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(flatten(),app(app(node(),x),xs)) -> app(app(cons(),x),app(concat(),app(app(map(),flatten()),xs)))
  app(concat(),nil()) -> nil()
  app(concat(),app(app(cons(),x),xs)) -> app(app(append(),x),app(concat(),xs))
  app(app(append(),nil()),xs) -> xs
  app(app(append(),app(app(cons(),x),xs)),ys) -> app(app(cons(),x),app(app(append(),xs),ys))

Instead we prove termination of the uncurried system:

  app(append_0(),y) -> append_1(y)
  app(append_1(x1),y) -> append_2(x1,y)
  app(concat_0(),y) -> concat_1(y)
  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(flatten_0(),y) -> flatten_1(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)
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
  concat_1(nil_0()) -> nil_0()
  concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
  append_2(nil_0(),xs) -> xs
  append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))


-- SCC decomposition.

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

p1: app#(append_1(x1),y) -> append_2#(x1,y)
p2: app#(concat_0(),y) -> concat_1#(y)
p3: app#(flatten_0(),y) -> flatten_1#(y)
p4: app#(map_1(x1),y) -> map_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: flatten_1#(node_2(x,xs)) -> concat_1#(map_2(flatten_0(),xs))
p8: flatten_1#(node_2(x,xs)) -> map_2#(flatten_0(),xs)
p9: concat_1#(cons_2(x,xs)) -> append_2#(x,concat_1(xs))
p10: concat_1#(cons_2(x,xs)) -> concat_1#(xs)
p11: append_2#(cons_2(x,xs),ys) -> append_2#(xs,ys)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The estimated dependency graph contains the following SCCs:

  {p3, p4, p5, p6, p8}
  {p10}
  {p11}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: app#(flatten_0(),y) -> flatten_1#(y)
p5: flatten_1#(node_2(x,xs)) -> map_2#(flatten_0(),xs)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The set of usable rules consists of

  r1, r4, r5, r9

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{5, x1 - 3}
        cons_2_A(x1,x2) = max{26, x1 - 73, x2 - 73}
        app#_A(x1,x2) = max{5, x1 - 3}
        map_1_A(x1) = x1 + 9
        flatten_0_A = 2
        flatten_1#_A(x1) = 5
        node_2_A(x1,x2) = max{40, x1 + 1, x2 - 10}
        app_A(x1,x2) = max{63, x1 - 10, x2 - 61}
        append_0_A = 8
        append_1_A(x1) = max{9, x1 - 139}
        cons_0_A = 8
        cons_1_A(x1) = max{9, x1 - 62}
        node_0_A = 8
        node_1_A(x1) = max{9, x1 - 62}
  
    base order 2:
  
      precedence:
      
        node_2 > map_1 > app > map_2# = cons_2 = app# = flatten_1# = node_1 > flatten_0 = append_1 = cons_1 = node_0 > append_0 = cons_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(flatten_0) = []
      pi(flatten_1#) = []
      pi(node_2) = [1]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(node_0) = []
      pi(node_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)) -> app#(f,x)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p3: app#(flatten_0(),y) -> flatten_1#(y)
p4: flatten_1#(node_2(x,xs)) -> map_2#(flatten_0(),xs)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

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)) -> app#(f,x)
p2: app#(flatten_0(),y) -> flatten_1#(y)
p3: flatten_1#(node_2(x,xs)) -> map_2#(flatten_0(),xs)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The set of usable rules consists of

  r1, r4, r7, r9

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{3, x1 - 1, x2 - 8}
        cons_2_A(x1,x2) = max{38, x1 + 16, x2 + 18}
        app#_A(x1,x2) = max{x1 - 2, x2 + 7}
        flatten_0_A = 4
        flatten_1#_A(x1) = x1 + 6
        node_2_A(x1,x2) = max{38, x1 - 1, x2 - 7}
        app_A(x1,x2) = max{38, x1 - 35, x2 - 54}
        append_0_A = 34
        append_1_A(x1) = max{16, x1 - 87}
        cons_0_A = 33
        cons_1_A(x1) = max{34, x1 - 55}
        map_0_A = 33
        map_1_A(x1) = max{34, x1 - 55}
        node_0_A = 33
        node_1_A(x1) = max{34, x1 - 55}
  
    base order 2:
  
      precedence:
      
        node_2 > flatten_1# = app > app# = map_1 = node_1 > map_2# = cons_2 = flatten_0 = cons_0 = map_0 = node_0 > append_1 = cons_1 > append_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(flatten_0) = []
      pi(flatten_1#) = [1]
      pi(node_2) = []
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(flatten_0(),y) -> flatten_1#(y)
p2: flatten_1#(node_2(x,xs)) -> map_2#(flatten_0(),xs)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

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(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The set of usable rules consists of

  r1, r4, r7, r9, r10

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{19, x1 + 1, x2 + 10}
        cons_2_A(x1,x2) = max{x1 - 8, x2 + 9}
        app_A(x1,x2) = max{7, x1 - 8, x2 - 7}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 12}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 8}
        map_0_A = 10
        map_1_A(x1) = max{1, x1 - 8}
        node_0_A = 6
        node_1_A(x1) = max{7, x1 - 16}
        node_2_A(x1,x2) = max{7, x1 - 24, x2 - 24}
  
    base order 2:
  
      precedence:
      
        map_2# > cons_2 = app > cons_0 = node_2 > append_1 = cons_1 = node_1 > append_0 = node_0 > map_1 > map_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_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: concat_1#(cons_2(x,xs)) -> concat_1#(xs)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The set of usable rules consists of

  r1, r4, r7, r9, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        concat_1#_A(x1) = max{2, x1 - 7}
        cons_2_A(x1,x2) = max{10, x1 + 6, x2 + 6}
        app_A(x1,x2) = max{11, x1 - 7, x2 - 7}
        append_0_A = 6
        append_1_A(x1) = max{0, x1 - 8}
        cons_0_A = 5
        cons_1_A(x1) = max{6, x1 - 8}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 8}
        node_0_A = 5
        node_1_A(x1) = max{6, x1 - 8}
        node_2_A(x1,x2) = max{9, x1 - 23, x2 - 23}
  
    base order 2:
  
      precedence:
      
        app > cons_2 = append_1 > concat_1# = append_0 = cons_1 = node_2 > cons_0 = node_1 > map_1 = node_0 > map_0
      
  
    partial status:
  
      pi(concat_1#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_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: append_2#(cons_2(x,xs),ys) -> append_2#(xs,ys)

and R consists of:

r1: app(append_0(),y) -> append_1(y)
r2: app(append_1(x1),y) -> append_2(x1,y)
r3: app(concat_0(),y) -> concat_1(y)
r4: app(cons_0(),y) -> cons_1(y)
r5: app(cons_1(x1),y) -> cons_2(x1,y)
r6: app(flatten_0(),y) -> flatten_1(y)
r7: app(map_0(),y) -> map_1(y)
r8: app(map_1(x1),y) -> map_2(x1,y)
r9: app(node_0(),y) -> node_1(y)
r10: app(node_1(x1),y) -> node_2(x1,y)
r11: map_2(f,nil_0()) -> nil_0()
r12: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r13: flatten_1(node_2(x,xs)) -> cons_2(x,concat_1(map_2(flatten_0(),xs)))
r14: concat_1(nil_0()) -> nil_0()
r15: concat_1(cons_2(x,xs)) -> append_2(x,concat_1(xs))
r16: append_2(nil_0(),xs) -> xs
r17: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))

The set of usable rules consists of

  r1, r4, r7, r9, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        append_2#_A(x1,x2) = max{35, x1 + 31, x2 + 1}
        cons_2_A(x1,x2) = max{4, x1 - 30, x2}
        app_A(x1,x2) = max{22, x1 - 7, x2 - 1}
        append_0_A = 1
        append_1_A(x1) = max{2, x1 - 2}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 31}
        map_0_A = 6
        map_1_A(x1) = max{0, x1 - 2}
        node_0_A = 5
        node_1_A(x1) = max{6, x1 - 2}
        node_2_A(x1,x2) = max{7, x1 - 11, x2 - 11}
  
    base order 2:
  
      precedence:
      
        append_2# > cons_2 = app > append_0 = cons_0 = node_2 > append_1 = cons_1 = node_1 > node_0 > map_1 > map_0
      
  
    partial status:
  
      pi(append_2#) = [1]
      pi(cons_2) = [2]
      pi(app) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(node_0) = []
      pi(node_1) = []
      pi(node_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.