs)) -> map_2#(size_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The set of usable rules consists of

  r1, r2, r5, r6, r7, r8, r9, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{13, x1 - 77}
        map_1_A(x1) = x1 + 91
        map_2#_A(x1,x2) = max{13, x1 - 77}
        cons_2_A(x1,x2) = max{12, x1 - 16, x2 - 16}
        size_0_A = 1
        size_1#_A(x1) = 13
        node_2_A(x1,x2) = max{14, x1 - 75, x2 - 75}
        app_A(x1,x2) = max{15, x1 - 6, x2 + 4}
        cons_0_A = 4
        cons_1_A(x1) = max{5, x1 - 1}
        node_0_A = 4
        node_1_A(x1) = max{5, x1 - 1}
        plus_0_A = 7
        plus_1_A(x1) = max{8, x1 - 1}
        plus_2_A(x1,x2) = max{10, x1 - 34, x2 - 26}
        s_0_A = 7
        s_1_A(x1) = max{8, x1 - 35}
        |0|_0_A = 0
  
    base order 2:
  
      precedence:
      
        app# = map_2# = size_1# > node_2 = s_0 > size_0 > map_1 > app > cons_2 = node_1 > cons_1 = node_0 > cons_0 = plus_2 > plus_1 = s_1 = |0|_0 > plus_0
      
  
    partial status:
  
      pi(app#) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(cons_2) = []
      pi(size_0) = []
      pi(size_1#) = []
      pi(node_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(plus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []

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#(size_0(),y) -> size_1#(y)
p4: size_1#(node_2(x,xs)) -> map_2#(size_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

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: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(size_0(),y) -> size_1#(y)
p4: size_1#(node_2(x,xs)) -> map_2#(size_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The set of usable rules consists of

  r1, r5, r7, r8, r16, r17

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, x1 - 10, x2 - 15}
        cons_2_A(x1,x2) = max{x1 + 26, x2 + 5}
        app#_A(x1,x2) = max{11, x1 - 10, x2 + 9}
        size_0_A = 9
        size_1#_A(x1) = max{10, x1 - 12}
        node_2_A(x1,x2) = max{10, x1 - 50, x2 - 2}
        app_A(x1,x2) = max{27, x1 - 27, x2 - 27}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 28}
        node_0_A = 1
        node_1_A(x1) = 9
        plus_0_A = 4
        plus_1_A(x1) = max{5, x1 - 28}
        plus_2_A(x1,x2) = max{6, x1 - 56, x2 - 28}
        |0|_0_A = 0
        s_1_A(x1) = max{2, x1}
  
    base order 2:
  
      precedence:
      
        size_1# = node_2 = app > map_2# = size_0 > app# = node_1 > node_0 > cons_2 > cons_1 > cons_0 = plus_2 > plus_1 = |0|_0 = s_1 > plus_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1, 2]
      pi(app#) = [2]
      pi(size_0) = []
      pi(size_1#) = []
      pi(node_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(plus_2) = []
      pi(|0|_0) = []
      pi(s_1) = []

The following rules are removed:

  p2



-- 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: app#(size_0(),y) -> size_1#(y)
p3: size_1#(node_2(x,xs)) -> map_2#(size_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r8, r9, r16, r17

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{x1 + 8, x2 + 20}
        cons_2_A(x1,x2) = max{6, x1 - 12, x2}
        app_A(x1,x2) = max{9, x1 - 10, x2 + 7}
        cons_0_A = 8
        cons_1_A(x1) = max{9, x1 - 1}
        map_0_A = 8
        map_1_A(x1) = max{9, x1 - 1}
        node_0_A = 1
        node_1_A(x1) = max{2, x1 - 1}
        node_2_A(x1,x2) = max{3, x1 - 30, x2 - 51}
        plus_0_A = 3
        plus_1_A(x1) = max{4, x1 - 1}
        plus_2_A(x1,x2) = max{5, x1 - 20, x2 - 17}
        s_0_A = 2
        s_1_A(x1) = max{3, x1 - 21}
        |0|_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > map_2# > cons_2 = map_1 > node_2 = plus_2 > s_1 = |0|_0 > cons_0 > cons_1 = map_0 = node_1 = plus_1 = s_0 > node_0 = plus_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(plus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_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: sum_1#(cons_2(x,xs)) -> sum_1#(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The set of usable rules consists of

  r1, r3, r5, r6, r7, r8, r9, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        sum_1#_A(x1) = x1 + 20
        cons_2_A(x1,x2) = max{x1 - 16, x2 + 21}
        app_A(x1,x2) = max{15, x1 - 3, x2 - 1}
        cons_0_A = 2
        cons_1_A(x1) = max{1, x1 - 2}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 1}
        node_0_A = 1
        node_1_A(x1) = max{2, x1 - 80}
        node_2_A(x1,x2) = max{14, x1 - 84, x2 - 84}
        plus_0_A = 4
        plus_1_A(x1) = max{12, x1 - 2}
        plus_2_A(x1,x2) = max{13, x1 - 39, x2 - 28}
        s_0_A = 2
        s_1_A(x1) = max{11, x1 - 40}
        |0|_0_A = 0
  
    base order 2:
  
      precedence:
      
        node_0 = node_2 > app > cons_2 = map_1 = node_1 > sum_1# = cons_1 = map_0 > cons_0 = plus_2 > plus_1 = s_1 = |0|_0 > plus_0 = s_0
      
  
    partial status:
  
      pi(sum_1#) = [1]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(plus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_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: plus_2#(s_1(x),y) -> plus_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(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(plus_0(),y) -> plus_1(y)
r8: app(plus_1(x1),y) -> plus_2(x1,y)
r9: app(s_0(),y) -> s_1(y)
r10: app(size_0(),y) -> size_1(y)
r11: app(sum_0(),y) -> sum_1(y)
r12: map_2(f,nil_0()) -> nil_0()
r13: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r14: sum_1(cons_2(x,xs)) -> plus_2(x,sum_1(xs))
r15: size_1(node_2(x,xs)) -> s_1(sum_1(map_2(size_0(),xs)))
r16: plus_2(|0|_0(),x) -> |0|_0()
r17: plus_2(s_1(x),y) -> s_1(plus_2(x,y))

The set of usable rules consists of

  r1, r2, r5, r6, r7

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus_2#_A(x1,x2) = max{x1 + 6, x2 + 13}
        s_1_A(x1) = max{11, x1 + 7}
        app_A(x1,x2) = max{11, x1 - 13, x2 - 1}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 1}
        cons_2_A(x1,x2) = max{4, x1 - 18, x2 - 6}
        node_0_A = 1
        node_1_A(x1) = max{2, x1 - 2}
        node_2_A(x1,x2) = max{11, x1 - 44, x2 - 44}
        plus_0_A = 28
        plus_1_A(x1) = max{14, x1 - 18}
  
    base order 2:
  
      precedence:
      
        cons_2 > app > plus_2# = cons_1 = node_0 = node_2 > s_1 = cons_0 = node_1 > plus_1 > plus_0
      
  
    partial status:
  
      pi(plus_2#) = [1]
      pi(s_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []
      pi(plus_0) = []
      pi(plus_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.