YES
We show the termination of the TRS R:

  app(f(),app(app(cons(),nil()),y)) -> y
  app(f(),app(app(cons(),app(f(),app(app(cons(),nil()),y))),z)) -> app(app(app(copy(),n()),y),z)
  app(app(app(copy(),|0|()),y),z) -> app(f(),z)
  app(app(app(copy(),app(s(),x)),y),z) -> app(app(app(copy(),x),y),app(app(cons(),app(f(),y)),z))
  app(app(map(),|fun|),nil()) -> nil()
  app(app(map(),|fun|),app(app(cons(),x),xs)) -> app(app(cons(),app(|fun|,x)),app(app(map(),|fun|),xs))
  app(app(filter(),|fun|),nil()) -> nil()
  app(app(filter(),|fun|),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(|fun|,x)),|fun|),x),xs)
  app(app(app(app(filter2(),true()),|fun|),x),xs) -> app(app(cons(),x),app(app(filter(),|fun|),xs))
  app(app(app(app(filter2(),false()),|fun|),x),xs) -> app(app(filter(),|fun|),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(copy_0(),y) -> copy_1(y)
  app(copy_1(x1),y) -> copy_2(x1,y)
  app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
  app(f_0(),y) -> f_1(y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filter2_0(),y) -> filter2_1(y)
  app(filter2_1(x1),y) -> filter2_2(x1,y)
  app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(s_0(),y) -> s_1(y)
  f_1(cons_2(nil_0(),y)) -> y
  f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
  copy_3(|0|_0(),y,z) -> f_1(z)
  copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
  map_2(|fun|,nil_0()) -> nil_0()
  map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
  filter_2(|fun|,nil_0()) -> nil_0()
  filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
  filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
  filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)


-- SCC decomposition.

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

p1: app#(copy_2(x1,x2),y) -> copy_3#(x1,x2,y)
p2: app#(f_0(),y) -> f_1#(y)
p3: app#(filter_1(x1),y) -> filter_2#(x1,y)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: f_1#(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3#(n_0(),y,z)
p7: copy_3#(|0|_0(),y,z) -> f_1#(z)
p8: copy_3#(s_1(x),y,z) -> copy_3#(x,y,cons_2(f_1(y),z))
p9: copy_3#(s_1(x),y,z) -> f_1#(y)
p10: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p11: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p12: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p13: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p14: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p15: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p3, p4, p5, p10, p11, p12, p13, p14, p15}
  {p8}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p7: app#(filter_1(x1),y) -> filter_2#(x1,y)
p8: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p9: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r9, r10, r15, r17, r22

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 39
        cons_2_A(x1,x2) = max{38, x1 - 61, x2 - 61}
        app#_A(x1,x2) = max{37, x1 + 36}
        map_1_A(x1) = x1 + 39
        filter2_3_A(x1,x2,x3) = max{x1 - 33, x2 + 37, x3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 37
        false_0_A = 12
        filter_2#_A(x1,x2) = x1 + 37
        filter_1_A(x1) = x1 + 38
        app_A(x1,x2) = max{60, x1 + 11, x2 - 41}
        true_0_A = 12
        cons_0_A = 16
        cons_1_A(x1) = max{26, x1 - 60}
        copy_0_A = 1
        copy_1_A(x1) = max{13, x1 - 42}
        copy_2_A(x1,x2) = max{60, x1 - 43, x2 - 209}
        filter2_0_A = 1
        filter2_1_A(x1) = max{13, x1 - 42}
        filter2_2_A(x1,x2) = max{25, x1 - 43, x2 - 42}
        s_0_A = 1
        s_1_A(x1) = max{13, x1 - 45}
        f_1_A(x1) = max{15, x1 - 42}
        nil_0_A = 13
        copy_3_A(x1,x2,x3) = max{14, x1 - 44, x2 - 208, x3 - 104}
        n_0_A = 12
        filter_2_A(x1,x2) = max{50, x1 - 171, x2 - 59}
  
    base order 2:
  
      precedence:
      
        copy_0 = filter_2 > filter_1 = copy_1 > app > false_0 > cons_2 = map_1 = filter2_3 = f_1 = copy_3 > true_0 = cons_1 = copy_2 = filter2_2 = s_1 = nil_0 > map_2# = filter2_4# = filter_2# = cons_0 = filter2_1 = s_0 = n_0 > app# = filter2_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = []
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(app) = [1]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = []
      pi(copy_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)
p7: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p8: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r7, r9, r10, r11, r13, r15, r17, r22

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x2 + 25
        cons_2_A(x1,x2) = max{x1 - 23, x2 + 26}
        app_A(x1,x2) = max{x1 - 9, x2 + 8}
        cons_0_A = 8
        cons_1_A(x1) = max{3, x1 - 24}
        copy_0_A = 1
        copy_1_A(x1) = max{2, x1 - 1}
        copy_2_A(x1,x2) = max{3, x1 - 11, x2 - 93}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 30}
        filter2_0_A = 1
        filter2_1_A(x1) = max{5, x1 - 1}
        filter2_2_A(x1,x2) = max{6, x1 - 11, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 20, x2 - 10, x3 - 1}
        map_0_A = 7
        map_1_A(x1) = max{8, x1 - 1}
        s_0_A = 1
        s_1_A(x1) = max{6, x1 - 50}
        f_1_A(x1) = max{9, x1 - 52}
        nil_0_A = 2
        copy_3_A(x1,x2,x3) = max{7, x1 - 49, x2 - 103, x3 - 27}
        n_0_A = 1
        filter_2_A(x1,x2) = max{3, x1 - 40, x2 - 53}
  
    base order 2:
  
      precedence:
      
        app > cons_2 = f_1 = copy_3 > map_2# = cons_1 = copy_1 = copy_2 = map_1 = s_1 > cons_0 = copy_0 = filter2_3 = map_0 = s_0 = nil_0 = filter_2 > filter_1 = filter2_2 = n_0 > filter_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = []
      pi(copy_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []
      pi(filter_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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r9, r10, r13, r15, r17, r22

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 6
        filter2_3_A(x1,x2,x3) = max{x2 + 7, x3 + 2}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 7
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 7
        cons_2_A(x1,x2) = max{5, x1 - 44, x2 - 44}
        app_A(x1,x2) = max{42, x1 - 12, x2 - 12}
        false_0_A = 1
        filter_1_A(x1) = x1 + 8
        cons_0_A = 11
        cons_1_A(x1) = max{4, x1 - 13}
        copy_0_A = 1
        copy_1_A(x1) = max{2, x1 - 12}
        copy_2_A(x1,x2) = max{3, x1 - 25, x2 - 140}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 13}
        filter2_2_A(x1,x2) = max{13, x1 - 26, x2 - 13}
        map_0_A = 10
        map_1_A(x1) = max{11, x1 - 57}
        s_0_A = 2
        s_1_A(x1) = max{3, x1 - 38}
        f_1_A(x1) = max{5, x1 - 13}
        nil_0_A = 40
        copy_3_A(x1,x2,x3) = max{4, x1 - 37, x2 - 139, x3 - 81}
        n_0_A = 0
        filter_2_A(x1,x2) = max{41, x1 - 90, x2 - 46}
  
    base order 2:
  
      precedence:
      
        app > app# = filter2_3 = filter_1 = map_1 = filter_2 > filter2_4# = filter_2# = filter2_2 = map_0 = f_1 = copy_3 > cons_2 = false_0 = copy_2 = filter2_1 = nil_0 > true_0 = cons_1 = copy_1 = filter2_0 = s_1 = n_0 > cons_0 = copy_0 = s_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = []
      pi(copy_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []
      pi(filter_2) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p5: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r7, r10, r13, r15, r17, r22

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{6, x1 - 8}
        filter2_3_A(x1,x2,x3) = max{x1 + 29, x2 + 19, x3 + 19}
        filter2_4#_A(x1,x2,x3,x4) = max{10, x2 - 1}
        false_0_A = 11
        filter_2#_A(x1,x2) = max{10, x1 - 1}
        cons_2_A(x1,x2) = max{5, x1 - 77, x2 - 46}
        app_A(x1,x2) = max{32, x1 - 22, x2 - 53}
        true_0_A = 4
        cons_0_A = 1
        cons_1_A(x1) = max{4, x1 - 54}
        copy_0_A = 1
        copy_1_A(x1) = max{2, x1 - 54}
        copy_2_A(x1,x2) = max{3, x1 - 77, x2 - 208}
        filter_0_A = 1
        filter_1_A(x1) = max{21, x1 - 54}
        filter2_1_A(x1) = max{33, x1 - 54}
        filter2_2_A(x1,x2) = max{21, x1 - 77, x2 - 54}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 54}
        s_0_A = 2
        s_1_A(x1) = max{3, x1 - 178}
        f_1_A(x1) = max{32, x1 - 53}
        nil_0_A = 3
        copy_3_A(x1,x2,x3) = max{4, x1 - 177, x2 - 231, x3 - 100}
        n_0_A = 0
        filter_2_A(x1,x2) = max{45, x1 - 77, x2 - 54}
  
    base order 2:
  
      precedence:
      
        filter2_3 = filter_2 > false_0 = app > filter2_4# = filter_2# > app# = f_1 = copy_3 > cons_2 = copy_2 = filter_1 = filter2_2 = map_1 = s_1 = nil_0 > true_0 = cons_1 = copy_1 = filter_0 = filter2_1 = map_0 = s_0 = n_0 > cons_0 = copy_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = []
      pi(copy_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4}


-- Reduction pair.

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

p1: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r7, r9, r10, r11, r13, r17, r22

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{28, x2 - 1, x3 - 21, x4 + 23}
        false_0_A = 1
        filter_2#_A(x1,x2) = max{28, x1 - 1, x2 + 21}
        cons_2_A(x1,x2) = max{6, x1 - 20, x2 + 2}
        app_A(x1,x2) = max{4, x1 - 44, x2 - 44}
        true_0_A = 4
        cons_0_A = 1
        cons_1_A(x1) = max{3, x1 - 45}
        copy_0_A = 1
        copy_1_A(x1) = max{2, x1 - 45}
        copy_2_A(x1,x2) = max{3, x1 - 90, x2 - 45}
        filter_0_A = 2
        filter_1_A(x1) = max{3, x1 - 45}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 45}
        filter2_2_A(x1,x2) = max{3, x1 - 90, x2 - 45}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 134, x2 - 89, x3 - 44}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 45}
        f_1_A(x1) = max{6, x1 - 14}
        nil_0_A = 4
        copy_3_A(x1,x2,x3) = max{5, x1 - 5, x2 - 90, x3 - 13}
        n_0_A = 0
        filter_2_A(x1,x2) = max{5, x1 - 43, x2 - 7}
  
    base order 2:
  
      precedence:
      
        nil_0 = filter_2 > app > false_0 = cons_2 = cons_0 = filter_1 = filter2_0 = filter2_2 = filter2_3 = map_1 = f_1 = copy_3 > filter2_4# = filter_2# = cons_1 = copy_2 = filter_0 = filter2_1 = map_0 = n_0 > true_0 = copy_1 > copy_0
      
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = []
      pi(copy_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(f_1) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,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(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: copy_3#(s_1(x),y,z) -> copy_3#(x,y,cons_2(f_1(y),z))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(copy_0(),y) -> copy_1(y)
r4: app(copy_1(x1),y) -> copy_2(x1,y)
r5: app(copy_2(x1,x2),y) -> copy_3(x1,x2,y)
r6: app(f_0(),y) -> f_1(y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: f_1(cons_2(nil_0(),y)) -> y
r17: f_1(cons_2(f_1(cons_2(nil_0(),y)),z)) -> copy_3(n_0(),y,z)
r18: copy_3(|0|_0(),y,z) -> f_1(z)
r19: copy_3(s_1(x),y,z) -> copy_3(x,y,cons_2(f_1(y),z))
r20: map_2(|fun|,nil_0()) -> nil_0()
r21: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r22: filter_2(|fun|,nil_0()) -> nil_0()
r23: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r24: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r25: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r7, r9, r10, r11, r13, r15, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        copy_3#_A(x1,x2,x3) = max{3, x1 + 2}
        s_1_A(x1) = max{3, x1 + 2}
        cons_2_A(x1,x2) = max{14, x1 - 33, x2 - 48}
        f_1_A(x1) = max{27, x1 - 15}
        app_A(x1,x2) = max{28, x1 - 30, x2 + 19}
        cons_0_A = 12
        cons_1_A(x1) = max{13, x1 - 1}
        copy_0_A = 19
        copy_1_A(x1) = max{20, x1 + 18}
        copy_2_A(x1,x2) = max{21, x1 - 13, x2 - 81}
        filter_0_A = 26
        filter_1_A(x1) = max{27, x1 - 97}
        filter2_0_A = 19
        filter2_1_A(x1) = max{20, x1 - 1}
        filter2_2_A(x1,x2) = max{21, x1 - 32, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{27, x1 - 63, x2 - 32, x3 - 1}
        map_0_A = 60
        map_1_A(x1) = max{29, x1 - 1}
        s_0_A = 1
        nil_0_A = 18
        copy_3_A(x1,x2,x3) = max{13, x1 - 75, x2 - 112, x3 - 65}
        n_0_A = 13
  
    base order 2:
  
      precedence:
      
        app > cons_2 = f_1 = copy_3 > s_1 = cons_1 = copy_2 = filter2_3 = map_1 = nil_0 > copy_3# = cons_0 = copy_1 = filter2_2 = map_0 = s_0 = n_0 > copy_0 = filter_1 = filter2_1 > filter_0 = filter2_0
      
  
    partial status:
  
      pi(copy_3#) = [1]
      pi(s_1) = [1]
      pi(cons_2) = []
      pi(f_1) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(copy_0) = []
      pi(copy_1) = [1]
      pi(copy_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(nil_0) = []
      pi(copy_3) = []
      pi(n_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.