YES
We show the termination of the TRS R:

  app(f(),app(s(),x)) -> app(f(),x)
  app(g(),app(app(cons(),|0|()),y)) -> app(g(),y)
  app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
  app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
  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(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(g_0(),y) -> g_1(y)
  app(h_0(),y) -> h_1(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(s_1(x)) -> f_1(x)
  g_1(cons_2(|0|_0(),y)) -> g_1(y)
  g_1(cons_2(s_1(x),y)) -> s_1(x)
  h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
  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#(f_0(),y) -> f_1#(y)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: app#(g_0(),y) -> g_1#(y)
p5: app#(h_0(),y) -> h_1#(y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: f_1#(s_1(x)) -> f_1#(x)
p8: g_1#(cons_2(|0|_0(),y)) -> g_1#(y)
p9: h_1#(cons_2(x,y)) -> h_1#(g_1(cons_2(x,y)))
p10: h_1#(cons_2(x,y)) -> g_1#(cons_2(x,y))
p11: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p12: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p13: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p14: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p15: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p16: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p11, p12, p13, p14, p15, p16}
  {p7}
  {p9}
  {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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r6, r7, r14, r21

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 + 11
        cons_2_A(x1,x2) = max{10, x1 - 13, x2 - 13}
        app#_A(x1,x2) = x1 + 7
        map_1_A(x1) = max{16, x1 + 11}
        filter2_3_A(x1,x2,x3) = max{x1 - 3, x2 + 8, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 8
        false_0_A = 3
        filter_2#_A(x1,x2) = x1 + 8
        filter_1_A(x1) = x1 + 9
        app_A(x1,x2) = max{16, x1, x2}
        true_0_A = 1
        cons_0_A = 0
        cons_1_A(x1) = max{0, x1 - 1}
        filter2_0_A = 0
        filter2_1_A(x1) = max{1, x1 - 1}
        filter2_2_A(x1,x2) = max{0, x1 - 2, x2 - 1}
        s_0_A = 0
        s_1_A(x1) = max{11, x1 - 1}
        filter_2_A(x1,x2) = max{10, x1 - 4, x2 - 9}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app > filter2_3 = false_0 = filter_1 = filter2_0 = s_1 > cons_2 = map_1 = filter2_2 = s_0 = nil_0 > app# = true_0 = cons_1 = filter2_1 > map_2# = filter2_4# = filter_2# = cons_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      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(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r8, r12, r14, r21

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 + 6, x2 + 18}
        cons_2_A(x1,x2) = max{4, x1 - 12, x2 + 1}
        app_A(x1,x2) = max{7, x1 - 8, x2 - 18}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 19}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 19}
        filter2_0_A = 1
        filter2_1_A(x1) = max{5, x1 - 19}
        filter2_2_A(x1,x2) = max{6, x1 - 27, x2 - 18}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 35, x2 - 27, x3 - 18}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 19}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 19}
        filter_2_A(x1,x2) = max{3, x1 - 28, x2 - 19}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > map_2# = cons_0 = filter2_3 = map_1 = s_1 = filter_2 > cons_2 = filter_1 = filter2_2 = map_0 = s_0 = nil_0 > cons_1 = 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(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(filter_2) = []
      pi(nil_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: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 8
        filter2_3_A(x1,x2,x3) = max{29, x1 - 3, x2 + 10, x3 - 3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 9
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 9
        cons_2_A(x1,x2) = max{3, x1 - 31, x2 - 18}
        app_A(x1,x2) = max{66, x1 - 30}
        false_0_A = 1
        filter_1_A(x1) = max{29, x1 + 9}
        filter_2_A(x1,x2) = max{0, x1 - 93, x2 - 20}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        true_0 = false_0 > filter_2 > app > filter2_3 = filter_1 > filter2_4# = filter_2# > cons_2 > app# > nil_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r12, r14, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{1, x1 - 2}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 18, x2 + 3, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = max{4, x2 - 1}
        false_0_A = 3
        filter_2#_A(x1,x2) = max{4, x1 - 1}
        cons_2_A(x1,x2) = max{11, x1 + 9, x2 - 18}
        app_A(x1,x2) = max{13, x1 - 15, x2 + 1}
        true_0_A = 3
        cons_0_A = 9
        cons_1_A(x1) = max{10, x1 - 1}
        filter_0_A = 14
        filter_1_A(x1) = max{7, x1 - 19}
        filter2_0_A = 14
        filter2_1_A(x1) = max{13, x1 - 1}
        filter2_2_A(x1,x2) = max{2, x1 - 17, x2 - 1}
        map_0_A = 6
        map_1_A(x1) = max{7, x1 - 1}
        s_0_A = 11
        s_1_A(x1) = max{12, x1 - 10}
        filter_2_A(x1,x2) = max{8, x1 - 34, x2 - 17}
        nil_0_A = 7
  
    base order 2:
  
      precedence:
      
        app > filter2_4# = filter_2# > app# > cons_2 = map_1 > filter2_3 = cons_1 = map_0 = s_1 = filter_2 > false_0 = true_0 = cons_0 = filter_1 = filter2_2 = s_0 = nil_0 > filter_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = [1]
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r4, r6, r7, r8, r10, r12, r14, r16, r17, r21

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{x3 + 2, x4 + 17}
        false_0_A = 20
        filter_2#_A(x1,x2) = x2 + 16
        cons_2_A(x1,x2) = max{x1 + 4, x2 + 18}
        app_A(x1,x2) = max{x1 - 7, x2 + 19}
        true_0_A = 20
        cons_0_A = 3
        cons_1_A(x1) = x1 + 17
        filter_0_A = 3
        filter_1_A(x1) = max{4, x1 - 1}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 1}
        filter2_2_A(x1,x2) = max{4, x1 - 9, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 17, x2 - 9, x3 - 1}
        g_0_A = 6
        g_1_A(x1) = max{17, x1 - 17}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 20}
        s_0_A = 6
        s_1_A(x1) = 2
        |0|_0_A = 14
        filter_2_A(x1,x2) = max{29, x1 + 14, x2 - 20}
        nil_0_A = 15
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app > filter2_3 = s_0 > filter_1 = filter2_2 = map_1 = s_1 > cons_2 = filter_0 = filter2_1 = g_1 = map_0 > filter2_4# = filter_2# = cons_1 = filter2_0 = g_0 = |0|_0 > false_0 = true_0 = cons_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r8, r12, r14, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{47, x1 - 9, x2 + 40}
        cons_2_A(x1,x2) = max{x1 - 48, x2 + 6}
        filter2_4#_A(x1,x2,x3,x4) = max{47, x2 - 9, x3 - 49, x4 + 40}
        app_A(x1,x2) = max{37, x1 - 9, x2 - 9}
        true_0_A = 2
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 49}
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 38}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 10}
        filter2_2_A(x1,x2) = max{3, x1 - 20, x2 - 10}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 29, x2 - 19, x3 - 9}
        map_0_A = 5
        map_1_A(x1) = max{6, x1 - 10}
        s_0_A = 8
        s_1_A(x1) = max{37, x1 - 9}
        filter_2_A(x1,x2) = max{9, x1 - 48, x2 - 16}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2# = filter2_4# > app > s_1 > cons_2 = filter2_3 = map_1 = s_0 = filter_2 > true_0 = cons_1 = filter_1 = filter2_1 = filter2_2 = map_0 = nil_0 > cons_0 = filter_0 = filter2_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(filter2_4#) = [4]
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      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(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: 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: f_1#(s_1(x)) -> f_1#(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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r4, r6, r7, r8, r12, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_1#_A(x1) = max{15, x1 + 5}
        s_1_A(x1) = max{16, x1 + 11}
        app_A(x1,x2) = max{13, x1 - 14, x2 - 5}
        cons_0_A = 11
        cons_1_A(x1) = max{12, x1 - 6}
        cons_2_A(x1,x2) = max{3, x1 - 21, x2 - 26}
        filter_0_A = 10
        filter_1_A(x1) = max{11, x1 - 6}
        filter2_0_A = 12
        filter2_1_A(x1) = max{13, x1 - 5}
        filter2_2_A(x1,x2) = max{13, x1 - 19, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{13, x1 - 34, x2 - 19, x3 - 6}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 21}
        filter_2_A(x1,x2) = max{12, x1 + 1, x2 - 16}
        nil_0_A = 3
  
    base order 2:
  
      precedence:
      
        s_1 = nil_0 > f_1# = app = filter_2 > filter_1 = filter2_3 > cons_2 = filter_0 = filter2_2 = map_1 > cons_1 = filter2_1 = map_0 > cons_0 = filter2_0
      
  
    partial status:
  
      pi(f_1#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_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(filter_2) = [1]
      pi(nil_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: h_1#(cons_2(x,y)) -> h_1#(g_1(cons_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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r8, r10, r12, r14, r16, r17, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        h_1#_A(x1) = x1 + 14
        cons_2_A(x1,x2) = max{4, x1 - 8, x2 - 8}
        g_1_A(x1) = 2
        app_A(x1,x2) = max{6, x1 - 7, x2 - 16}
        cons_0_A = 6
        cons_1_A(x1) = max{3, x1 - 17}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 17}
        filter2_0_A = 3
        filter2_1_A(x1) = max{4, x1 - 17}
        filter2_2_A(x1,x2) = max{5, x1 - 25, x2 - 17}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 32, x2 - 24, x3 - 16}
        g_0_A = 6
        map_0_A = 4
        map_1_A(x1) = max{5, x1 - 17}
        s_0_A = 6
        s_1_A(x1) = 1
        |0|_0_A = 7
        filter_2_A(x1,x2) = max{7, x1 - 3, x2 - 5}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        h_1# = filter_2 > cons_2 = app = cons_1 = g_0 = s_0 = |0|_0 = nil_0 > cons_0 = filter_1 = filter2_3 = map_1 > g_1 = filter_0 = filter2_2 = map_0 > filter2_1 = s_1 > filter2_0
      
  
    partial status:
  
      pi(h_1#) = []
      pi(cons_2) = []
      pi(g_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_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: g_1#(cons_2(|0|_0(),y)) -> g_1#(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(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(h_0(),y) -> h_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_1(s_1(x)) -> f_1(x)
r16: g_1(cons_2(|0|_0(),y)) -> g_1(y)
r17: g_1(cons_2(s_1(x),y)) -> s_1(x)
r18: h_1(cons_2(x,y)) -> h_1(g_1(cons_2(x,y)))
r19: map_2(|fun|,nil_0()) -> nil_0()
r20: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r21: filter_2(|fun|,nil_0()) -> nil_0()
r22: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r23: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r24: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r8, r12, r14, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g_1#_A(x1) = x1 + 6
        cons_2_A(x1,x2) = max{x1 - 8, x2 + 7}
        |0|_0_A = 4
        app_A(x1,x2) = max{7, x1 - 11, x2 - 26}
        cons_0_A = 4
        cons_1_A(x1) = max{5, x1 - 27}
        filter_0_A = 6
        filter_1_A(x1) = max{7, x1 - 27}
        filter2_0_A = 10
        filter2_1_A(x1) = max{4, x1 - 27}
        filter2_2_A(x1,x2) = max{5, x1 - 39, x2 - 27}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 51, x2 - 39, x3 - 27}
        map_0_A = 10
        map_1_A(x1) = max{6, x1 - 27}
        s_0_A = 6
        s_1_A(x1) = max{7, x1 - 27}
        filter_2_A(x1,x2) = max{8, x1 - 10, x2 - 17}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app = nil_0 > cons_0 = filter_1 = filter2_2 = filter2_3 = map_1 = s_1 > cons_2 = filter_0 = filter2_1 = map_0 = s_0 > g_1# = cons_1 = filter2_0 > |0|_0
      
  
    partial status:
  
      pi(g_1#) = [1]
      pi(cons_2) = [2]
      pi(|0|_0) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      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(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.