YES
We show the termination of the TRS R:

  app(app(minus(),|0|()),y) -> |0|()
  app(app(minus(),app(s(),x)),|0|()) -> app(s(),x)
  app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
  app(app(le(),|0|()),y) -> true()
  app(app(le(),app(s(),x)),|0|()) -> false()
  app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
  app(app(app(if(),true()),x),y) -> x
  app(app(app(if(),false()),x),y) -> y
  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)
  minus_2(|0|_0(),y) -> |0|_0()
  minus_2(s_1(x),|0|_0()) -> s_1(x)
  minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
  le_2(|0|_0(),y) -> true_0()
  le_2(s_1(x),|0|_0()) -> false_0()
  le_2(s_1(x),s_1(y)) -> le_2(x,y)
  if_3(true_0(),x,y) -> x
  if_3(false_0(),x,y) -> 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#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p5: app#(le_1(x1),y) -> le_2#(x1,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: app#(minus_1(x1),y) -> minus_2#(x1,y)
p8: app#(perfectp_0(),y) -> perfectp_1#(y)
p9: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p10: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p11: perfectp_1#(s_1(x)) -> f_4#(x,s_1(|0|_0()),s_1(x),s_1(x))
p12: f_4#(s_1(x),|0|_0(),z,u) -> f_4#(x,u,minus_2(z,s_1(x)),u)
p13: f_4#(s_1(x),|0|_0(),z,u) -> minus_2#(z,s_1(x))
p14: 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))
p15: f_4#(s_1(x),s_1(y),z,u) -> le_2#(x,y)
p16: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),z,u)
p17: f_4#(s_1(x),s_1(y),z,u) -> minus_2#(y,x)
p18: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p19: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p20: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p21: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p22: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p23: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p24: 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p19, p20, p21, p22, p23, p24}
  {p12, p16, p18}
  {p10}
  {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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r9, r10, r13, r14, r16, r20, r23, r24, r27, r28, r32, r34, r35, r40

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{20, x1}
        cons_2_A(x1,x2) = max{18, x1 - 42, x2 - 21}
        app#_A(x1,x2) = max{19, x1 - 19}
        map_1_A(x1) = x1 + 40
        filter2_3_A(x1,x2,x3) = max{x1 - 152, x2 + 40, x3 - 3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 19
        false_0_A = 1
        filter_2#_A(x1,x2) = x1 + 19
        filter_1_A(x1) = x1 + 39
        app_A(x1,x2) = max{17, x1 - 43, x2 - 63}
        true_0_A = 1
        cons_0_A = 16
        cons_1_A(x1) = max{17, x1 - 63}
        f_0_A = 1
        f_1_A(x1) = max{2, x1 - 63}
        f_2_A(x1,x2) = max{3, x1 - 106, x2 - 64}
        f_3_A(x1,x2,x3) = max{8, x1 - 149, x2 - 107, x3 - 63}
        filter2_0_A = 13
        filter2_1_A(x1) = max{14, x1 - 64}
        filter2_2_A(x1,x2) = max{15, x1 - 108, x2 - 64}
        if_0_A = 15
        if_1_A(x1) = max{16, x1 - 64}
        if_2_A(x1,x2) = max{15, x1 - 108}
        le_0_A = 1
        le_1_A(x1) = max{9, x1 - 396}
        minus_0_A = 7
        minus_1_A(x1) = max{8, x1 - 397}
        s_0_A = 1
        s_1_A(x1) = max{13, x1 - 143}
        minus_2_A(x1,x2) = max{9, x1 - 441, x2 - 453}
        |0|_0_A = 8
        le_2_A(x1,x2) = max{8, x1 - 440, x2 - 295}
        perfectp_1_A(x1) = max{10, x1 - 152}
        f_4_A(x1,x2,x3,x4) = max{9, x1 - 296, x2 - 151, x3 - 441, x4 - 441}
        filter_2_A(x1,x2) = max{20, x1 - 3, x2 - 20}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > filter2_3 = filter_1 = f_4 > app > map_1 = f_0 = f_3 = le_0 = le_2 = nil_0 > map_2# = filter2_4# = filter_2# = f_1 = f_2 = filter2_2 = if_1 = if_2 = le_1 = s_1 = minus_2 > app# = filter2_1 = if_0 = minus_1 = s_0 = perfectp_1 > cons_2 = filter2_0 = minus_0 = |0|_0 > false_0 = true_0 = cons_1 > cons_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      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) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(perfectp_1) = []
      pi(f_4) = []
      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)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)
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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {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)) -> 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r9, r11, r13, r14, r16, r18, r20, r23, r24, r27, r28, r32, r34, r35, r40

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 + 8, x2 + 16}
        cons_2_A(x1,x2) = max{6, x1 - 8, x2}
        app_A(x1,x2) = max{7, x1 - 9, x2 - 17}
        cons_0_A = 4
        cons_1_A(x1) = max{5, x1 - 17}
        f_0_A = 1
        f_1_A(x1) = max{2, x1 - 18}
        f_2_A(x1,x2) = max{3, x1 - 28, x2 - 18}
        f_3_A(x1,x2,x3) = max{4, x1 - 38, x2 - 31, x3 - 18}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 18}
        filter2_0_A = 5
        filter2_1_A(x1) = max{6, x1 - 18}
        filter2_2_A(x1,x2) = max{8, x1 - 28, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 37, x2 - 10, x3 - 17}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 18}
        if_2_A(x1,x2) = max{3, x1 - 27, x2 - 17}
        le_0_A = 2
        le_1_A(x1) = max{3, x1 - 17}
        map_0_A = 5
        map_1_A(x1) = max{6, x1 - 18}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 18}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 64}
        minus_2_A(x1,x2) = max{4, x1 - 105, x2 - 176}
        |0|_0_A = 3
        le_2_A(x1,x2) = max{4, x1 - 139, x2 - 105}
        true_0_A = 1
        false_0_A = 1
        perfectp_1_A(x1) = max{8, x1 - 16}
        f_4_A(x1,x2,x3,x4) = max{5, x1 - 111, x2 - 40, x3 - 61, x4 - 35}
        filter_2_A(x1,x2) = max{6, x1 - 28, x2 - 18}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_2# > cons_2 = app = cons_0 > cons_1 = filter2_3 = map_1 = le_2 = filter_2 = nil_0 > filter_1 = filter2_2 = le_1 = map_0 = f_4 > f_3 = filter_0 = filter2_1 = le_0 > f_2 = filter2_0 = if_2 = s_1 = minus_2 > f_1 = if_1 = minus_1 = s_0 = perfectp_1 > f_0 = if_0 = minus_0 = |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      pi(f_4) = []
      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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_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#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r9, r10, r13, r14, r16, r18, r20, r23, r24, r27, r28, r32, r34, r35, r40

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{4, x1 - 9}
        filter2_3_A(x1,x2,x3) = max{x1 - 7, x2 + 17, x3 - 4}
        filter2_4#_A(x1,x2,x3,x4) = max{7, x2 + 5}
        true_0_A = 1
        filter_2#_A(x1,x2) = max{7, x1 + 5}
        cons_2_A(x1,x2) = max{3, x1 - 46, x2 - 46}
        app_A(x1,x2) = max{25, x1 - 19, x2 - 19}
        false_0_A = 1
        filter_1_A(x1) = max{17, x1 + 15}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 20}
        f_0_A = 15
        f_1_A(x1) = max{16, x1 - 20}
        f_2_A(x1,x2) = max{17, x1 - 40, x2 - 20}
        f_3_A(x1,x2,x3) = max{18, x1 - 154, x2 - 39, x3 - 20}
        filter2_0_A = 5
        filter2_1_A(x1) = max{6, x1 - 20}
        filter2_2_A(x1,x2) = max{7, x1 - 40, x2 - 20}
        if_0_A = 16
        if_1_A(x1) = max{17, x1 - 20}
        if_2_A(x1,x2) = max{18, x1 - 40, x2 - 20}
        le_0_A = 17
        le_1_A(x1) = max{18, x1 - 238}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 46}
        minus_0_A = 18
        minus_1_A(x1) = max{20, x1 - 129}
        s_0_A = 18
        s_1_A(x1) = max{24, x1 - 84}
        minus_2_A(x1,x2) = max{21, x1 - 149, x2 - 258}
        |0|_0_A = 2
        le_2_A(x1,x2) = max{21, x1 - 258, x2 - 149}
        perfectp_1_A(x1) = max{23, x1 - 41}
        f_4_A(x1,x2,x3,x4) = max{22, x1 - 173, x2 - 64, x3 - 42, x4 - 42}
        filter_2_A(x1,x2) = max{24, x1 - 46, x2 - 24}
        nil_0_A = 3
  
    base order 2:
  
      precedence:
      
        app > f_4 > app# = f_3 = le_0 = le_2 = filter_2 > filter2_4# = filter_2# = f_2 = filter2_0 = if_1 = if_2 = le_1 = map_1 = s_1 = minus_2 = nil_0 > filter2_3 = cons_2 = filter_1 = cons_1 = f_1 = if_0 = map_0 = minus_1 = s_0 = perfectp_1 > cons_0 = f_0 = filter2_2 = minus_0 = |0|_0 > true_0 = false_0 = filter2_1
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(perfectp_1) = []
      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: 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)
p4: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p5: app#(filter_1(x1),y) -> filter_2#(x1,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p3: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)

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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r9, r10, r13, r14, r16, r18, r20, r23, r24, r27, r28, r32, r34, r35, r40

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) = x2 + 4
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 4
        cons_2_A(x1,x2) = max{5, x1 - 34, x2 - 34}
        app#_A(x1,x2) = max{4, x1 + 1}
        filter_1_A(x1) = max{5, x1 + 4}
        app_A(x1,x2) = max{11, x1 - 6, x2 - 6}
        false_0_A = 1
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 7}
        f_0_A = 5
        f_1_A(x1) = max{1, x1 - 7}
        f_2_A(x1,x2) = max{2, x1 - 14, x2 - 7}
        f_3_A(x1,x2,x3) = max{3, x1 - 61, x2 - 18, x3 - 12}
        filter2_0_A = 5
        filter2_1_A(x1) = max{7, x1 - 7}
        filter2_2_A(x1,x2) = max{4, x1 - 13, x2 - 7}
        if_0_A = 2
        if_1_A(x1) = max{1, x1 - 7}
        if_2_A(x1,x2) = max{2, x1 - 14, x2 - 7}
        le_0_A = 1
        le_1_A(x1) = max{2, x1 - 97}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 34}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 54}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 36}
        minus_2_A(x1,x2) = max{3, x1 - 61, x2 - 104}
        |0|_0_A = 2
        le_2_A(x1,x2) = max{3, x1 - 104, x2 - 61}
        perfectp_1_A(x1) = max{5, x1 - 7}
        f_4_A(x1,x2,x3,x4) = max{4, x1 - 67, x2 - 24, x3 - 18, x4 - 18}
        filter_2_A(x1,x2) = max{9, x1 - 34, x2 - 27}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > if_0 = f_4 > f_3 = if_1 = if_2 = le_2 = filter_2 = nil_0 > filter_1 = f_0 = f_2 = filter2_0 = filter2_2 = le_1 = map_1 = s_1 = minus_2 > f_1 = filter2_1 = le_0 = map_0 = minus_1 = s_0 = perfectp_1 > minus_0 = |0|_0 > cons_2 = false_0 > filter2_4# = filter_2# = cons_1 > true_0 = app# = cons_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filter_1) = []
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(f_3) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(perfectp_1) = []
      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)) -> app#(|fun|,x)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4}


-- 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r2, r3, r4, r5, r7, r9, r10, r11, r13, r14, r16, r18, r20, r24, r27, r28, r32, r34, r35, r40

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{11, x3 + 2, x4 + 6}
        true_0_A = 7
        filter_2#_A(x1,x2) = x2 + 5
        cons_2_A(x1,x2) = max{x1 + 20, x2 + 7}
        app_A(x1,x2) = max{x1 - 21, x2 + 8}
        false_0_A = 9
        cons_1_A(x1) = x1 + 42
        f_0_A = 20
        f_1_A(x1) = max{0, x1 - 1}
        f_2_A(x1,x2) = max{0, x1 - 23, x2 - 1}
        f_3_A(x1,x2,x3) = max{8, x1 - 52, x3 - 86}
        filter_0_A = 31
        filter_1_A(x1) = max{9, x1 - 46}
        filter2_0_A = 1
        filter2_1_A(x1) = max{5, x1 - 1}
        filter2_2_A(x1,x2) = max{6, x1 - 23, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 45, x2 - 59, x3 - 2}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 1}
        if_2_A(x1,x2) = max{3, x1 - 23, x2 - 1}
        le_0_A = 6
        le_1_A(x1) = max{7, x1 - 1}
        map_0_A = 1
        map_1_A(x1) = max{0, x1 - 9}
        minus_0_A = 1
        minus_1_A(x1) = max{0, x1 - 64}
        minus_2_A(x1,x2) = max{28, x1 - 84, x2 - 13}
        |0|_0_A = 10
        le_2_A(x1,x2) = max{69, x1 - 13, x2 - 83}
        s_1_A(x1) = max{12, x1 - 84}
        perfectp_1_A(x1) = max{11, x1 - 156}
        f_4_A(x1,x2,x3,x4) = max{x1 - 51, x2 - 69, x3 - 86, x4 + 11}
        filter_2_A(x1,x2) = max{15, x1 - 66, x2 - 65}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        le_1 = nil_0 > cons_2 = app = filter2_3 = filter_2 > cons_1 = filter_1 = filter2_2 = le_0 = map_1 = le_2 = s_1 = f_4 > f_3 = filter_0 = filter2_1 = if_2 = map_0 = minus_1 = minus_2 = perfectp_1 > f_2 = filter2_0 = if_1 = minus_0 = |0|_0 > filter2_4# = true_0 = f_1 = if_0 > filter_2# = f_0 > false_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(false_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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(s_1) = []
      pi(perfectp_1) = []
      pi(f_4) = [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)
p2: 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p2: 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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, r16, r17, r18, r20, r23, r24, r27, r28, r29, r32, r34, r35, r40

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x2 + 36
        cons_2_A(x1,x2) = max{x1 + 22, x2 + 38}
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 2, x4 + 37}
        app_A(x1,x2) = max{x1 + 12, x2 + 57}
        false_0_A = 2
        cons_0_A = 1
        cons_1_A(x1) = max{14, x1 + 11}
        f_0_A = 30
        f_1_A(x1) = max{41, x1}
        f_2_A(x1,x2) = max{x1 - 13, x2 + 56}
        f_3_A(x1,x2,x3) = max{55, x1 - 21, x2 - 14}
        filter_0_A = 47
        filter_1_A(x1) = 58
        filter2_0_A = 46
        filter2_1_A(x1) = max{58, x1 - 70}
        filter2_2_A(x1,x2) = max{0, x1 - 58, x2 - 14}
        filter2_3_A(x1,x2,x3) = max{x1 - 57, x2 - 13, x3}
        if_0_A = 1
        if_1_A(x1) = max{14, x1 - 13}
        le_0_A = 47
        le_1_A(x1) = max{58, x1 + 1}
        le_2_A(x1,x2) = max{68, x1 - 11, x2 - 11}
        map_0_A = 47
        map_1_A(x1) = max{58, x1}
        minus_0_A = 47
        minus_1_A(x1) = max{58, x1}
        s_0_A = 1
        s_1_A(x1) = x1 + 10
        minus_2_A(x1,x2) = max{2, x1 - 12, x2 - 10}
        |0|_0_A = 0
        true_0_A = 0
        perfectp_1_A(x1) = max{3, x1}
        f_4_A(x1,x2,x3,x4) = max{68, x1 - 9, x2 - 14, x3 - 11, x4 + 1}
        filter_2_A(x1,x2) = max{0, x1 - 39, x2 - 20}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app = cons_0 = le_0 = minus_2 = f_4 > cons_2 = f_2 = filter_1 = le_1 = le_2 = map_1 = minus_1 = nil_0 > filter2_4# = cons_1 = f_1 = f_3 = filter_0 = filter2_3 = map_0 = minus_0 = s_1 > filter_2# = f_0 = filter2_2 = if_1 = s_0 = perfectp_1 > filter2_1 = if_0 = |0|_0 > false_0 = filter2_0 = true_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(filter2_4#) = [3]
      pi(app) = [2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      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) = [3]
      pi(if_0) = []
      pi(if_1) = []
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(perfectp_1) = [1]
      pi(f_4) = [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: 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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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)
p3: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24, r27, r28, r32, r34, r35, r40

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) = x1
        s_1_A(x1) = x1
        |0|_0_A = 6
        minus_2_A(x1,x2) = 13
        le_2_A(x1,x2) = max{8, x1 - 7, x2 - 7}
        true_0_A = 4
        false_0_A = 0
        perfectp_1_A(x1) = max{18, x1 - 7}
        f_4_A(x1,x2,x3,x4) = max{5, x1 - 16, x2 - 2, x3 - 8, x4 - 8}
        filter_2_A(x1,x2) = max{19, x1 - 2, x2 - 2}
        nil_0_A = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > true_0 > le_2 > minus_2 = f_4 > |0|_0 > f_4# = s_1 > perfectp_1 > false_0 > nil_0
      
  
    partial status:
  
      pi(f_4#) = [1]
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(minus_2) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      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: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p2: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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: f_4#(s_1(x),s_1(y),z,u) -> f_4#(x,u,z,u)
p2: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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, r16, r18, r20, r23, r24, r27, r28, r32, r34, r35, r40

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) = x1 + 22
        s_1_A(x1) = x1 + 23
        minus_2_A(x1,x2) = max{6, x1 - 75, x2 - 75}
        app_A(x1,x2) = max{108, x1 - 27, x2 + 24}
        cons_0_A = 26
        cons_1_A(x1) = max{25, x1 - 1}
        cons_2_A(x1,x2) = max{22, x1 - 136, x2 - 27}
        f_0_A = 24
        f_1_A(x1) = max{25, x1 - 1}
        f_2_A(x1,x2) = max{26, x1 - 29, x2 - 1}
        f_3_A(x1,x2,x3) = max{9, x1 - 57, x2 - 29, x3 - 21}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 85}
        filter2_0_A = 19
        filter2_1_A(x1) = max{20, x1 - 1}
        filter2_2_A(x1,x2) = max{21, x1 - 29, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{22, x1 - 57, x2 - 28, x3 - 1}
        if_0_A = 1
        if_1_A(x1) = max{25, x1 - 1}
        if_2_A(x1,x2) = max{8, x1 - 29, x2 - 1}
        le_0_A = 9
        le_1_A(x1) = max{10, x1 - 1}
        map_0_A = 24
        map_1_A(x1) = max{25, x1 - 1}
        minus_0_A = 4
        minus_1_A(x1) = max{5, x1 - 47}
        s_0_A = 17
        |0|_0_A = 5
        le_2_A(x1,x2) = max{11, x1 - 28, x2 - 6}
        true_0_A = 4
        false_0_A = 4
        perfectp_1_A(x1) = max{25, x1 - 19}
        f_4_A(x1,x2,x3,x4) = max{10, x1 - 74, x2 - 27, x3 - 20, x4 - 20}
        filter_2_A(x1,x2) = max{8, x1 - 113, x2 - 21}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > filter2_2 = filter2_3 = map_0 = le_2 > cons_2 = filter2_1 = le_1 = f_4 = filter_2 > minus_2 = cons_1 = f_3 = filter_1 = filter2_0 = if_1 = le_0 = map_1 = nil_0 > f_4# = cons_0 = f_2 = filter_0 = if_0 = if_2 = minus_1 > s_1 = f_1 = minus_0 > f_0 = s_0 = perfectp_1 > |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(f_4#) = [1]
      pi(s_1) = [1]
      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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      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: f_4#(s_1(x),s_1(y),z,u) -> f_4#(s_1(x),minus_2(y,x),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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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#(s_1(x),minus_2(y,x),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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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, r16, r18, r20, r24, r25, r26, r27, r28, r32, r34, r35

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{1, x2 - 9}
        s_1_A(x1) = x1 + 11
        minus_2_A(x1,x2) = x1 + 4
        app_A(x1,x2) = max{37, x1 - 4, x2 - 4}
        cons_0_A = 6
        cons_1_A(x1) = max{1, x1 - 4}
        cons_2_A(x1,x2) = max{36, x1 - 75, x2 - 37}
        f_0_A = 2
        f_1_A(x1) = max{3, x1 - 5}
        f_2_A(x1,x2) = max{5, x1 - 17, x2 - 5}
        f_3_A(x1,x2,x3) = max{0, x1 - 21, x3 - 4}
        filter_0_A = 2
        filter_1_A(x1) = 3
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 5}
        filter2_2_A(x1,x2) = max{1, x1 - 9, x2 - 4}
        filter2_3_A(x1,x2,x3) = max{3, x1 - 14, x2 - 9, x3 - 5}
        if_0_A = 1
        if_1_A(x1) = max{2, x1 - 4}
        if_2_A(x1,x2) = max{3, x1 - 33, x2 - 4}
        le_0_A = 3
        le_1_A(x1) = max{36, x1 - 5}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 5}
        minus_0_A = 43
        minus_1_A(x1) = max{38, x1 - 5}
        |0|_0_A = 6
        le_2_A(x1,x2) = max{31, x1 - 10, x2 - 12}
        true_0_A = 5
        false_0_A = 5
        perfectp_1_A(x1) = max{7, x1 - 33}
        f_4_A(x1,x2,x3,x4) = max{x1 - 21, x2 + 20, x3 + 3, x4 + 4}
  
    base order 2:
  
      precedence:
      
        f_4# = s_1 = minus_2 = map_1 = le_2 = f_4 > app > f_3 = filter_1 = if_2 = le_1 = map_0 = minus_1 = perfectp_1 > cons_1 = cons_2 = f_0 = f_2 = filter_0 = filter2_3 = if_1 = le_0 = minus_0 = |0|_0 > cons_0 = f_1 = filter2_2 = if_0 = true_0 = false_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(f_4#) = []
      pi(s_1) = [1]
      pi(minus_2) = [1]
      pi(app) = []
      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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      pi(f_4) = [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: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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, r16, r18, r20, r24, r27, r28, r32, r34, r35, r40

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = max{7, x2 + 2}
        s_1_A(x1) = max{9, x1 + 6}
        app_A(x1,x2) = max{8, x1 - 11, x2 - 11}
        cons_0_A = 7
        cons_1_A(x1) = max{8, x1 - 12}
        cons_2_A(x1,x2) = max{5, x1 - 60, x2 - 12}
        f_0_A = 1
        f_1_A(x1) = max{2, x1 - 12}
        f_2_A(x1,x2) = max{3, x1 - 24, x2 - 12}
        f_3_A(x1,x2,x3) = max{4, x1 - 36, x2 - 24, x3 - 12}
        filter_0_A = 5
        filter_1_A(x1) = max{6, x1 - 40}
        filter2_0_A = 4
        filter2_1_A(x1) = max{5, x1 - 12}
        filter2_2_A(x1,x2) = max{6, x1 - 24, x2 - 12}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 36, x2 - 24, x3 - 12}
        if_0_A = 10
        if_1_A(x1) = max{8, x1 - 11}
        if_2_A(x1,x2) = max{8, x1 - 23, x2 - 11}
        le_0_A = 6
        le_1_A(x1) = max{7, x1 - 12}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 60}
        minus_0_A = 1
        minus_1_A(x1) = max{2, x1 - 21}
        minus_2_A(x1,x2) = max{3, x1 - 33, x2 - 67}
        |0|_0_A = 2
        le_2_A(x1,x2) = max{25, x1 - 10, x2 - 10}
        true_0_A = 1
        false_0_A = 1
        perfectp_1_A(x1) = max{6, x1 - 31}
        f_4_A(x1,x2,x3,x4) = max{5, x1 - 66, x2 - 36, x3 - 32, x4 - 36}
        filter_2_A(x1,x2) = max{7, x1 - 41, x2 - 46}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        le_2 > f_4 > minus_2 > s_1 = minus_1 = |0|_0 = perfectp_1 = filter_2 > app = le_0 = false_0 > le_2# = f_0 = f_2 = f_3 = filter_1 = filter2_3 = if_2 = le_1 = minus_0 > cons_1 = cons_2 = f_1 = filter_0 = filter2_0 = filter2_2 = if_1 = map_1 = nil_0 > cons_0 = filter2_1 = if_0 = map_0 = true_0
      
  
    partial status:
  
      pi(le_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(le_2) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      pi(f_4) = []
      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: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(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: minus_2(|0|_0(),y) -> |0|_0()
r25: minus_2(s_1(x),|0|_0()) -> s_1(x)
r26: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r27: le_2(|0|_0(),y) -> true_0()
r28: le_2(s_1(x),|0|_0()) -> false_0()
r29: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r30: if_3(true_0(),x,y) -> x
r31: if_3(false_0(),x,y) -> y
r32: perfectp_1(|0|_0()) -> false_0()
r33: perfectp_1(s_1(x)) -> f_4(x,s_1(|0|_0()),s_1(x),s_1(x))
r34: f_4(|0|_0(),y,|0|_0(),u) -> true_0()
r35: f_4(|0|_0(),y,s_1(z),u) -> false_0()
r36: f_4(s_1(x),|0|_0(),z,u) -> f_4(x,u,minus_2(z,s_1(x)),u)
r37: 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))
r38: map_2(|fun|,nil_0()) -> nil_0()
r39: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r40: filter_2(|fun|,nil_0()) -> nil_0()
r41: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r42: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r43: 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, r16, r18, r20, r27, r28, r32, r34, r35, r40

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = max{x1 - 7, x2 + 8}
        s_1_A(x1) = max{15, x1 + 9}
        app_A(x1,x2) = max{11, x1 - 13, x2 - 10}
        cons_0_A = 12
        cons_1_A(x1) = max{11, x1 - 10}
        cons_2_A(x1,x2) = max{8, x1 - 24, x2 - 11}
        f_0_A = 1
        f_1_A(x1) = max{2, x1 - 11}
        f_2_A(x1,x2) = max{3, x1 - 25, x2 - 11}
        f_3_A(x1,x2,x3) = max{4, x1 - 39, x2 - 67, x3 - 17}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 27}
        filter2_0_A = 7
        filter2_1_A(x1) = max{8, x1 - 11}
        filter2_2_A(x1,x2) = max{9, x1 - 25, x2 - 11}
        filter2_3_A(x1,x2,x3) = max{10, x1 - 39, x2 - 25, x3 - 11}
        if_0_A = 12
        if_1_A(x1) = max{10, x1 - 11}
        le_0_A = 10
        le_1_A(x1) = max{11, x1 - 10}
        map_0_A = 8
        map_1_A(x1) = max{9, x1 - 11}
        minus_0_A = 1
        minus_1_A(x1) = max{0, x1 - 11}
        le_2_A(x1,x2) = max{10, x1 - 24, x2 - 33}
        |0|_0_A = 7
        true_0_A = 1
        false_0_A = 6
        perfectp_1_A(x1) = max{12, x1 - 67}
        f_4_A(x1,x2,x3,x4) = max{8, x1 - 52, x2 - 66, x3 - 16, x4 - 59}
        filter_2_A(x1,x2) = max{10, x1 - 26, x2 - 11}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        le_2 > f_4 > minus_2# = s_1 > f_3 = minus_1 = perfectp_1 = filter_2 > app = le_0 = |0|_0 > cons_0 = f_2 = filter_1 = filter2_3 = if_1 = le_1 = minus_0 = false_0 > cons_1 = cons_2 = f_1 = filter_0 = filter2_2 = if_0 = map_1 = nil_0 > f_0 = filter2_1 = map_0 = true_0 > filter2_0
      
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      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(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(perfectp_1) = []
      pi(f_4) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.