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:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x1}
        cons_2_A(x1,x2) = 0
        app#_A(x1,x2) = max{4, x1}
        map_1_A(x1) = max{3, x1}
        filter2_3_A(x1,x2,x3) = max{x2 + 2, x3 + 7}
        filter2_4#_A(x1,x2,x3,x4) = max{4, x2 + 1}
        false_0_A = 15
        filter_2#_A(x1,x2) = max{4, x1 + 1}
        filter_1_A(x1) = max{9, x1 + 3}
        app_A(x1,x2) = max{x1 + 8, x2 + 8}
        true_0_A = 15
        cons_0_A = 2
        cons_1_A(x1) = 1
        f_0_A = 2
        f_1_A(x1) = 1
        f_2_A(x1,x2) = max{5, x2 - 8}
        f_3_A(x1,x2,x3) = max{9, x3 + 7}
        f_4_A(x1,x2,x3,x4) = 16
        filter_0_A = 1
        filter_2_A(x1,x2) = max{14, x1 + 8}
        filter2_0_A = 1
        filter2_1_A(x1) = max{0, x1 - 8}
        filter2_2_A(x1,x2) = max{4, x2 - 5}
        filter2_4_A(x1,x2,x3,x4) = max{14, x2 + 8}
        if_0_A = 8
        if_1_A(x1) = max{7, x1 - 8}
        if_2_A(x1,x2) = 2
        if_3_A(x1,x2,x3) = max{9, x3 - 17}
        le_0_A = 2
        le_1_A(x1) = 1
        le_2_A(x1,x2) = max{5, x2 - 1}
        map_0_A = 4
        map_2_A(x1,x2) = 10
        minus_0_A = 12
        minus_1_A(x1) = 11
        minus_2_A(x1,x2) = max{18, x2 - 17}
        perfectp_0_A = 11
        perfectp_1_A(x1) = max{18, x1 - 18}
        s_0_A = 1
        s_1_A(x1) = max{0, x1 - 17}
        |0|_0_A = 17
        nil_0_A = 9
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x1 - 70, x2 - 145}
          cons_2_A(x1,x2) = max{70, x1 + 6, x2}
          app#_A(x1,x2) = max{0, x1 - 70, x2 - 139}
          map_1_A(x1) = x1 + 6
          filter2_3_A(x1,x2,x3) = max{x1 - 132, x2 + 7, x3 + 69}
          filter2_4#_A(x1,x2,x3,x4) = max{0, x1 - 202, x2 - 70, x3 - 133, x4 - 139}
          false_0_A = 70
          filter_2#_A(x1,x2) = max{0, x1 - 70, x2 - 139}
          filter_1_A(x1) = max{8, x1 + 7}
          app_A(x1,x2) = max{133, x1 - 9, x2 + 69}
          true_0_A = 70
          cons_0_A = 0
          cons_1_A(x1) = x1 + 15
          f_0_A = 0
          f_1_A(x1) = max{8, x1 - 114}
          f_2_A(x1,x2) = max{2, x1 - 123, x2 - 114}
          f_3_A(x1,x2,x3) = max{8, x1 - 132, x2 - 123, x3 - 1}
          f_4_A(x1,x2,x3,x4) = max{70, x1 - 141, x2 - 213, x3 - 107, x4 - 107}
          filter_0_A = 0
          filter_2_A(x1,x2) = max{72, x1 - 9, x2 + 1}
          filter2_0_A = 0
          filter2_1_A(x1) = max{0, x1 - 114}
          filter2_2_A(x1,x2) = max{x1 - 123, x2 + 16}
          filter2_4_A(x1,x2,x3,x4) = max{72, x1 - 141, x2 - 9, x3 + 6, x4 + 1}
          if_0_A = 0
          if_1_A(x1) = max{8, x1 - 1}
          if_2_A(x1,x2) = max{8, x1 - 10, x2 - 1}
          if_3_A(x1,x2,x3) = max{70, x1 - 176, x2 - 71, x3 - 71}
          le_0_A = 0
          le_1_A(x1) = max{8, x1 - 1}
          le_2_A(x1,x2) = max{70, x1 - 10, x2 - 3}
          map_0_A = 0
          map_2_A(x1,x2) = max{76, x1 - 3, x2 + 69}
          minus_0_A = 0
          minus_1_A(x1) = max{8, x1 - 114}
          minus_2_A(x1,x2) = max{106, x1 - 146, x2 - 35}
          perfectp_0_A = 0
          perfectp_1_A(x1) = max{70, x1 - 107}
          s_0_A = 0
          s_1_A(x1) = x1 + 34
          |0|_0_A = 34
          nil_0_A = 389
          map_2#!_A(x1,x2) = max{0, x2 - 2}
          cons_2!_A(x1,x2) = max{179, x1 - 134, x2 - 211}
          app#!_A(x1,x2) = max{0, x2 - 1}
          map_1!_A(x1) = max{2, x1 - 216}
          filter2_3!_A(x1,x2,x3) = max{100, x1 - 11, x2 - 1, x3 - 133}
          filter2_4#!_A(x1,x2,x3,x4) = max{x1 + 111, x2 + 102, x3 - 133, x4 + 174}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{x1 + 102, x2 + 174}
          filter_1!_A(x1) = max{2, x1 - 1}
          app!_A(x1,x2) = max{179, x1 + 113, x2 + 115}
          true_0!_A = 1
          cons_0!_A = 179
          cons_1!_A(x1) = max{178, x1 - 133}
          f_0!_A = 2
          f_1!_A(x1) = x1 + 115
          f_2!_A(x1,x2) = max{117, x1 - 2, x2 - 287}
          f_3!_A(x1,x2,x3) = max{116, x2 - 286, x3 - 393}
          f_4!_A(x1,x2,x3,x4) = max{178, x1 - 34, x2 - 285, x3 - 392, x4 - 285}
          filter_0!_A = 1
          filter_2!_A(x1,x2) = max{181, x1 + 114, x2 + 110}
          filter2_0!_A = 97
          filter2_1!_A(x1) = max{98, x1 - 13}
          filter2_2!_A(x1,x2) = max{99, x1 - 12, x2 - 2}
          filter2_4!_A(x1,x2,x3,x4) = max{181, x2 + 114, x3 - 133, x4 + 110}
          if_0!_A = 1
          if_1!_A(x1) = max{0, x1 - 1}
          if_2!_A(x1,x2) = x2 + 116
          if_3!_A(x1,x2,x3) = max{x1 - 247, x2 + 107, x3 - 71}
          le_0!_A = 115
          le_1!_A(x1) = 114
          le_2!_A(x1,x2) = max{111, x2 - 1}
          map_0!_A = 1
          map_2!_A(x1,x2) = max{179, x1 - 215, x2 - 206}
          minus_0!_A = 1
          minus_1!_A(x1) = max{2, x1 - 253}
          minus_2!_A(x1,x2) = max{178, x1 - 252, x2 - 35}
          perfectp_0!_A = 1
          perfectp_1!_A(x1) = max{116, x1 - 393}
          s_0!_A = 0
          s_1!_A(x1) = 0
          |0|_0!_A = 2
          nil_0!_A = 182
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = [3]
      pi(f_4) = []
      pi(filter_0) = []
      pi(filter_2) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_4) = [2]
      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_2) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(perfectp_0) = []
      pi(perfectp_1) = []
      pi(s_0) = []
      pi(s_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:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1
        cons_2_A(x1,x2) = max{5, x1 - 274, x2 - 134}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x2 - 13}
          cons_2_A(x1,x2) = max{x1 - 8, x2 + 12}
          map_2#!_A(x1,x2) = max{7, x2 - 4}
          cons_2!_A(x1,x2) = max{6, x1 - 189, x2 - 152}
  
    partial status:
  
      pi(map_2#) = [1]
      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, r9, r10, r13, r16, r17, r18, r21, r24, r26, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = x2 + 2
        false_0_A = 40
        filter_2#_A(x1,x2) = x1 + 2
        cons_2_A(x1,x2) = 10
        app_A(x1,x2) = max{35, x1 - 32}
        true_0_A = 37
        f_0_A = 32
        f_1_A(x1) = 32
        filter2_0_A = 32
        filter2_1_A(x1) = 31
        filter2_2_A(x1,x2) = 32
        if_0_A = 31
        if_1_A(x1) = 32
        le_0_A = 32
        le_1_A(x1) = 0
        le_2_A(x1,x2) = 34
        map_0_A = 0
        map_1_A(x1) = 0
        minus_1_A(x1) = x1 + 33
        minus_2_A(x1,x2) = 31
        perfectp_1_A(x1) = 47
        |0|_0_A = 41
        f_4_A(x1,x2,x3,x4) = max{38, x3 - 7, x4 + 7}
        filter_2_A(x1,x2) = max{33, x1 - 1}
        nil_0_A = 1
  
    base order 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 - 64, x4 - 36}
          false_0_A = 27
          filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 36}
          cons_2_A(x1,x2) = max{x1 - 28, x2 + 35}
          app_A(x1,x2) = max{63, x1 - 4, x2 + 27}
          true_0_A = 27
          f_0_A = 0
          f_1_A(x1) = 3
          filter2_0_A = 0
          filter2_1_A(x1) = max{3, x1 - 1}
          filter2_2_A(x1,x2) = max{x1 - 5, x2 + 27}
          if_0_A = 0
          if_1_A(x1) = 0
          le_0_A = 0
          le_1_A(x1) = 3
          le_2_A(x1,x2) = 28
          map_0_A = 0
          map_1_A(x1) = x1 + 27
          minus_1_A(x1) = max{3, x1 - 9}
          minus_2_A(x1,x2) = 15
          perfectp_1_A(x1) = max{83, x1 - 20}
          |0|_0_A = 0
          f_4_A(x1,x2,x3,x4) = max{28, x1 - 6, x3}
          filter_2_A(x1,x2) = max{x1 + 35, x2 - 18}
          nil_0_A = 35
          filter2_4#!_A(x1,x2,x3,x4) = max{x3 + 3, x4 + 32}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = x2 + 31
          cons_2!_A(x1,x2) = max{74, x1 - 27, x2 + 33}
          app!_A(x1,x2) = max{81, x1 + 9, x2 + 77}
          true_0!_A = 79
          f_0!_A = 1
          f_1!_A(x1) = max{76, x1 - 5}
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{0, x1 - 1}
          filter2_2!_A(x1,x2) = 76
          if_0!_A = 1
          if_1!_A(x1) = max{10, x1 - 1}
          le_0!_A = 90
          le_1!_A(x1) = max{89, x1 - 2}
          le_2!_A(x1,x2) = max{88, x1 - 1, x2 - 1}
          map_0!_A = 83
          map_1!_A(x1) = max{82, x1 + 76}
          minus_1!_A(x1) = max{82, x1 - 1}
          minus_2!_A(x1,x2) = max{76, x2 - 2}
          perfectp_1!_A(x1) = max{94, x1 - 20}
          |0|_0!_A = 82
          f_4!_A(x1,x2,x3,x4) = max{80, x2 + 68, x3 - 17, x4 + 75}
          filter_2!_A(x1,x2) = x2 + 11
          nil_0!_A = 146
  
    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(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(f_4) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p3



-- SCC decomposition.

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

p1: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,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:
  
      max/plus interpretation on natural numbers:
      
        f_4#_A(x1,x2,x3,x4) = 1
        s_1_A(x1) = max{10, x1 - 14}
        |0|_0_A = 2
        minus_2_A(x1,x2) = max{43, x1 - 14, x2 - 39}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f_4#_A(x1,x2,x3,x4) = max{0, x1 - 41, x2 - 81, x3 - 29, x4 - 47}
          s_1_A(x1) = x1 + 39
          |0|_0_A = 34
          minus_2_A(x1,x2) = max{28, x1 - 3, x2 - 42}
          f_4#!_A(x1,x2,x3,x4) = max{x1 - 1, x3 + 2}
          s_1!_A(x1) = 38
          |0|_0!_A = 31
          minus_2!_A(x1,x2) = max{39, x1 - 234, x2 - 30}
  
    partial status:
  
      pi(f_4#) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(minus_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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:
  
      max/plus interpretation on natural numbers:
      
        f_4#_A(x1,x2,x3,x4) = max{21, x2 - 21, x3, x4 - 1}
        s_1_A(x1) = max{20, x1 - 79}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f_4#_A(x1,x2,x3,x4) = max{0, x1 - 4, x2 - 1, x4 - 1}
          s_1_A(x1) = max{9, x1 + 3}
          f_4#!_A(x1,x2,x3,x4) = max{7, x1 - 1}
          s_1!_A(x1) = max{6, x1 - 59}
  
    partial status:
  
      pi(f_4#) = [3]
      pi(s_1) = []

The following rules are removed:

  p1

Then no dependency pair remains.