YES
We show the termination of the TRS R:

  app(app(neq(),|0|()),|0|()) -> false()
  app(app(neq(),|0|()),app(s(),y)) -> true()
  app(app(neq(),app(s(),x)),|0|()) -> true()
  app(app(neq(),app(s(),x)),app(s(),y)) -> app(app(neq(),x),y)
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
  app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
  app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)
  nonzero() -> app(filter(),app(neq(),|0|()))

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(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filtersub_0(),y) -> filtersub_1(y)
  app(filtersub_1(x1),y) -> filtersub_2(x1,y)
  app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
  app(neq_0(),y) -> neq_1(y)
  app(neq_1(x1),y) -> neq_2(x1,y)
  app(s_0(),y) -> s_1(y)
  neq_2(|0|_0(),|0|_0()) -> false_0()
  neq_2(|0|_0(),s_1(y)) -> true_0()
  neq_2(s_1(x),|0|_0()) -> true_0()
  neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
  filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
  filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
  nonzero_0() -> filter_1(neq_1(|0|_0()))


-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p3: app#(neq_1(x1),y) -> neq_2#(x1,y)
p4: neq_2#(s_1(x),s_1(y)) -> neq_2#(x,y)
p5: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p6: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p7: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p8: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p5, p6, p7, p8}
  {p4}


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p3: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p4: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p5: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p6: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r12, r13, r15, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x2 + 15
        filter_1_A(x1) = max{7, x1 - 23}
        filter_2#_A(x1,x2) = max{8, x2 + 3}
        cons_2_A(x1,x2) = max{x1 + 16, x2 + 13}
        filtersub_2_A(x1,x2) = max{19, x1 - 23, x2 - 23}
        filtersub_3#_A(x1,x2,x3) = max{14, x3 - 9}
        false_0_A = 9
        app_A(x1,x2) = max{x1 - 20, x2 + 21}
        true_0_A = 9
        cons_0_A = 8
        cons_1_A(x1) = max{9, x1 - 1}
        filter_0_A = 19
        filtersub_0_A = 16
        filtersub_1_A(x1) = max{17, x1 - 1}
        neq_0_A = 18
        neq_1_A(x1) = max{19, x1 - 11}
        s_0_A = 18
        s_1_A(x1) = max{19, x1 - 33}
        neq_2_A(x1,x2) = max{20, x1 - 32, x2 - 20}
        |0|_0_A = 10
        filter_2_A(x1,x2) = max{17, x1 - 44, x2 - 24}
        nil_0_A = 0
        nonzero_0_A = 20
  
    base order 2:
  
      precedence:
      
        app > neq_2 = nonzero_0 > neq_1 = s_1 = |0|_0 > app# = false_0 = true_0 > filter_2# = cons_2 = neq_0 = s_0 = filter_2 > filter_1 = filtersub_3# = cons_1 = nil_0 > filtersub_2 = cons_0 = filter_0 > filtersub_1 > filtersub_0
      
  
    partial status:
  
      pi(app#) = [2]
      pi(filter_1) = []
      pi(filter_2#) = []
      pi(cons_2) = [1]
      pi(filtersub_2) = []
      pi(filtersub_3#) = []
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(neq_2) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []
      pi(nonzero_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p3: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p4: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p5: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

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#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p3: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p4: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p5: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r12, r13, r15, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{9, x2 + 5}
        filter_1_A(x1) = max{13, x1 - 3}
        filter_2#_A(x1,x2) = x2 + 4
        cons_2_A(x1,x2) = max{13, x1 + 6, x2 + 8}
        filtersub_3#_A(x1,x2,x3) = max{7, x3}
        app_A(x1,x2) = max{25, x1 - 25, x2 - 1}
        true_0_A = 76
        false_0_A = 13
        cons_0_A = 11
        cons_1_A(x1) = max{12, x1 - 2}
        filter_0_A = 12
        filtersub_0_A = 24
        filtersub_1_A(x1) = max{1, x1 - 2}
        filtersub_2_A(x1,x2) = max{24, x1 - 28, x2 - 55}
        neq_0_A = 1
        neq_1_A(x1) = max{2, x1 - 79}
        s_0_A = 23
        s_1_A(x1) = max{24, x1 - 79}
        neq_2_A(x1,x2) = max{77, x1 - 78, x2 - 78}
        |0|_0_A = 77
        filter_2_A(x1,x2) = max{14, x1 - 79, x2 - 42}
        nil_0_A = 0
        nonzero_0_A = 78
  
    base order 2:
  
      precedence:
      
        neq_2 = nonzero_0 > app = |0|_0 > true_0 = filtersub_2 > filter_1 = filter_2# = cons_2 = filtersub_1 = neq_1 = s_1 = filter_2 > app# = filtersub_3# = false_0 = cons_1 = filter_0 = filtersub_0 = neq_0 = s_0 = nil_0 > cons_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1]
      pi(filtersub_3#) = []
      pi(app) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filtersub_2) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(neq_2) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []
      pi(nonzero_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p2: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p3: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p4: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p2: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p3: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The set of usable rules consists of

  r1, r2, r3, r5, r6, r8, r10, r15, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x2 + 15
        cons_2_A(x1,x2) = max{x1 + 11, x2 + 24}
        filtersub_3#_A(x1,x2,x3) = max{0, x3 - 9}
        app_A(x1,x2) = max{x1 - 13, x2 + 25}
        false_0_A = 28
        true_0_A = 30
        cons_0_A = 11
        cons_1_A(x1) = x1 + 25
        filter_0_A = 5
        filter_1_A(x1) = x1 + 21
        filtersub_0_A = 10
        filtersub_1_A(x1) = max{11, x1 - 1}
        filtersub_2_A(x1,x2) = max{12, x1 - 14, x2 - 1}
        neq_0_A = 12
        neq_1_A(x1) = x1 + 14
        s_0_A = 10
        s_1_A(x1) = max{11, x1 - 13}
        filter_2_A(x1,x2) = max{7, x1 - 20, x2 - 15}
        nil_0_A = 0
        nonzero_0_A = 75
        |0|_0_A = 39
  
    base order 2:
  
      precedence:
      
        app > nonzero_0 > cons_2 = cons_1 = filtersub_2 = filter_2 = |0|_0 > filtersub_3# = true_0 = filter_1 = nil_0 > cons_0 = filter_0 = filtersub_1 = s_1 > false_0 = filtersub_0 = neq_1 = s_0 > filter_2# = neq_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(filtersub_3#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filtersub_2) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []
      pi(nonzero_0) = []
      pi(|0|_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p2: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: neq_2#(s_1(x),s_1(y)) -> neq_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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: app(neq_0(),y) -> neq_1(y)
r9: app(neq_1(x1),y) -> neq_2(x1,y)
r10: app(s_0(),y) -> s_1(y)
r11: neq_2(|0|_0(),|0|_0()) -> false_0()
r12: neq_2(|0|_0(),s_1(y)) -> true_0()
r13: neq_2(s_1(x),|0|_0()) -> true_0()
r14: neq_2(s_1(x),s_1(y)) -> neq_2(x,y)
r15: filter_2(f,nil_0()) -> nil_0()
r16: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r17: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r18: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)
r19: nonzero_0() -> filter_1(neq_1(|0|_0()))

The set of usable rules consists of

  r1, r2, r3, r5, r6, r8, r9, r11, r12, r13, r14, r15, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        neq_2#_A(x1,x2) = max{9, x1 + 7, x2 + 7}
        s_1_A(x1) = max{8, x1 + 3}
        app_A(x1,x2) = max{7, x1 - 4, x2 - 1}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 2}
        cons_2_A(x1,x2) = max{5, x1 - 7, x2 - 14}
        filter_0_A = 2
        filter_1_A(x1) = max{3, x1 - 15}
        filtersub_0_A = 1
        filtersub_1_A(x1) = max{2, x1 - 2}
        filtersub_2_A(x1,x2) = max{3, x1 - 23, x2 - 2}
        neq_0_A = 19
        neq_1_A(x1) = max{14, x1 - 6}
        neq_2_A(x1,x2) = max{9, x1 - 11, x2 - 9}
        |0|_0_A = 5
        false_0_A = 1
        true_0_A = 4
        filter_2_A(x1,x2) = max{6, x1 + 1, x2 - 6}
        nil_0_A = 2
        nonzero_0_A = 15
  
    base order 2:
  
      precedence:
      
        neq_2# = neq_2 > s_1 = filtersub_0 = nonzero_0 > app = cons_2 = |0|_0 = filter_2 > cons_1 = filter_1 = filtersub_2 = neq_1 = false_0 = true_0 = nil_0 > cons_0 = filter_0 = filtersub_1 = neq_0
      
  
    partial status:
  
      pi(neq_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filtersub_2) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(neq_2) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []
      pi(nonzero_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.