YES
We show the termination of the TRS R:

  a(a(append(),nil()),ys) -> ys
  a(a(append(),a(a(cons(),x),xs)),ys) -> a(a(cons(),x),a(a(append(),xs),ys))
  a(a(filter(),f),nil()) -> nil()
  a(a(filter(),f),a(a(cons(),x),xs)) -> a(a(a(if(),a(f,x)),x),a(a(filter(),f),xs))
  a(a(le(),|0|()),y) -> true()
  a(a(le(),a(s(),x)),|0|()) -> false()
  a(a(le(),a(s(),x)),a(s(),y)) -> a(a(le(),x),y)
  a(a(a(if(),true()),x),xs) -> a(a(cons(),x),xs)
  a(a(a(if(),false()),x),xs) -> xs
  a(a(not(),f),b) -> a(not2(),a(f,b))
  a(not2(),true()) -> false()
  a(not2(),false()) -> true()
  a(qs(),nil()) -> nil()
  a(qs(),a(a(cons(),x),xs)) -> a(a(append(),a(qs(),a(a(filter(),a(le(),x)),xs))),a(a(cons(),x),a(qs(),a(a(filter(),a(not(),a(le(),x))),xs))))

Instead we prove termination of the uncurried system:

  a(append_0(),y) -> append_1(y)
  a(append_1(x1),y) -> append_2(x1,y)
  a(cons_0(),y) -> cons_1(y)
  a(cons_1(x1),y) -> cons_2(x1,y)
  a(filter_0(),y) -> filter_1(y)
  a(filter_1(x1),y) -> filter_2(x1,y)
  a(if_0(),y) -> if_1(y)
  a(if_1(x1),y) -> if_2(x1,y)
  a(if_2(x1,x2),y) -> if_3(x1,x2,y)
  a(le_0(),y) -> le_1(y)
  a(le_1(x1),y) -> le_2(x1,y)
  a(not_0(),y) -> not_1(y)
  a(not_1(x1),y) -> not_2(x1,y)
  a(not2_0(),y) -> not2_1(y)
  a(qs_0(),y) -> qs_1(y)
  a(s_0(),y) -> s_1(y)
  append_2(nil_0(),ys) -> ys
  append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
  le_2(|0|_0(),y) -> true_0()
  le_2(s_1(x),|0|_0()) -> false_0()
  le_2(s_1(x),s_1(y)) -> le_2(x,y)
  if_3(true_0(),x,xs) -> cons_2(x,xs)
  if_3(false_0(),x,xs) -> xs
  not_2(f,b) -> not2_1(a(f,b))
  not2_1(true_0()) -> false_0()
  not2_1(false_0()) -> true_0()
  qs_1(nil_0()) -> nil_0()
  qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))


-- SCC decomposition.

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

p1: a#(append_1(x1),y) -> append_2#(x1,y)
p2: a#(filter_1(x1),y) -> filter_2#(x1,y)
p3: a#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p4: a#(le_1(x1),y) -> le_2#(x1,y)
p5: a#(not_1(x1),y) -> not_2#(x1,y)
p6: a#(not2_0(),y) -> not2_1#(y)
p7: a#(qs_0(),y) -> qs_1#(y)
p8: append_2#(cons_2(x,xs),ys) -> append_2#(xs,ys)
p9: filter_2#(f,cons_2(x,xs)) -> if_3#(a(f,x),x,filter_2(f,xs))
p10: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p11: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p12: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p13: not_2#(f,b) -> not2_1#(a(f,b))
p14: not_2#(f,b) -> a#(f,b)
p15: qs_1#(cons_2(x,xs)) -> append_2#(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))
p16: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))
p17: qs_1#(cons_2(x,xs)) -> filter_2#(le_1(x),xs)
p18: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))
p19: qs_1#(cons_2(x,xs)) -> filter_2#(not_1(le_1(x)),xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The estimated dependency graph contains the following SCCs:

  {p2, p5, p7, p10, p11, p14, p16, p17, p18, p19}
  {p8}
  {p12}


-- Reduction pair.

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

p1: not_2#(f,b) -> a#(f,b)
p2: a#(qs_0(),y) -> qs_1#(y)
p3: qs_1#(cons_2(x,xs)) -> filter_2#(not_1(le_1(x)),xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p6: a#(not_1(x1),y) -> not_2#(x1,y)
p7: a#(filter_1(x1),y) -> filter_2#(x1,y)
p8: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))
p9: qs_1#(cons_2(x,xs)) -> filter_2#(le_1(x),xs)
p10: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r5, r7, r8, r10, r12, r14, r17, r18, r19, r20, r21, r22, r23, r24, r25, r27, r28

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:
          not_2#_A(x1,x2) = x1 + x2 + 23
          a#_A(x1,x2) = x1 + x2 + 22
          qs_0_A() = 1
          qs_1#_A(x1) = x1 + 6
          cons_2_A(x1,x2) = x1 + x2 + 21
          filter_2#_A(x1,x2) = x1 + x2 + 22
          not_1_A(x1) = x1 + 2
          le_1_A(x1) = x1 + 2
          filter_1_A(x1) = x1 + 8
          filter_2_A(x1,x2) = x1 + x2 + 10
          a_A(x1,x2) = x2 + 9
          append_0_A() = 1
          append_1_A(x1) = 2
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          le_0_A() = 1
          not_0_A() = 1
          not2_0_A() = 4
          not2_1_A(x1) = 3
          append_2_A(x1,x2) = x1 + x2 + 3
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 21
          le_2_A(x1,x2) = x1 + x2 + 11
          |0|_0_A() = 3
          true_0_A() = 1
          s_1_A(x1) = x1 + 10
          false_0_A() = 2
  
    base order 2:
  
      precedence:
      
        if_0 = nil_0 > a# = if_1 > filter_2# = le_1 = filter_2 = if_2 = append_2 > cons_2 = if_3 = le_2 > qs_1# = append_0 = s_1 > not_2# = qs_0 = a = append_1 = cons_0 = le_0 = not2_0 = |0|_0 > not_1 = filter_1 = cons_1 = not2_1 = false_0 > filter_0 = not_0 = true_0
      
  
    partial status:
  
      pi(not_2#) = []
      pi(a#) = [2]
      pi(qs_0) = []
      pi(qs_1#) = [1]
      pi(cons_2) = []
      pi(filter_2#) = [1]
      pi(not_1) = [1]
      pi(le_1) = [1]
      pi(filter_1) = [1]
      pi(filter_2) = []
      pi(a) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(append_2) = [1]
      pi(nil_0) = []
      pi(if_3) = []
      pi(le_2) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: a#(qs_0(),y) -> qs_1#(y)
p2: qs_1#(cons_2(x,xs)) -> filter_2#(not_1(le_1(x)),xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p5: a#(not_1(x1),y) -> not_2#(x1,y)
p6: a#(filter_1(x1),y) -> filter_2#(x1,y)
p7: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))
p8: qs_1#(cons_2(x,xs)) -> filter_2#(le_1(x),xs)
p9: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p6, p7, p8, p9}


-- Reduction pair.

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

p1: a#(qs_0(),y) -> qs_1#(y)
p2: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))
p3: qs_1#(cons_2(x,xs)) -> filter_2#(le_1(x),xs)
p4: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p5: a#(filter_1(x1),y) -> filter_2#(x1,y)
p6: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p7: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))
p8: qs_1#(cons_2(x,xs)) -> filter_2#(not_1(le_1(x)),xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r5, r7, r8, r10, r12, r14, r16, r17, r19, r20, r21, r22, r23, r24, r25, r27, r28

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:
          a#_A(x1,x2) = x1 + x2 + 18
          qs_0_A() = 1
          qs_1#_A(x1) = x1 + 2
          cons_2_A(x1,x2) = x1 + x2 + 17
          filter_2_A(x1,x2) = x1 + x2 + 7
          le_1_A(x1) = x1 + 2
          filter_2#_A(x1,x2) = x1 + x2 + 9
          filter_1_A(x1) = x1 + 5
          not_1_A(x1) = x1 + 5
          a_A(x1,x2) = x2 + 6
          append_0_A() = 2
          append_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          le_0_A() = 1
          not_0_A() = 1
          not2_0_A() = 1
          not2_1_A(x1) = 3
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          append_2_A(x1,x2) = x1 + x2 + 18
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 17
          le_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 3
          true_0_A() = 2
          false_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > a# = if_3 > cons_2 = append_0 = if_0 = not2_0 = le_2 = |0|_0 > qs_1# > qs_0 = filter_2# = not_1 = append_2 = nil_0 > a = cons_0 = not_0 > le_1 = filter_1 = append_1 = cons_1 = if_1 = if_2 = not2_1 = s_1 > filter_0 = le_0 = s_0 = true_0 > false_0
      
  
    partial status:
  
      pi(a#) = [2]
      pi(qs_0) = []
      pi(qs_1#) = [1]
      pi(cons_2) = []
      pi(filter_2) = [1]
      pi(le_1) = [1]
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(not_1) = [1]
      pi(a) = [2]
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(not_0) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(append_2) = [1]
      pi(nil_0) = []
      pi(if_3) = [3]
      pi(le_2) = [2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))
p2: qs_1#(cons_2(x,xs)) -> filter_2#(le_1(x),xs)
p3: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p4: a#(filter_1(x1),y) -> filter_2#(x1,y)
p5: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)
p6: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))
p7: qs_1#(cons_2(x,xs)) -> filter_2#(not_1(le_1(x)),xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The estimated dependency graph contains the following SCCs:

  {p1, p6}
  {p3, p4, p5}


-- Reduction pair.

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

p1: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))
p2: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(not_1(le_1(x)),xs))

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r5, r7, r8, r11, r14, r17, r18, r19, r20, r21, r22, r23, r24, r25, r27, r28

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:
          qs_1#_A(x1) = x1
          cons_2_A(x1,x2) = x1 + x2 + 13
          filter_2_A(x1,x2) = x1 + x2 + 2
          le_1_A(x1) = x1 + 5
          not_1_A(x1) = x1 + 5
          a_A(x1,x2) = 4
          append_0_A() = 2
          append_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          le_2_A(x1,x2) = 3
          not2_0_A() = 4
          not2_1_A(x1) = 3
          append_2_A(x1,x2) = x1 + x2 + 14
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 13
          |0|_0_A() = 3
          true_0_A() = 1
          s_1_A(x1) = x1 + 5
          false_0_A() = 2
  
    base order 2:
  
      precedence:
      
        qs_1# = le_1 > not_1 = append_2 > cons_0 = filter_0 = if_0 = not2_0 > filter_2 = not2_1 = if_3 = |0|_0 = s_1 > cons_2 = nil_0 = false_0 > a = append_0 = le_2 = true_0 > append_1 = cons_1 = filter_1 = if_1 = if_2
      
  
    partial status:
  
      pi(qs_1#) = [1]
      pi(cons_2) = [1]
      pi(filter_2) = []
      pi(le_1) = [1]
      pi(not_1) = []
      pi(a) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_2) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(append_2) = [1, 2]
      pi(nil_0) = []
      pi(if_3) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: qs_1#(cons_2(x,xs)) -> qs_1#(filter_2(le_1(x),xs))

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r5, r7, r8, r14, r17, r18, r19, r20, r21, r22, r23, r24, r25, r27, r28

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:
          qs_1#_A(x1) = x1
          cons_2_A(x1,x2) = x1 + x2 + 12
          filter_2_A(x1,x2) = x2 + 5
          le_1_A(x1) = 6
          a_A(x1,x2) = 4
          append_0_A() = 1
          append_1_A(x1) = 2
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not2_0_A() = 4
          not2_1_A(x1) = 3
          append_2_A(x1,x2) = x1 + x2 + 13
          nil_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 12
          le_2_A(x1,x2) = x2 + 5
          |0|_0_A() = 3
          true_0_A() = 2
          s_1_A(x1) = x1 + 6
          false_0_A() = 1
  
    base order 2:
  
      precedence:
      
        nil_0 > filter_2 > not2_0 = append_2 = if_3 > cons_2 = not2_1 > le_1 = |0|_0 > qs_1# = append_1 = le_2 = true_0 = s_1 > a = append_0 = cons_0 = false_0 > cons_1 = filter_1 = if_1 > filter_0 = if_0 = if_2
      
  
    partial status:
  
      pi(qs_1#) = [1]
      pi(cons_2) = [1]
      pi(filter_2) = []
      pi(le_1) = []
      pi(a) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(append_2) = []
      pi(nil_0) = []
      pi(if_3) = [2]
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_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: filter_2#(f,cons_2(x,xs)) -> a#(f,x)
p2: a#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r7, r8, r14, r17, r18, r21, r22, r23, r24, r25, r27, r28, r29

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 + 27
          cons_2_A(x1,x2) = x1 + x2 + 26
          a#_A(x1,x2) = x1 + x2 + 22
          filter_1_A(x1) = x1 + 6
          a_A(x1,x2) = 5
          append_0_A() = 1
          append_1_A(x1) = 2
          cons_0_A() = 2
          cons_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          not2_0_A() = 4
          not2_1_A(x1) = 3
          append_2_A(x1,x2) = x1 + x2 + 27
          nil_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 3
          true_0_A() = 1
          s_1_A(x1) = x1 + 6
          false_0_A() = 2
          if_3_A(x1,x2,x3) = x2 + x3 + 26
          qs_1_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        a# > filter_2# = if_0 = append_2 = if_3 = qs_1 > cons_2 = if_1 = nil_0 = le_2 = s_1 > filter_1 = if_2 > a = append_0 = cons_0 = not2_0 > append_1 = cons_1 > not2_1 = |0|_0 > false_0 > true_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(a#) = [2]
      pi(filter_1) = [1]
      pi(a) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(append_2) = [1]
      pi(nil_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []
      pi(if_3) = []
      pi(qs_1) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: a#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),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#(f,cons_2(x,xs)) -> filter_2#(f,xs)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r10, r12, r14, r16, r17, r18, r21, r22, r23, r24, r25, r27, r28, r29

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) = x2 + 2
          a_A(x1,x2) = x2 + 6
          append_0_A() = 2
          append_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 2
          if_1_A(x1) = 1
          if_2_A(x1,x2) = 2
          if_3_A(x1,x2,x3) = x3 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 5
          not_0_A() = 1
          not_1_A(x1) = x1 + 2
          not2_0_A() = 1
          not2_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          append_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 1
          le_2_A(x1,x2) = x1 + 2
          |0|_0_A() = 3
          true_0_A() = 1
          false_0_A() = 2
          qs_1_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        cons_0 = le_1 > cons_1 = if_2 = not_1 = s_1 = append_2 = le_2 > cons_2 = a = append_0 = filter_0 = if_0 = le_0 = not_0 = s_0 = nil_0 = qs_1 > filter_2# = append_1 = filter_1 = if_1 = if_3 = not2_1 = |0|_0 > not2_0 = false_0 > true_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(a) = [2]
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = [3]
      pi(le_0) = []
      pi(le_1) = []
      pi(not_0) = []
      pi(not_1) = [1]
      pi(not2_0) = []
      pi(not2_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(append_2) = []
      pi(nil_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(qs_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: append_2#(cons_2(x,xs),ys) -> append_2#(xs,ys)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r5, r8, r14, r17, r18, r21, r22, r23, r24, r25, r27, r28, r29

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:
          append_2#_A(x1,x2) = x1 + x2 + 4
          cons_2_A(x1,x2) = x2 + 3
          a_A(x1,x2) = 5
          append_0_A() = 0
          append_1_A(x1) = 1
          cons_0_A() = 0
          cons_1_A(x1) = 0
          filter_0_A() = 0
          filter_1_A(x1) = 0
          if_1_A(x1) = x1 + 6
          if_2_A(x1,x2) = 0
          not2_0_A() = 0
          not2_1_A(x1) = 3
          append_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 11
          |0|_0_A() = 3
          true_0_A() = 2
          s_1_A(x1) = x1 + 6
          false_0_A() = 1
          if_3_A(x1,x2,x3) = x2 + x3 + 4
          qs_1_A(x1) = x1 + 6
  
    base order 2:
  
      precedence:
      
        cons_0 = if_1 = le_2 > append_2# = cons_1 = if_2 = append_2 = nil_0 > cons_2 = a = filter_0 = |0|_0 = qs_1 > append_1 = filter_1 = true_0 = if_3 > append_0 = false_0 > not2_1 = s_1 > not2_0
      
  
    partial status:
  
      pi(append_2#) = [1, 2]
      pi(cons_2) = []
      pi(a) = []
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_1) = [1]
      pi(if_2) = []
      pi(not2_0) = []
      pi(not2_1) = []
      pi(append_2) = [1]
      pi(nil_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = [1]
      pi(false_0) = []
      pi(if_3) = [3]
      pi(qs_1) = [1]

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: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)

and R consists of:

r1: a(append_0(),y) -> append_1(y)
r2: a(append_1(x1),y) -> append_2(x1,y)
r3: a(cons_0(),y) -> cons_1(y)
r4: a(cons_1(x1),y) -> cons_2(x1,y)
r5: a(filter_0(),y) -> filter_1(y)
r6: a(filter_1(x1),y) -> filter_2(x1,y)
r7: a(if_0(),y) -> if_1(y)
r8: a(if_1(x1),y) -> if_2(x1,y)
r9: a(if_2(x1,x2),y) -> if_3(x1,x2,y)
r10: a(le_0(),y) -> le_1(y)
r11: a(le_1(x1),y) -> le_2(x1,y)
r12: a(not_0(),y) -> not_1(y)
r13: a(not_1(x1),y) -> not_2(x1,y)
r14: a(not2_0(),y) -> not2_1(y)
r15: a(qs_0(),y) -> qs_1(y)
r16: a(s_0(),y) -> s_1(y)
r17: append_2(nil_0(),ys) -> ys
r18: append_2(cons_2(x,xs),ys) -> cons_2(x,append_2(xs,ys))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> if_3(a(f,x),x,filter_2(f,xs))
r21: le_2(|0|_0(),y) -> true_0()
r22: le_2(s_1(x),|0|_0()) -> false_0()
r23: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r24: if_3(true_0(),x,xs) -> cons_2(x,xs)
r25: if_3(false_0(),x,xs) -> xs
r26: not_2(f,b) -> not2_1(a(f,b))
r27: not2_1(true_0()) -> false_0()
r28: not2_1(false_0()) -> true_0()
r29: qs_1(nil_0()) -> nil_0()
r30: qs_1(cons_2(x,xs)) -> append_2(qs_1(filter_2(le_1(x),xs)),cons_2(x,qs_1(filter_2(not_1(le_1(x)),xs))))

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r10, r12, r14, r16, r17, r18, r21, r22, r23, r24, r25, r27, r28, r29

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:
          le_2#_A(x1,x2) = x1 + x2 + 4
          s_1_A(x1) = x1 + 3
          a_A(x1,x2) = x2 + 6
          append_0_A() = 2
          append_1_A(x1) = 1
          cons_0_A() = 4
          cons_1_A(x1) = 3
          cons_2_A(x1,x2) = x2 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          if_0_A() = 1
          if_1_A(x1) = x1 + 2
          if_2_A(x1,x2) = 1
          if_3_A(x1,x2,x3) = x3 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 5
          not_0_A() = 1
          not_1_A(x1) = x1 + 2
          not2_0_A() = 1
          not2_1_A(x1) = x1 + 2
          s_0_A() = 1
          append_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 3
          true_0_A() = 1
          false_0_A() = 2
          qs_1_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        le_2# > s_1 = le_2 > cons_0 = le_1 > a = append_0 = cons_1 = cons_2 = filter_0 = not_1 = append_2 = qs_1 > append_1 = filter_1 = if_1 = if_3 = le_0 = not_0 = not2_1 = s_0 = nil_0 = |0|_0 > if_0 = if_2 = not2_0 = false_0 > true_0
      
  
    partial status:
  
      pi(le_2#) = [1]
      pi(s_1) = [1]
      pi(a) = [2]
      pi(append_0) = []
      pi(append_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = []
      pi(if_3) = [3]
      pi(le_0) = []
      pi(le_1) = []
      pi(not_0) = []
      pi(not_1) = []
      pi(not2_0) = []
      pi(not2_1) = [1]
      pi(s_0) = []
      pi(append_2) = []
      pi(nil_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(qs_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.