YES
We show the termination of the TRS R:

  app(perfectp(),|0|()) -> false()
  app(perfectp(),app(s(),x)) -> app(app(app(app(f(),x),app(s(),|0|())),app(s(),x)),app(s(),x))
  app(app(app(app(f(),|0|()),y),|0|()),u) -> true()
  app(app(app(app(f(),|0|()),y),app(s(),z)),u) -> false()
  app(app(app(app(f(),app(s(),x)),|0|()),z),u) -> app(app(app(app(f(),x),u),app(app(minus(),z),app(s(),x))),u)
  app(app(app(app(f(),app(s(),x)),app(s(),y)),z),u) -> app(app(app(if(),app(app(le(),x),y)),app(app(app(app(f(),app(s(),x)),app(app(minus(),y),x)),z),u)),app(app(app(app(f(),x),u),z),u))
  app(app(map(),|fun|),nil()) -> nil()
  app(app(map(),|fun|),app(app(cons(),x),xs)) -> app(app(cons(),app(|fun|,x)),app(app(map(),|fun|),xs))
  app(app(filter(),|fun|),nil()) -> nil()
  app(app(filter(),|fun|),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(|fun|,x)),|fun|),x),xs)
  app(app(app(app(filter2(),true()),|fun|),x),xs) -> app(app(cons(),x),app(app(filter(),|fun|),xs))
  app(app(app(app(filter2(),false()),|fun|),x),xs) -> app(app(filter(),|fun|),xs)

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(f_0(),y) -> f_1(y)
  app(f_1(x1),y) -> f_2(x1,y)
  app(f_2(x1,x2),y) -> f_3(x1,x2,y)
  app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filter2_0(),y) -> filter2_1(y)
  app(filter2_1(x1),y) -> filter2_2(x1,y)
  app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
  app(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(le_0(),y) -> le_1(y)
  app(le_1(x1),y) -> le_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(minus_0(),y) -> minus_1(y)
  app(minus_1(x1),y) -> minus_2(x1,y)
  app(perfectp_0(),y) -> perfectp_1(y)
  app(s_0(),y) -> s_1(y)
  perfectp_1(|0|_0()) -> false_0()
  perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
  f_4(|0|_0(),y,|0|_0(),u) -> true_0()
  f_4(|0|_0(),y,s_1(z),u) -> false_0()
  f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
  f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
  map_2(|fun|,nil_0()) -> nil_0()
  map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
  filter_2(|fun|,nil_0()) -> nil_0()
  filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
  filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
  filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)


-- SCC decomposition.

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

p1: app#(f_3(x1,x2,x3),y) -> f_4#(x1,x2,x3,y)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: app#(map_1(x1),y) -> map_2#(x1,y)
p5: app#(perfectp_0(),y) -> perfectp_1#(y)
p6: perfectp_1#(s_1(x)) -> f_4#(x,s_1(|0|_0()),s_1(x),s_1(x))
p7: f_4#(s_1(x),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),u)
p8: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)
p9: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p10: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p11: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p12: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p13: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p14: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p15: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: perfectp_1(|0|_0()) -> false_0()
r25: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r26: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r27: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r28: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r29: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r30: map_2(|fun|,nil_0()) -> nil_0()
r31: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r32: filter_2(|fun|,nil_0()) -> nil_0()
r33: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r34: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r35: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p10, p11, p12, p13, p14, p15}
  {p7, p9}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p7: app#(filter_1(x1),y) -> filter_2#(x1,y)
p8: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p9: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: perfectp_1(|0|_0()) -> false_0()
r25: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r26: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r27: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r28: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r29: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r30: map_2(|fun|,nil_0()) -> nil_0()
r31: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r32: filter_2(|fun|,nil_0()) -> nil_0()
r33: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r34: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r35: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x1 + x2
              cons_2_A(x1,x2) = x1 + x2 + 5
              app#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 3
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 3
              false_0_A() = 16
              filter_2#_A(x1,x2) = x1 + x2 + 3
              filter_1_A(x1) = x1 + 3
              app_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 11
              map_2#!_A(x1,x2) = 8
              cons_2!_A(x1,x2) = x1 + 8
              app#!_A(x1,x2) = x2 + 8
              map_1!_A(x1) = 1
              filter2_3!_A(x1,x2,x3) = x1 + x2 + x3 + 2
              filter2_4#!_A(x1,x2,x3,x4) = x4 + 3
              false_0!_A() = 9
              filter_2#!_A(x1,x2) = x2 + 3
              filter_1!_A(x1) = x1 + 15
              app!_A(x1,x2) = 14
              true_0!_A() = 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              map_2#_A(x1,x2) = x1 + x2
              cons_2_A(x1,x2) = x1 + x2 + 5
              app#_A(x1,x2) = x1 + x2
              map_1_A(x1) = x1 + 3
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
              filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 3
              false_0_A() = 13
              filter_2#_A(x1,x2) = x1 + x2 + 3
              filter_1_A(x1) = x1 + 3
              app_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 11
              map_2#!_A(x1,x2) = x2 + 8
              cons_2!_A(x1,x2) = x1 + x2 + 8
              app#!_A(x1,x2) = x2 + 8
              map_1!_A(x1) = 9
              filter2_3!_A(x1,x2,x3) = x1 + x2 + x3 + 17
              filter2_4#!_A(x1,x2,x3,x4) = x4 + 3
              false_0!_A() = 2
              filter_2#!_A(x1,x2) = x2 + 3
              filter_1!_A(x1) = x1 + 16
              app!_A(x1,x2) = 15
              true_0!_A() = 0
        
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app#) = [2]
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(app) = []
      pi(true_0) = []

The following rules are removed:

  p1, p3, p4, p6, p7, p8



-- SCC decomposition.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: perfectp_1(|0|_0()) -> false_0()
r25: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r26: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r27: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r28: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r29: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r30: map_2(|fun|,nil_0()) -> nil_0()
r31: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r32: filter_2(|fun|,nil_0()) -> nil_0()
r33: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r34: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r35: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: f_4#(s_1(x),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),u)
p2: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(f_3(x1,x2,x3),y) -> f_4(x1,x2,x3,y)
r7: app(filter_0(),y) -> filter_1(y)
r8: app(filter_1(x1),y) -> filter_2(x1,y)
r9: app(filter2_0(),y) -> filter2_1(y)
r10: app(filter2_1(x1),y) -> filter2_2(x1,y)
r11: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r12: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(le_0(),y) -> le_1(y)
r17: app(le_1(x1),y) -> le_2(x1,y)
r18: app(map_0(),y) -> map_1(y)
r19: app(map_1(x1),y) -> map_2(x1,y)
r20: app(minus_0(),y) -> minus_1(y)
r21: app(minus_1(x1),y) -> minus_2(x1,y)
r22: app(perfectp_0(),y) -> perfectp_1(y)
r23: app(s_0(),y) -> s_1(y)
r24: perfectp_1(|0|_0()) -> false_0()
r25: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r26: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r27: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r28: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r29: f_4(s_1(x),s_1(y),z,u) -> if_3(le_2(x,y),f_4(s_1(x),minus_2(y,x),z,u),f_4(x,u,z,u))
r30: map_2(|fun|,nil_0()) -> nil_0()
r31: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r32: filter_2(|fun|,nil_0()) -> nil_0()
r33: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r34: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r35: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f_4#_A(x1,x2,x3,x4) = x1
              s_1_A(x1) = x1 + 3
              |0|_0_A() = 1
              minus_2_A(x1,x2) = x1 + 2
              f_4#!_A(x1,x2,x3,x4) = x1 + 5
              s_1!_A(x1) = x1 + 8
              |0|_0!_A() = 4
              minus_2!_A(x1,x2) = x1 + 7
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              f_4#_A(x1,x2,x3,x4) = x1
              s_1_A(x1) = x1 + 3
              |0|_0_A() = 1
              minus_2_A(x1,x2) = x1 + 2
              f_4#!_A(x1,x2,x3,x4) = x1 + 7
              s_1!_A(x1) = 10
              |0|_0!_A() = 4
              minus_2!_A(x1,x2) = x2 + 11
        
  
    partial status:
  
      pi(f_4#) = [1]
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(minus_2) = [2]

The following rules are removed:

  p1, p2

Then no dependency pair remains.