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, r3, r4, r5, r7, r9, r10, r13, r16, r17, r18, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{3, x1 - 1}
        cons_2_A(x1,x2) = max{x1 + 19, x2 - 23}
        app#_A(x1,x2) = max{2, x1 - 2}
        map_1_A(x1) = x1 + 6
        filter2_3_A(x1,x2,x3) = max{x1 + 37, x2 + 6, x3 + 6}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 3
        false_0_A = 14
        filter_2#_A(x1,x2) = x1 + 3
        filter_1_A(x1) = max{19, x1 + 6}
        app_A(x1,x2) = max{20, x1 - 36, x2 + 8}
        true_0_A = 2
        cons_0_A = 8
        cons_1_A(x1) = max{9, x1 - 1}
        f_0_A = 5
        f_1_A(x1) = max{6, x1 - 1}
        f_2_A(x1,x2) = max{7, x1 - 38, x2 - 1}
        f_3_A(x1,x2,x3) = max{20, x1 - 75, x2 - 37, x3 - 1}
        filter_0_A = 1
        filter2_0_A = 19
        filter2_1_A(x1) = max{20, x1 - 1}
        filter2_2_A(x1,x2) = max{20, x1 - 37, x2 - 1}
        if_0_A = 35
        if_1_A(x1) = max{9, x1 - 2}
        le_0_A = 1
        le_1_A(x1) = max{17, x1 - 176}
        le_2_A(x1,x2) = max{16, x1 - 213, x2 - 134}
        map_0_A = 1
        minus_0_A = 35
        minus_1_A(x1) = max{7, x1 - 1}
        minus_2_A(x1,x2) = max{16, x1 - 134, x2 - 213}
        s_0_A = 1
        s_1_A(x1) = max{19, x1 - 59}
        perfectp_1_A(x1) = max{18, x1 - 56}
        |0|_0_A = 15
        f_4_A(x1,x2,x3,x4) = max{17, x1 - 153, x2 - 74, x3 - 57, x4 - 77}
        filter_2_A(x1,x2) = max{22, x1 - 25, x2 - 10}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_2 = nil_0 > filter2_3 = f_4 = filter_2 > app > filter_1 = f_3 = if_1 = le_2 = minus_0 = minus_2 > f_0 = f_2 = if_0 = le_1 = minus_1 = s_1 > map_1 = cons_1 = f_1 = filter_0 = filter2_2 = le_0 = s_0 = perfectp_1 > cons_0 = filter2_1 = map_0 = |0|_0 > false_0 = true_0 = filter2_0 > filter2_4# = filter_2# > app# > map_2#
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app#) = []
      pi(map_1) = []
      pi(filter2_3) = [1]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      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(filter_0) = []
      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(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(f_4) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p7



-- SCC decomposition.

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: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p8: 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, p5, p6, p7, p8}


-- 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#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
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#(map_1(x1),y) -> map_2#(x1,y)
p8: 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

  r1, r2, r3, r5, r7, r9, r10, r13, r14, r15, r16, r17, r20, r21, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{6, x1 - 2}
        cons_2_A(x1,x2) = max{4, x1 - 27, x2 - 27}
        app#_A(x1,x2) = max{5, x1 - 3}
        filter2_3_A(x1,x2,x3) = max{x1 + 5, x2 + 10, x3 + 5}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 6
        true_0_A = 3
        filter_2#_A(x1,x2) = x1 + 6
        app_A(x1,x2) = max{11, x1 - 13, x2 - 13}
        false_0_A = 1
        map_1_A(x1) = max{12, x1 + 2}
        cons_0_A = 1
        cons_1_A(x1) = max{3, x1 - 13}
        f_0_A = 25
        f_1_A(x1) = max{12, x1 - 13}
        f_2_A(x1,x2) = max{x1 - 27, x2 + 12}
        f_3_A(x1,x2,x3) = max{8, x1 - 41, x2 - 182, x3 - 13}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 40}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 14}
        filter2_2_A(x1,x2) = max{9, x1 - 28, x2 - 14}
        if_0_A = 8
        if_1_A(x1) = max{9, x1 - 14}
        if_2_A(x1,x2) = max{10, x1 - 215, x2 - 14}
        if_3_A(x1,x2,x3) = max{11, x1 - 228, x2 - 27, x3 - 13}
        le_0_A = 8
        le_1_A(x1) = max{9, x1 - 14}
        le_2_A(x1,x2) = max{10, x1 - 47, x2 - 14}
        minus_0_A = 5
        minus_1_A(x1) = max{6, x1 - 14}
        minus_2_A(x1,x2) = max{7, x1 - 46, x2 - 277}
        perfectp_1_A(x1) = max{10, x1 - 137}
        |0|_0_A = 8
        f_4_A(x1,x2,x3,x4) = max{9, x1 - 229, x2 - 196, x3 - 138, x4 - 217}
        s_1_A(x1) = max{90, x1 - 47}
        filter_2_A(x1,x2) = max{10, x1 - 53, x2 - 33}
        nil_0_A = 5
  
    base order 2:
  
      precedence:
      
        app > if_3 = le_2 > filter2_3 = filter_0 = minus_2 = f_4 = nil_0 > f_0 = f_3 = filter_1 = filter2_2 = if_2 = le_1 = minus_1 = s_1 = filter_2 > f_2 = filter2_1 = if_1 = le_0 = minus_0 = perfectp_1 > f_1 = filter2_0 = if_0 = |0|_0 > false_0 = map_1 > map_2# = filter2_4# = filter_2# > app# > cons_2 > true_0 = cons_1 > cons_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(app) = []
      pi(false_0) = []
      pi(map_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = [2]
      pi(f_3) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(f_4) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p5: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: 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 estimated dependency graph contains the following SCCs:

  {p1, p6, p7}
  {p3, p4, p5}


-- 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)

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, r7, r9, r10, r11, r13, r14, r15, r16, r17, r18, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x1 - 6}
        cons_2_A(x1,x2) = max{2, x1 - 51, x2 - 64}
        app#_A(x1,x2) = max{3, x1 - 6}
        map_1_A(x1) = x1 + 11
        app_A(x1,x2) = max{50, x1 - 20, x2 + 16}
        cons_0_A = 18
        cons_1_A(x1) = max{19, x1 - 1}
        f_0_A = 16
        f_1_A(x1) = max{17, x1 - 1}
        f_2_A(x1,x2) = max{18, x1 - 22, x2 - 1}
        f_3_A(x1,x2,x3) = max{19, x1 - 43, x2 - 22, x3 - 1}
        filter_0_A = 13
        filter_1_A(x1) = max{14, x1 - 141}
        filter2_0_A = 17
        filter2_1_A(x1) = max{18, x1 - 1}
        filter2_2_A(x1,x2) = max{19, x1 - 22, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{18, x1 - 99, x2 - 30, x3 - 30}
        if_0_A = 16
        if_1_A(x1) = max{17, x1 - 1}
        if_2_A(x1,x2) = max{18, x1 - 22, x2 - 1}
        if_3_A(x1,x2,x3) = max{19, x1 - 43, x2 - 24, x3 - 24}
        le_0_A = 18
        le_1_A(x1) = max{19, x1 - 1}
        le_2_A(x1,x2) = max{22, x1 - 96, x2 - 143}
        map_0_A = 19
        minus_0_A = 16
        minus_1_A(x1) = max{17, x1 - 1}
        minus_2_A(x1,x2) = max{18, x1 - 143, x2 - 96}
        s_0_A = 19
        s_1_A(x1) = max{49, x1 - 22}
        perfectp_1_A(x1) = max{24, x1 - 50}
        |0|_0_A = 2
        false_0_A = 1
        f_4_A(x1,x2,x3,x4) = max{23, x1 - 73, x2 - 120, x3 - 70, x4 - 51}
        true_0_A = 1
        filter_2_A(x1,x2) = max{15, x1 - 140, x2 - 51}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_2# = f_4 > app# = app = cons_0 = f_0 = le_0 = minus_1 = minus_2 > cons_2 = f_3 = filter2_3 = if_3 = le_2 = minus_0 > map_1 = cons_1 = f_2 = filter2_2 = if_2 = le_1 = s_1 = filter_2 = nil_0 > f_1 = filter_1 = filter2_1 = if_1 = map_0 = s_0 = perfectp_1 > filter_0 = filter2_0 = if_0 = |0|_0 > false_0 = true_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      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(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(f_4) = []
      pi(true_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: 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 estimated dependency graph contains the following SCCs:

  {p2}


-- 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

  r1, r4, r5, r7, r9, r10, r11, r13, r14, r16, r17, r18, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 11, x2 + 22}
        cons_2_A(x1,x2) = max{9, x1 - 11, x2}
        app_A(x1,x2) = max{10, x1 - 11, x2 - 23}
        cons_0_A = 7
        cons_1_A(x1) = max{8, x1 - 24}
        f_1_A(x1) = x1 + 8
        f_2_A(x1,x2) = max{9, x1 - 138, x2 - 24}
        f_3_A(x1,x2,x3) = max{10, x1 - 149, x2 - 35, x3 - 23}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 24}
        filter2_0_A = 7
        filter2_1_A(x1) = max{8, x1 - 24}
        filter2_2_A(x1,x2) = max{9, x1 - 35, x2 - 23}
        filter2_3_A(x1,x2,x3) = max{10, x1 - 46, x2 - 35, x3 - 23}
        if_0_A = 9
        if_1_A(x1) = max{10, x1 - 23}
        if_2_A(x1,x2) = max{1, x1 - 163}
        le_0_A = 7
        le_1_A(x1) = max{8, x1 - 24}
        le_2_A(x1,x2) = max{9, x1 - 95, x2 - 95}
        map_0_A = 9
        map_1_A(x1) = max{10, x1 - 23}
        minus_0_A = 2
        minus_1_A(x1) = max{3, x1 - 24}
        minus_2_A(x1,x2) = max{4, x1 - 36, x2 - 24}
        s_0_A = 4
        s_1_A(x1) = max{5, x1 - 96}
        perfectp_1_A(x1) = max{4, x1 - 31}
        |0|_0_A = 2
        false_0_A = 1
        f_4_A(x1,x2,x3,x4) = max{3, x1 - 161, x2 - 59, x3 - 37, x4 - 32}
        true_0_A = 1
        filter_2_A(x1,x2) = max{11, x1 - 10, x2 - 11}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        f_1 = filter_2 > map_2# = cons_2 = app = nil_0 > filter_0 = if_1 = le_2 = map_0 = minus_2 > cons_1 = filter_1 = filter2_3 = if_0 = if_2 = le_1 = map_1 = minus_1 = s_1 = f_4 > cons_0 = f_3 = filter2_0 = filter2_2 = le_0 = minus_0 = s_0 = perfectp_1 > f_2 = filter2_1 = |0|_0 > false_0 = true_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_1) = [1]
      pi(f_2) = []
      pi(f_3) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(f_4) = []
      pi(true_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(true_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#(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 set of usable rules consists of

  r2, r3, r4, r9, r10, r11, r13, r14, r15, r16, r17, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 4, x4 + 3}
        true_0_A = 1
        filter_2#_A(x1,x2) = max{2, x2 + 1}
        cons_2_A(x1,x2) = max{x1 + 5, x2 + 5}
        app_A(x1,x2) = max{11, x1 - 22, x2 + 5}
        false_0_A = 6
        cons_1_A(x1) = x1 + 28
        f_0_A = 5
        f_1_A(x1) = max{6, x1 - 1}
        f_2_A(x1,x2) = max{7, x1 - 24, x2 - 1}
        filter2_0_A = 3
        filter2_1_A(x1) = max{4, x1 - 1}
        filter2_2_A(x1,x2) = max{9, x1 - 24, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{10, x1 - 47, x2 - 24, x3 - 1}
        if_0_A = 1
        if_1_A(x1) = max{4, x1 - 1}
        if_2_A(x1,x2) = max{6, x1 - 24, x2 - 1}
        if_3_A(x1,x2,x3) = max{9, x1 - 47, x2 - 54, x3 - 48}
        le_0_A = 23
        le_1_A(x1) = max{7, x1 - 1}
        le_2_A(x1,x2) = max{8, x1 - 185, x2 - 94}
        minus_0_A = 1
        minus_1_A(x1) = max{7, x1 - 1}
        minus_2_A(x1,x2) = max{8, x1 - 94, x2 - 185}
        s_0_A = 1
        s_1_A(x1) = max{10, x1 - 43}
        perfectp_1_A(x1) = max{22, x1 - 97}
        |0|_0_A = 7
        f_4_A(x1,x2,x3,x4) = max{9, x1 - 141, x2 - 50, x3 - 41, x4 - 98}
        filter_2_A(x1,x2) = max{24, x1 + 2}
        nil_0_A = 5
  
    base order 2:
  
      precedence:
      
        f_4 > minus_2 = filter_2 > app > cons_2 = if_3 = le_2 > filter2_4# = cons_1 = f_1 = f_2 = filter2_3 = if_2 = le_1 = minus_1 = s_1 = nil_0 > f_0 = filter2_0 = filter2_2 = if_1 = le_0 = minus_0 = s_0 = perfectp_1 > filter2_1 = if_0 = |0|_0 > false_0 > filter_2# > true_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = [2]
      pi(false_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter2_0) = []
      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(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(f_4) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: 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:

  {p1, p2}


-- Reduction pair.

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

p1: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: 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 set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r9, r10, r11, r13, r14, r15, r16, r17, r18, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 2, x4 + 6}
        true_0_A = 10
        filter_2#_A(x1,x2) = x2 + 5
        cons_2_A(x1,x2) = max{x1 + 3, x2 + 7}
        app_A(x1,x2) = max{223, x1 + 9, x2 + 9}
        cons_0_A = 2
        cons_1_A(x1) = max{10, x1 + 4}
        f_0_A = 1
        f_1_A(x1) = max{11, x1 - 10}
        f_2_A(x1,x2) = max{21, x1 - 11, x2 - 19}
        f_3_A(x1,x2,x3) = max{31, x1 - 12, x2 - 10, x3 - 314}
        filter_0_A = 1
        filter_1_A(x1) = max{210, x1 - 20}
        filter2_0_A = 1
        filter2_1_A(x1) = max{11, x1 - 19}
        filter2_2_A(x1,x2) = max{204, x1 - 10, x2 - 10}
        filter2_3_A(x1,x2,x3) = max{214, x1 - 243, x2 - 11, x3 - 10}
        if_0_A = 11
        if_1_A(x1) = max{10, x1 - 10}
        if_2_A(x1,x2) = max{20, x1 - 11, x2 - 10}
        if_3_A(x1,x2,x3) = max{30, x1 - 267, x2 - 267, x3 - 134}
        le_0_A = 1
        le_1_A(x1) = max{11, x1 - 131}
        le_2_A(x1,x2) = max{223, x1 - 132, x2 - 132}
        map_0_A = 213
        map_1_A(x1) = max{212, x1 - 18}
        minus_0_A = 1
        minus_1_A(x1) = max{11, x1 - 10}
        minus_2_A(x1,x2) = max{42, x1 - 11, x2 - 44}
        s_0_A = 1
        s_1_A(x1) = max{43, x1 - 133}
        perfectp_1_A(x1) = max{42, x1 - 44}
        |0|_0_A = 11
        false_0_A = 1
        f_4_A(x1,x2,x3,x4) = max{41, x1 - 265, x2 - 88, x3 - 314, x4 - 45}
        filter_2_A(x1,x2) = max{218, x1 - 20, x2 - 23}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_2 > app > cons_1 = filter_1 = if_3 = le_2 = minus_2 = filter_2 = nil_0 > filter_2# = cons_0 = f_3 = filter_0 = filter2_3 = if_2 = le_1 = map_1 = minus_1 = s_1 = f_4 > filter2_4# = f_2 = filter2_2 = if_1 = le_0 = map_0 = minus_0 = s_0 = perfectp_1 > f_1 = filter2_1 = if_0 = |0|_0 > true_0 = f_0 = filter2_0 = false_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app) = [1, 2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      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(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(f_4) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- 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

  r1, r2, r3, r4, r5, r7, r9, r10, r11, r13, r14, r15, r16, r17, r18, r20, r21, r23, r24, r26, r27, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_4#_A(x1,x2,x3,x4) = max{38, x1 + 15, x2 - 34, x3 - 15, x4 - 1}
        s_1_A(x1) = max{23, x1 + 14}
        |0|_0_A = 2
        minus_2_A(x1,x2) = max{13, x1 - 7, x2 - 39}
        app_A(x1,x2) = max{x1 - 5, x2 + 25}
        cons_0_A = 3
        cons_1_A(x1) = max{4, x1 - 1}
        cons_2_A(x1,x2) = max{23, x1 - 26, x2 - 30}
        f_0_A = 3
        f_1_A(x1) = max{4, x1 - 1}
        f_2_A(x1,x2) = max{3, x1 - 6, x2 - 1}
        f_3_A(x1,x2,x3) = max{6, x1 - 69, x2 - 7, x3 - 57}
        filter_0_A = 4
        filter_1_A(x1) = max{3, x1 - 50}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 1}
        filter2_2_A(x1,x2) = max{3, x1 - 7, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 13, x2 - 7, x3 - 1}
        if_0_A = 2
        if_1_A(x1) = max{3, x1 - 1}
        if_2_A(x1,x2) = max{4, x1 - 7, x2 - 1}
        if_3_A(x1,x2,x3) = max{24, x1 - 25, x2 - 8, x3 - 69}
        le_0_A = 10
        le_1_A(x1) = max{4, x1 - 1}
        le_2_A(x1,x2) = max{24, x1 - 7, x2 - 1}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 26}
        minus_0_A = 6
        minus_1_A(x1) = max{7, x1 - 1}
        s_0_A = 4
        perfectp_1_A(x1) = max{25, x1 - 50}
        false_0_A = 1
        f_4_A(x1,x2,x3,x4) = max{7, x1 - 75, x2 - 22, x3 - 63, x4 - 51}
        true_0_A = 1
        filter_2_A(x1,x2) = max{6, x1 - 56, x2 - 66}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        minus_2 > f_4 > filter_0 = if_0 = if_3 = minus_0 > if_2 > if_1 > app = filter2_1 = filter2_2 = filter2_3 = le_1 = le_2 > f_4# = s_1 = f_2 = f_3 = filter2_0 = le_0 = minus_1 > cons_2 = f_1 = map_1 = s_0 = perfectp_1 = filter_2 = nil_0 > |0|_0 = cons_1 = f_0 = filter_1 = map_0 > cons_0 = false_0 = true_0
      
  
    partial status:
  
      pi(f_4#) = [1]
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(minus_2) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      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(s_0) = []
      pi(perfectp_1) = []
      pi(false_0) = []
      pi(f_4) = []
      pi(true_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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)

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),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),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

  r1, r2, r3, r4, r5, r6, r7, r9, r10, r11, r13, r14, r15, r16, r17, r18, r20, r21, r22, r24, r25, r26, r27, r28, r29, r32

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_4#_A(x1,x2,x3,x4) = max{x1 + 18, x3 + 81}
        s_1_A(x1) = max{67, x1 + 48}
        |0|_0_A = 46
        minus_2_A(x1,x2) = max{3, x1 - 70, x2 - 155}
        app_A(x1,x2) = max{8, x1 - 9, x2 - 7}
        cons_0_A = 3
        cons_1_A(x1) = max{4, x1 - 8}
        cons_2_A(x1,x2) = max{5, x1 - 45, x2 - 38}
        f_0_A = 1
        f_1_A(x1) = max{4, x1 - 8}
        f_2_A(x1,x2) = max{5, x1 - 17, x2 - 8}
        f_3_A(x1,x2,x3) = max{6, x1 - 27, x2 - 18, x3 - 59}
        f_4_A(x1,x2,x3,x4) = max{7, x1 - 137, x2 - 93, x3 - 69, x4 - 69}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 8}
        filter2_0_A = 7
        filter2_1_A(x1) = max{8, x1 - 7}
        filter2_2_A(x1,x2) = max{0, x1 - 17, x2 - 8}
        filter2_3_A(x1,x2,x3) = max{0, x1 - 27, x2 - 18, x3 - 8}
        if_0_A = 8
        if_1_A(x1) = max{4, x1 - 8}
        if_2_A(x1,x2) = max{5, x1 - 18, x2 - 8}
        if_3_A(x1,x2,x3) = max{6, x1 - 28, x2 - 18, x3 - 8}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 8}
        le_2_A(x1,x2) = max{3, x1 - 90, x2 - 46}
        map_0_A = 10
        map_1_A(x1) = max{8, x1 - 7}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 8}
        perfectp_0_A = 57
        perfectp_1_A(x1) = max{47, x1 - 68}
        false_0_A = 2
        true_0_A = 1
        filter_2_A(x1,x2) = max{3, x1 + 1, x2 - 6}
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        f_4# = f_4 > minus_2 = if_3 = le_2 > s_1 = f_3 > app > f_2 = if_0 = if_2 = le_1 = minus_1 = perfectp_1 > |0|_0 = cons_2 = f_1 = if_1 = le_0 = map_1 = minus_0 = perfectp_0 > cons_1 = f_0 = map_0 > cons_0 = filter2_3 = filter_2 > filter_1 = filter2_2 = false_0 = true_0 = nil_0 > filter_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(f_4#) = [1, 3]
      pi(s_1) = []
      pi(|0|_0) = []
      pi(minus_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(f_4) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      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_0) = []
      pi(perfectp_1) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.