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, r5, r8, r11, r12, r13, r14, r15, r16, r17, r18, r19

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:
          app#_A(x1,x2) = x1 + 3
          filter_1_A(x1) = x1 + 5
          filter_2#_A(x1,x2) = x1 + 4
          cons_2_A(x1,x2) = x1 + x2 + 5
          filtersub_2_A(x1,x2) = x1 + x2 + 5
          filtersub_3#_A(x1,x2,x3) = x2 + 4
          false_0_A() = 3
          app_A(x1,x2) = 3
          true_0_A() = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filtersub_0_A() = 2
          filtersub_1_A(x1) = 1
          neq_0_A() = 2
          neq_1_A(x1) = 1
          neq_2_A(x1,x2) = x1 + x2 + 4
          |0|_0_A() = 2
          s_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x2 + 6
          nil_0_A() = 1
          filtersub_3_A(x1,x2,x3) = x3 + 6
          nonzero_0_A() = 7
  
    base order 2:
  
      precedence:
      
        filter_2 = filtersub_3 > cons_2 = nil_0 > filter_2# = filtersub_3# = neq_0 = neq_2 = nonzero_0 > app# = filter_1 = filtersub_2 = false_0 = cons_0 = s_1 > app = cons_1 = filtersub_0 = neq_1 = |0|_0 > true_0 = filtersub_1
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(filtersub_2) = [2]
      pi(filtersub_3#) = [2]
      pi(false_0) = []
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(neq_2) = [2]
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []
      pi(filtersub_3) = [3]
      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)) -> app#(f,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,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: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p3: filtersub_3#(true_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#(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, r8, r10, r11, r12, r13, r14, r15, r19

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 + 6
          cons_2_A(x1,x2) = x1 + x2 + 4
          app#_A(x1,x2) = x1 + 5
          filtersub_2_A(x1,x2) = x1 + x2 + 7
          filtersub_3#_A(x1,x2,x3) = x2 + 6
          true_0_A() = 1
          app_A(x1,x2) = x2 + 3
          false_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filtersub_0_A() = 2
          filtersub_1_A(x1) = 1
          neq_0_A() = 2
          neq_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          neq_2_A(x1,x2) = x1 + x2 + 4
          |0|_0_A() = 2
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
          nonzero_0_A() = 3
  
    base order 2:
  
      precedence:
      
        filter_0 = filtersub_0 = neq_0 = neq_2 > s_1 > app# = app = s_0 = nonzero_0 > cons_2 = cons_1 = neq_1 = filter_2 > cons_0 = filter_1 = |0|_0 = nil_0 > filtersub_2 = false_0 > filter_2# = filtersub_3# = filtersub_1 > true_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(filtersub_2) = [2]
      pi(filtersub_3#) = []
      pi(true_0) = []
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(neq_0) = []
      pi(neq_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(neq_2) = [1]
      pi(|0|_0) = []
      pi(filter_2) = [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: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p2: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p3: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
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:

  {p2, p3, p4}


-- Reduction pair.

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

p1: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p2: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p3: 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, r14, r15, r19

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:
          filtersub_3#_A(x1,x2,x3) = x3 + 1
          true_0_A() = 1
          cons_2_A(x1,x2) = x1 + x2 + 4
          filter_2#_A(x1,x2) = x2 + 3
          app_A(x1,x2) = x2 + 6
          false_0_A() = 1
          cons_0_A() = 6
          cons_1_A(x1) = 5
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filtersub_0_A() = 3
          filtersub_1_A(x1) = 2
          filtersub_2_A(x1,x2) = 1
          neq_0_A() = 2
          neq_1_A(x1) = 1
          s_0_A() = 3
          s_1_A(x1) = x1 + 2
          neq_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 2
          filter_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 1
          nonzero_0_A() = 3
  
    base order 2:
  
      precedence:
      
        filtersub_3# = cons_0 > filter_2# = cons_1 = neq_0 = nonzero_0 > cons_2 = app = filter_0 = filtersub_0 = neq_1 = |0|_0 = filter_2 > filter_1 = filtersub_1 = s_0 = neq_2 = nil_0 > false_0 = filtersub_2 = s_1 > true_0
      
  
    partial status:
  
      pi(filtersub_3#) = [3]
      pi(true_0) = []
      pi(cons_2) = []
      pi(filter_2#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_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(neq_2) = [2]
      pi(|0|_0) = []
      pi(filter_2) = [1]
      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#(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}


-- 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)

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, r14, r15, r19

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
          filtersub_3#_A(x1,x2,x3) = x3
          app_A(x1,x2) = x2 + 7
          false_0_A() = 4
          cons_0_A() = 2
          cons_1_A(x1) = x1 + 1
          filter_0_A() = 1
          filter_1_A(x1) = 2
          filtersub_0_A() = 7
          filtersub_1_A(x1) = 6
          filtersub_2_A(x1,x2) = 5
          neq_0_A() = 6
          neq_1_A(x1) = 5
          s_0_A() = 2
          s_1_A(x1) = x1 + 1
          neq_2_A(x1,x2) = x1 + x2 + 6
          |0|_0_A() = 5
          true_0_A() = 4
          filter_2_A(x1,x2) = 3
          nil_0_A() = 2
          nonzero_0_A() = 6
  
    base order 2:
  
      precedence:
      
        neq_2 = nonzero_0 > s_1 = |0|_0 > false_0 = cons_0 = filtersub_0 = s_0 = true_0 > filter_2# = cons_2 > filtersub_3# = app = neq_0 > cons_1 = neq_1 = filter_2 > filter_1 = filtersub_1 = nil_0 > filter_0 = filtersub_2
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(filtersub_3#) = [3]
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_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) = [1]
      pi(neq_2) = [1, 2]
      pi(|0|_0) = []
      pi(true_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: 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:

  (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, r3, r5, r6, r8, r11, r12, r13, r14, r15, r19

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:
          neq_2#_A(x1,x2) = x1 + x2 + 5
          s_1_A(x1) = x1 + 4
          app_A(x1,x2) = 3
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filtersub_0_A() = 3
          filtersub_1_A(x1) = 2
          filtersub_2_A(x1,x2) = 1
          neq_0_A() = 2
          neq_1_A(x1) = 1
          neq_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 3
          false_0_A() = 1
          true_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
          nonzero_0_A() = 2
  
    base order 2:
  
      precedence:
      
        nonzero_0 > |0|_0 > neq_2# = true_0 > s_1 = cons_0 > app = cons_1 = filtersub_0 = neq_0 = filter_2 > filter_1 = filtersub_1 = neq_2 = nil_0 > filter_0 = filtersub_2 = neq_1 = false_0
      
  
    partial status:
  
      pi(neq_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      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) = [2]
      pi(nil_0) = []
      pi(nonzero_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.