YES
We show the termination of the TRS R:

  app(app(app(if(),true()),xs),ys) -> xs
  app(app(app(if(),false()),xs),ys) -> ys
  app(app(sub(),x),|0|()) -> x
  app(app(sub(),app(s(),x)),app(s(),y)) -> app(app(sub(),x),y)
  app(app(gtr(),|0|()),y) -> false()
  app(app(gtr(),app(s(),x)),|0|()) -> true()
  app(app(gtr(),app(s(),x)),app(s(),y)) -> app(app(gtr(),x),y)
  app(app(d(),x),|0|()) -> true()
  app(app(d(),app(s(),x)),app(s(),y)) -> app(app(app(if(),app(app(gtr(),x),y)),false()),app(app(d(),app(s(),x)),app(app(sub(),y),x)))
  app(len(),nil()) -> |0|()
  app(len(),app(app(cons(),x),xs)) -> app(s(),app(len(),xs))
  app(app(filter(),p),nil()) -> nil()
  app(app(filter(),p),app(app(cons(),x),xs)) -> app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(filter(),p),xs))),app(app(filter(),p),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(d_0(),y) -> d_1(y)
  app(d_1(x1),y) -> d_2(x1,y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(gtr_0(),y) -> gtr_1(y)
  app(gtr_1(x1),y) -> gtr_2(x1,y)
  app(if_0(),y) -> if_1(y)
  app(if_1(x1),y) -> if_2(x1,y)
  app(if_2(x1,x2),y) -> if_3(x1,x2,y)
  app(len_0(),y) -> len_1(y)
  app(s_0(),y) -> s_1(y)
  app(sub_0(),y) -> sub_1(y)
  app(sub_1(x1),y) -> sub_2(x1,y)
  if_3(true_0(),xs,ys) -> xs
  if_3(false_0(),xs,ys) -> ys
  sub_2(x,|0|_0()) -> x
  sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
  gtr_2(|0|_0(),y) -> false_0()
  gtr_2(s_1(x),|0|_0()) -> true_0()
  gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
  d_2(x,|0|_0()) -> true_0()
  d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
  len_1(nil_0()) -> |0|_0()
  len_1(cons_2(x,xs)) -> s_1(len_1(xs))
  filter_2(p,nil_0()) -> nil_0()
  filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))


-- SCC decomposition.

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

p1: app#(d_1(x1),y) -> d_2#(x1,y)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: app#(gtr_1(x1),y) -> gtr_2#(x1,y)
p4: app#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p5: app#(len_0(),y) -> len_1#(y)
p6: app#(sub_1(x1),y) -> sub_2#(x1,y)
p7: sub_2#(s_1(x),s_1(y)) -> sub_2#(x,y)
p8: gtr_2#(s_1(x),s_1(y)) -> gtr_2#(x,y)
p9: d_2#(s_1(x),s_1(y)) -> if_3#(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
p10: d_2#(s_1(x),s_1(y)) -> gtr_2#(x,y)
p11: d_2#(s_1(x),s_1(y)) -> d_2#(s_1(x),sub_2(y,x))
p12: d_2#(s_1(x),s_1(y)) -> sub_2#(y,x)
p13: len_1#(cons_2(x,xs)) -> len_1#(xs)
p14: filter_2#(p,cons_2(x,xs)) -> if_3#(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))
p15: filter_2#(p,cons_2(x,xs)) -> app#(p,x)
p16: filter_2#(p,cons_2(x,xs)) -> filter_2#(p,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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The estimated dependency graph contains the following SCCs:

  {p2, p15, p16}
  {p11}
  {p8}
  {p13}
  {p7}


-- Reduction pair.

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

p1: filter_2#(p,cons_2(x,xs)) -> app#(p,x)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(p,cons_2(x,xs)) -> filter_2#(p,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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r2, r3, r5, r7, r9, r10, r11, r12, r13, r14, r16, r17, r20, r21, r23, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 4
          cons_2_A(x1,x2) = x1 + x2 + 7
          app#_A(x1,x2) = x1 + x2
          filter_1_A(x1) = x1 + 5
          app_A(x1,x2) = x1 + x2 + 14
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          d_0_A() = 1
          d_1_A(x1) = x1 + 2
          filter_0_A() = 1
          gtr_0_A() = 1
          gtr_1_A(x1) = x1 + 2
          if_0_A() = 1
          if_1_A(x1) = x1 + 2
          if_2_A(x1,x2) = x1 + x2 + 3
          if_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          len_0_A() = 1
          len_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 4
          sub_0_A() = 1
          sub_1_A(x1) = x1 + 2
          true_0_A() = 5
          false_0_A() = 1
          gtr_2_A(x1,x2) = x2 + 8
          |0|_0_A() = 6
          d_2_A(x1,x2) = x2 + 3
          nil_0_A() = 7
  
    base order 2:
  
      precedence:
      
        if_3 = gtr_2 > filter_2# = cons_0 = d_0 = s_1 = d_2 > app# = app = s_0 > filter_1 = d_1 = gtr_1 = if_2 > filter_0 = gtr_0 = if_1 = sub_1 > cons_2 = if_0 = sub_0 = nil_0 > cons_1 = len_1 > len_0 = |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(filter_1) = [1]
      pi(app) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(d_0) = []
      pi(d_1) = [1]
      pi(filter_0) = []
      pi(gtr_0) = []
      pi(gtr_1) = [1]
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = []
      pi(if_3) = []
      pi(len_0) = []
      pi(len_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(sub_0) = []
      pi(sub_1) = [1]
      pi(true_0) = []
      pi(false_0) = []
      pi(gtr_2) = [2]
      pi(|0|_0) = []
      pi(d_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- 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#(p,cons_2(x,xs)) -> filter_2#(p,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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: filter_2#(p,cons_2(x,xs)) -> filter_2#(p,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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r5, r7, r9, r10, r14, r16, r17, r20, r21, r23, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 16
          app_A(x1,x2) = 10
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          gtr_0_A() = 10
          gtr_1_A(x1) = 9
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          sub_0_A() = 2
          sub_1_A(x1) = 1
          if_3_A(x1,x2,x3) = x1 + x2 + x3 + 11
          true_0_A() = 3
          false_0_A() = 1
          gtr_2_A(x1,x2) = x1 + x2 + 8
          |0|_0_A() = 2
          s_1_A(x1) = x1 + 4
          d_2_A(x1,x2) = x2 + 3
          len_1_A(x1) = x1 + 11
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        |0|_0 > len_1 > cons_0 = sub_0 = if_3 = s_1 > true_0 = gtr_2 > cons_2 = d_2 = nil_0 > filter_2# = false_0 > app = filter_0 = gtr_0 = if_0 = if_2 > cons_1 = filter_1 = gtr_1 = if_1 = sub_1
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(gtr_0) = []
      pi(gtr_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(sub_0) = []
      pi(sub_1) = []
      pi(if_3) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(gtr_2) = [1, 2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(d_2) = []
      pi(len_1) = [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: d_2#(s_1(x),s_1(y)) -> d_2#(s_1(x),sub_2(y,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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r12, r13, r16, r17, r18, r19, r21, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          d_2#_A(x1,x2) = x2 + 1
          s_1_A(x1) = x1 + 12
          sub_2_A(x1,x2) = x1 + 10
          app_A(x1,x2) = x2 + 13
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          d_0_A() = 4
          d_1_A(x1) = 3
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          gtr_0_A() = 2
          gtr_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          len_0_A() = 1
          len_1_A(x1) = x1
          s_0_A() = 1
          if_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          true_0_A() = 1
          false_0_A() = 4
          |0|_0_A() = 5
          gtr_2_A(x1,x2) = x2 + 3
          nil_0_A() = 6
          cons_2_A(x1,x2) = x1 + x2 + 14
  
    base order 2:
  
      precedence:
      
        len_1 = nil_0 > d_2# = |0|_0 > s_1 = sub_2 = d_0 = false_0 > d_1 = gtr_2 > app = cons_1 = gtr_0 = if_0 = len_0 = s_0 = true_0 = cons_2 > cons_0 = filter_1 = gtr_1 = if_1 = if_3 > filter_0 = if_2
      
  
    partial status:
  
      pi(d_2#) = []
      pi(s_1) = []
      pi(sub_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(d_0) = []
      pi(d_1) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(gtr_0) = []
      pi(gtr_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(len_0) = []
      pi(len_1) = [1]
      pi(s_0) = []
      pi(if_3) = [3]
      pi(true_0) = []
      pi(false_0) = []
      pi(|0|_0) = []
      pi(gtr_2) = [2]
      pi(nil_0) = []
      pi(cons_2) = [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: gtr_2#(s_1(x),s_1(y)) -> gtr_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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r12, r14, r16, r17, r20, r23, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gtr_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 6
          app_A(x1,x2) = x2 + 3
          cons_0_A() = 3
          cons_1_A(x1) = 2
          d_0_A() = 2
          d_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          gtr_0_A() = 1
          gtr_1_A(x1) = 2
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          len_0_A() = 1
          len_1_A(x1) = x1 + 2
          sub_0_A() = 2
          sub_1_A(x1) = 1
          if_3_A(x1,x2,x3) = x1 + x2 + x3
          true_0_A() = 5
          false_0_A() = 1
          gtr_2_A(x1,x2) = 4
          |0|_0_A() = 6
          d_2_A(x1,x2) = x2 + 2
          nil_0_A() = 7
          cons_2_A(x1,x2) = x1 + x2 + 9
  
    base order 2:
  
      precedence:
      
        if_3 > gtr_2 = nil_0 = cons_2 > gtr_2# = app = cons_0 = d_0 = filter_0 = gtr_1 = if_0 = sub_0 > cons_1 = filter_1 = gtr_0 = if_1 = sub_1 = |0|_0 = d_2 > s_1 = d_1 = if_2 = len_1 > len_0 = true_0 = false_0
      
  
    partial status:
  
      pi(gtr_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(d_0) = []
      pi(d_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(gtr_0) = []
      pi(gtr_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(len_0) = []
      pi(len_1) = [1]
      pi(sub_0) = []
      pi(sub_1) = []
      pi(if_3) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(gtr_2) = []
      pi(|0|_0) = []
      pi(d_2) = []
      pi(nil_0) = []
      pi(cons_2) = [1, 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: len_1#(cons_2(x,xs)) -> len_1#(xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r12, r13, r14, r16, r17, r20, r21, r23, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          len_1#_A(x1) = x1
          cons_2_A(x1,x2) = x1 + x2 + 9
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 8
          cons_1_A(x1) = 7
          d_0_A() = 1
          d_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          gtr_0_A() = 8
          gtr_1_A(x1) = 7
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          len_0_A() = 1
          len_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          sub_0_A() = 2
          sub_1_A(x1) = 1
          if_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          true_0_A() = 3
          false_0_A() = 1
          gtr_2_A(x1,x2) = x1 + x2 + 6
          |0|_0_A() = 4
          d_2_A(x1,x2) = x2 + 3
          nil_0_A() = 5
  
    base order 2:
  
      precedence:
      
        if_0 = nil_0 > if_1 = |0|_0 > app = gtr_0 = len_1 > gtr_1 > s_0 = sub_0 = gtr_2 > cons_0 = filter_0 = if_2 = len_0 = s_1 = sub_1 = if_3 = false_0 = d_2 > d_1 = true_0 > cons_2 = d_0 = filter_1 > len_1# = cons_1
      
  
    partial status:
  
      pi(len_1#) = [1]
      pi(cons_2) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(d_0) = []
      pi(d_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(gtr_0) = []
      pi(gtr_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(len_0) = []
      pi(len_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(sub_0) = []
      pi(sub_1) = []
      pi(if_3) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(gtr_2) = []
      pi(|0|_0) = []
      pi(d_2) = [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: sub_2#(s_1(x),s_1(y)) -> sub_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(d_0(),y) -> d_1(y)
r4: app(d_1(x1),y) -> d_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(gtr_0(),y) -> gtr_1(y)
r8: app(gtr_1(x1),y) -> gtr_2(x1,y)
r9: app(if_0(),y) -> if_1(y)
r10: app(if_1(x1),y) -> if_2(x1,y)
r11: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r12: app(len_0(),y) -> len_1(y)
r13: app(s_0(),y) -> s_1(y)
r14: app(sub_0(),y) -> sub_1(y)
r15: app(sub_1(x1),y) -> sub_2(x1,y)
r16: if_3(true_0(),xs,ys) -> xs
r17: if_3(false_0(),xs,ys) -> ys
r18: sub_2(x,|0|_0()) -> x
r19: sub_2(s_1(x),s_1(y)) -> sub_2(x,y)
r20: gtr_2(|0|_0(),y) -> false_0()
r21: gtr_2(s_1(x),|0|_0()) -> true_0()
r22: gtr_2(s_1(x),s_1(y)) -> gtr_2(x,y)
r23: d_2(x,|0|_0()) -> true_0()
r24: d_2(s_1(x),s_1(y)) -> if_3(gtr_2(x,y),false_0(),d_2(s_1(x),sub_2(y,x)))
r25: len_1(nil_0()) -> |0|_0()
r26: len_1(cons_2(x,xs)) -> s_1(len_1(xs))
r27: filter_2(p,nil_0()) -> nil_0()
r28: filter_2(p,cons_2(x,xs)) -> if_3(app(p,x),cons_2(x,filter_2(p,xs)),filter_2(p,xs))

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r12, r14, r16, r17, r20, r23, r25, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sub_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 6
          app_A(x1,x2) = x2 + 3
          cons_0_A() = 3
          cons_1_A(x1) = 2
          d_0_A() = 2
          d_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          gtr_0_A() = 1
          gtr_1_A(x1) = 2
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          len_0_A() = 1
          len_1_A(x1) = x1 + 2
          sub_0_A() = 2
          sub_1_A(x1) = 1
          if_3_A(x1,x2,x3) = x1 + x2 + x3
          true_0_A() = 5
          false_0_A() = 1
          gtr_2_A(x1,x2) = 4
          |0|_0_A() = 6
          d_2_A(x1,x2) = x2 + 2
          nil_0_A() = 7
          cons_2_A(x1,x2) = x1 + x2 + 9
  
    base order 2:
  
      precedence:
      
        if_3 > gtr_2 = nil_0 = cons_2 > sub_2# = app = cons_0 = d_0 = filter_0 = gtr_1 = if_0 = sub_0 > cons_1 = filter_1 = gtr_0 = if_1 = sub_1 = |0|_0 = d_2 > s_1 = d_1 = if_2 = len_1 > len_0 = true_0 = false_0
      
  
    partial status:
  
      pi(sub_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(d_0) = []
      pi(d_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(gtr_0) = []
      pi(gtr_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(len_0) = []
      pi(len_1) = [1]
      pi(sub_0) = []
      pi(sub_1) = []
      pi(if_3) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(gtr_2) = []
      pi(|0|_0) = []
      pi(d_2) = []
      pi(nil_0) = []
      pi(cons_2) = [1, 2]

The following rules are removed:

  p1

Then no dependency pair remains.