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

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35

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) = 0
              cons_2_A(x1,x2) = 1
              app#_A(x1,x2) = 0
              map_1_A(x1) = x1 + 3
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 5
              filter2_4#_A(x1,x2,x3,x4) = 0
              false_0_A() = 1
              filter_2#_A(x1,x2) = 0
              filter_1_A(x1) = x1 + 5
              app_A(x1,x2) = x1 + x2 + 2
              true_0_A() = 1
              cons_0_A() = 1
              cons_1_A(x1) = x1 + 1
              f_0_A() = 4
              f_1_A(x1) = x1 + 6
              f_2_A(x1,x2) = x1 + x2 + 8
              f_3_A(x1,x2,x3) = x1 + x2 + x3 + 7
              f_4_A(x1,x2,x3,x4) = x4 + 1
              filter_0_A() = 3
              filter_2_A(x1,x2) = x1 + 1
              filter2_0_A() = 1
              filter2_1_A(x1) = x1 + 3
              filter2_2_A(x1,x2) = x1 + x2 + 4
              filter2_4_A(x1,x2,x3,x4) = x2 + 1
              if_0_A() = 5
              if_1_A(x1) = x1 + 7
              if_2_A(x1,x2) = x1 + x2 + 6
              if_3_A(x1,x2,x3) = x2
              le_0_A() = 1
              le_1_A(x1) = x1 + 1
              le_2_A(x1,x2) = 1
              map_0_A() = 1
              map_2_A(x1,x2) = 1
              minus_0_A() = 7
              minus_1_A(x1) = x1 + 6
              minus_2_A(x1,x2) = x1 + 1
              perfectp_0_A() = 10
              perfectp_1_A(x1) = 3
              s_0_A() = 9
              s_1_A(x1) = 2
              |0|_0_A() = 1
              nil_0_A() = 1
              map_2#!_A(x1,x2) = x1 + 3
              cons_2!_A(x1,x2) = 2
              app#!_A(x1,x2) = x1 + 1
              map_1!_A(x1) = 4
              filter2_3!_A(x1,x2,x3) = 7
              filter2_4#!_A(x1,x2,x3,x4) = x2 + 5
              false_0!_A() = 9
              filter_2#!_A(x1,x2) = x1 + 5
              filter_1!_A(x1) = 4
              app!_A(x1,x2) = x1 + x2 + 4
              true_0!_A() = 1
              cons_0!_A() = 0
              cons_1!_A(x1) = 1
              f_0!_A() = 0
              f_1!_A(x1) = 1
              f_2!_A(x1,x2) = 2
              f_3!_A(x1,x2,x3) = x1 + x2 + 11
              f_4!_A(x1,x2,x3,x4) = 10
              filter_0!_A() = 0
              filter_2!_A(x1,x2) = x1 + 8
              filter2_0!_A() = 0
              filter2_1!_A(x1) = 5
              filter2_2!_A(x1,x2) = 6
              filter2_4!_A(x1,x2,x3,x4) = x2 + 8
              if_0!_A() = 0
              if_1!_A(x1) = 1
              if_2!_A(x1,x2) = 10
              if_3!_A(x1,x2,x3) = 9
              le_0!_A() = 2
              le_1!_A(x1) = x1 + 1
              le_2!_A(x1,x2) = 0
              map_0!_A() = 5
              map_2!_A(x1,x2) = x1 + 3
              minus_0!_A() = 0
              minus_1!_A(x1) = 10
              minus_2!_A(x1,x2) = 9
              perfectp_0!_A() = 14
              perfectp_1!_A(x1) = 13
              s_0!_A() = 11
              s_1!_A(x1) = 12
              |0|_0!_A() = 11
              nil_0!_A() = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{55, x1 - 79, x2 - 109}
            cons_2_A(x1,x2) = max{108, x1 + 55, x2}
            app#_A(x1,x2) = max{55, x1 - 134, x2 - 54}
            map_1_A(x1) = x1 + 55
            filter2_3_A(x1,x2,x3) = max{x1 - 278, x2, x3 + 133}
            filter2_4#_A(x1,x2,x3,x4) = max{55, x1 - 412, x2 - 134, x3 - 1, x4 - 54}
            false_0_A = 0
            filter_2#_A(x1,x2) = max{55, x1 - 134, x2 - 54}
            filter_1_A(x1) = max{77, x1 + 58}
            app_A(x1,x2) = max{196, x1 - 78, x2 + 143}
            true_0_A = 144
            cons_0_A = 0
            cons_1_A(x1) = x1 + 133
            f_0_A = 0
            f_1_A(x1) = max{79, x1 + 77}
            f_2_A(x1,x2) = max{82, x1 - 1, x2 + 6}
            f_3_A(x1,x2,x3) = max{77, x1 - 79, x2 - 72, x3 - 216}
            f_4_A(x1,x2,x3,x4) = max{144, x1 - 305, x2 - 465, x3 - 356, x4 - 429}
            filter_0_A = 0
            filter_2_A(x1,x2) = max{196, x1 - 295, x2 + 1}
            filter2_0_A = 0
            filter2_1_A(x1) = max{0, x1 - 81}
            filter2_2_A(x1,x2) = max{x1 - 159, x2 + 79}
            filter2_4_A(x1,x2,x3,x4) = max{196, x1 - 356, x2 - 295, x3 + 55, x4 + 1}
            if_0_A = 77
            if_1_A(x1) = max{77, x1 - 81}
            if_2_A(x1,x2) = max{77, x1 - 226, x2 - 81}
            if_3_A(x1,x2,x3) = max{144, x1 - 304, x2 - 159, x3 - 145}
            le_0_A = 0
            le_1_A(x1) = max{77, x1 - 81}
            le_2_A(x1,x2) = max{144, x1 - 159, x2 - 91}
            map_0_A = 76
            map_2_A(x1,x2) = max{196, x1 - 23, x2 + 143}
            minus_0_A = 0
            minus_1_A(x1) = max{77, x1 - 82}
            minus_2_A(x1,x2) = max{196, x1 - 160, x2 - 9}
            perfectp_0_A = 0
            perfectp_1_A(x1) = max{144, x1 - 356}
            s_0_A = 74
            s_1_A(x1) = max{144, x1 + 70}
            |0|_0_A = 73
            nil_0_A = 197
            map_2#!_A(x1,x2) = x2 + 157
            cons_2!_A(x1,x2) = max{163, x1 + 19, x2 + 19}
            app#!_A(x1,x2) = max{2, x1 - 56}
            map_1!_A(x1) = 155
            filter2_3!_A(x1,x2,x3) = max{81, x2 - 276, x3 + 21}
            filter2_4#!_A(x1,x2,x3,x4) = max{20, x2 + 1}
            false_0!_A = 79
            filter_2#!_A(x1,x2) = max{20, x1 + 1}
            filter_1!_A(x1) = max{155, x1 - 276}
            app!_A(x1,x2) = max{271, x1 + 80, x2 + 154}
            true_0!_A = 217
            cons_0!_A = 156
            cons_1!_A(x1) = max{155, x1 - 1}
            f_0!_A = 1
            f_1!_A(x1) = max{272, x1 - 3}
            f_2!_A(x1,x2) = x2 + 162
            f_3!_A(x1,x2,x3) = max{161, x1 - 1, x3 - 136}
            f_4!_A(x1,x2,x3,x4) = max{77, x2 + 8, x4 + 8}
            filter_0!_A = 156
            filter_2!_A(x1,x2) = max{216, x1 - 275, x2 + 162}
            filter2_0!_A = 272
            filter2_1!_A(x1) = max{271, x1 - 3}
            filter2_2!_A(x1,x2) = max{0, x1 - 2, x2 - 277}
            filter2_4!_A(x1,x2,x3,x4) = max{216, x1 - 197, x2 - 275, x3 + 214, x4 + 162}
            if_0!_A = 155
            if_1!_A(x1) = max{156, x1 - 1}
            if_2!_A(x1,x2) = max{270, x1 - 146, x2 - 146}
            if_3!_A(x1,x2,x3) = max{272, x1 - 145, x2 - 145, x3 - 145}
            le_0!_A = 1
            le_1!_A(x1) = max{272, x1 - 1}
            le_2!_A(x1,x2) = max{151, x1 - 2}
            map_0!_A = 156
            map_2!_A(x1,x2) = max{164, x1 - 60, x2 - 198}
            minus_0!_A = 82
            minus_1!_A(x1) = max{81, x1 - 1}
            minus_2!_A(x1,x2) = max{271, x1 - 2, x2 - 143}
            perfectp_0!_A = 79
            perfectp_1!_A(x1) = max{78, x1 + 6}
            s_0!_A = 1
            s_1!_A(x1) = max{153, x1 + 2}
            |0|_0!_A = 152
            nil_0!_A = 164
        
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(f_4) = []
      pi(filter_0) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_4) = []
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = []
      pi(if_3) = []
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(map_0) = []
      pi(map_2) = [2]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(perfectp_0) = []
      pi(perfectp_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p4, p6, p7



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p2: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: 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:

  {p1}
  {p2, p3, p4}


-- Reduction pair.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> map_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) = x2
              cons_2_A(x1,x2) = x1 + x2 + 5
              map_2#!_A(x1,x2) = x2 + 9
              cons_2!_A(x1,x2) = x1 + 8
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = 0
            cons_2_A(x1,x2) = max{8, x1 - 10, x2 - 8}
            map_2#!_A(x1,x2) = max{9, x1 - 1, x2 - 9}
            cons_2!_A(x1,x2) = max{8, x2 + 3}
        
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_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: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,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 set of usable rules consists of

  r3, r4, r7, r10, r11, r13, r14, r15, r16, r17, r18, r20, r24, r27, r30, r32

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:
              filter2_4#_A(x1,x2,x3,x4) = x4
              false_0_A() = 1
              filter_2#_A(x1,x2) = x2
              cons_2_A(x1,x2) = x2 + 8
              app_A(x1,x2) = x1 + x2 + 1
              true_0_A() = 1
              f_0_A() = 1
              f_1_A(x1) = x1 + 1
              f_2_A(x1,x2) = x1 + x2 + 1
              filter_0_A() = 1
              filter_1_A(x1) = x1 + 1
              filter2_1_A(x1) = x1 + 6
              filter2_2_A(x1,x2) = x1 + x2 + 7
              filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
              if_0_A() = 1
              if_1_A(x1) = x1 + 1
              if_2_A(x1,x2) = x1 + x2 + 1
              if_3_A(x1,x2,x3) = x2
              le_0_A() = 1
              le_1_A(x1) = x1 + 1
              le_2_A(x1,x2) = 1
              map_0_A() = 1
              map_1_A(x1) = 1
              minus_0_A() = 1
              minus_1_A(x1) = x1 + 1
              perfectp_1_A(x1) = 9
              |0|_0_A() = 4
              f_4_A(x1,x2,x3,x4) = x3 + x4 + 1
              s_1_A(x1) = 4
              map_2_A(x1,x2) = x2 + 1
              nil_0_A() = 1
              filter_2_A(x1,x2) = x1 + x2 + 1
              filter2_4#!_A(x1,x2,x3,x4) = x4 + 11
              false_0!_A() = 5
              filter_2#!_A(x1,x2) = x2 + 4
              cons_2!_A(x1,x2) = 12
              app!_A(x1,x2) = x2 + 11
              true_0!_A() = 7
              f_0!_A() = 0
              f_1!_A(x1) = 1
              f_2!_A(x1,x2) = 0
              filter_0!_A() = 1
              filter_1!_A(x1) = 0
              filter2_1!_A(x1) = 1
              filter2_2!_A(x1,x2) = 0
              filter2_3!_A(x1,x2,x3) = 1
              if_0!_A() = 0
              if_1!_A(x1) = 1
              if_2!_A(x1,x2) = 2
              if_3!_A(x1,x2,x3) = 3
              le_0!_A() = 0
              le_1!_A(x1) = 11
              le_2!_A(x1,x2) = 5
              map_0!_A() = 0
              map_1!_A(x1) = 1
              minus_0!_A() = 10
              minus_1!_A(x1) = 10
              perfectp_1!_A(x1) = 9
              |0|_0!_A() = 8
              f_4!_A(x1,x2,x3,x4) = x2 + 6
              s_1!_A(x1) = 4
              map_2!_A(x1,x2) = 13
              nil_0!_A() = 0
              filter_2!_A(x1,x2) = 6
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 1, x3 - 43, x4 - 1}
            false_0_A = 27
            filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 7}
            cons_2_A(x1,x2) = max{x1 - 36, x2 + 6}
            app_A(x1,x2) = max{27, x1 - 16, x2 - 16}
            true_0_A = 27
            f_0_A = 0
            f_1_A(x1) = max{0, x1 - 16}
            f_2_A(x1,x2) = max{0, x1 - 32, x2 - 16}
            filter_0_A = 2
            filter_1_A(x1) = max{0, x1 - 16}
            filter2_1_A(x1) = max{15, x1 + 13}
            filter2_2_A(x1,x2) = max{0, x1 - 28, x2 - 16}
            filter2_3_A(x1,x2,x3) = max{0, x1 - 44, x2 - 32, x3 - 16}
            if_0_A = 0
            if_1_A(x1) = max{0, x1 - 34}
            if_2_A(x1,x2) = max{0, x1 - 50, x2 - 16}
            if_3_A(x1,x2,x3) = max{27, x1 - 66, x2 - 32, x3 - 93}
            le_0_A = 0
            le_1_A(x1) = max{0, x1 - 16}
            le_2_A(x1,x2) = max{0, x1 - 193, x2 - 145}
            map_0_A = 10
            map_1_A(x1) = max{0, x1 - 62}
            minus_0_A = 0
            minus_1_A(x1) = max{2, x1 - 131}
            perfectp_1_A(x1) = max{27, x1 - 148}
            |0|_0_A = 9
            f_4_A(x1,x2,x3,x4) = max{27, x1 - 235, x2 - 187, x3 - 148, x4 - 178}
            s_1_A(x1) = max{120, x1 - 24}
            map_2_A(x1,x2) = max{x1 - 78, x2 + 3}
            nil_0_A = 11
            filter_2_A(x1,x2) = max{9, x1 - 54, x2 + 2}
            filter2_4#!_A(x1,x2,x3,x4) = max{12, x2 - 1, x3 - 1, x4 + 2}
            false_0!_A = 3
            filter_2#!_A(x1,x2) = max{11, x2 - 5}
            cons_2!_A(x1,x2) = max{10, x1 - 45}
            app!_A(x1,x2) = max{12, x1 - 1, x2 + 6}
            true_0!_A = 129
            f_0!_A = 4
            f_1!_A(x1) = max{5, x1 - 36}
            f_2!_A(x1,x2) = max{130, x1 - 35, x2 + 7}
            filter_0!_A = 1
            filter_1!_A(x1) = max{0, x1 - 1}
            filter2_1!_A(x1) = max{14, x1 + 7}
            filter2_2!_A(x1,x2) = 13
            filter2_3!_A(x1,x2,x3) = max{14, x1 - 29, x3 - 1}
            if_0!_A = 13
            if_1!_A(x1) = max{12, x1 - 133}
            if_2!_A(x1,x2) = max{13, x1 - 132, x2 - 48}
            if_3!_A(x1,x2,x3) = max{4, x1 - 131, x2 - 47, x3 - 158}
            le_0!_A = 13
            le_1!_A(x1) = max{12, x1 - 1}
            le_2!_A(x1,x2) = max{129, x1 - 18, x2 - 1}
            map_0!_A = 11
            map_1!_A(x1) = max{10, x1 - 63}
            minus_0!_A = 12
            minus_1!_A(x1) = 12
            perfectp_1!_A(x1) = x1 + 155
            |0|_0!_A = 130
            f_4!_A(x1,x2,x3,x4) = max{128, x1 + 5, x2 - 106, x3 + 28, x4 + 126}
            s_1!_A(x1) = x1 + 5
            map_2!_A(x1,x2) = max{11, x1 - 62, x2 - 26}
            nil_0!_A = 1
            filter_2!_A(x1,x2) = max{2, x2 - 9}
        
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(true_0) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(f_4) = []
      pi(s_1) = []
      pi(map_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- 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 + 1
              f_4#!_A(x1,x2,x3,x4) = x1
              s_1!_A(x1) = 4
              |0|_0!_A() = 10
              minus_2!_A(x1,x2) = 8
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            f_4#_A(x1,x2,x3,x4) = max{0, x1 - 29, x2 - 59, x3 - 1, x4 - 10}
            s_1_A(x1) = max{20, x1 + 6}
            |0|_0_A = 49
            minus_2_A(x1,x2) = max{0, x1 - 8, x2 - 28}
            f_4#!_A(x1,x2,x3,x4) = max{1, x1 - 29, x2 - 50, x4 - 1}
            s_1!_A(x1) = max{23, x1 + 2}
            |0|_0!_A = 57
            minus_2!_A(x1,x2) = max{22, x1 - 23, x2 - 10}
        
  
    partial status:
  
      pi(f_4#) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(minus_2) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.