YES
We show the termination of the TRS R:

  app(app(.(),|1|()),x) -> x
  app(app(.(),x),|1|()) -> x
  app(app(.(),app(i(),x)),x) -> |1|()
  app(app(.(),x),app(i(),x)) -> |1|()
  app(app(.(),app(i(),y)),app(app(.(),y),z)) -> z
  app(app(.(),y),app(app(.(),app(i(),y)),z)) -> z
  app(app(.(),app(app(.(),x),y)),z) -> app(app(.(),x),app(app(.(),y),z))
  app(i(),|1|()) -> |1|()
  app(i(),app(i(),x)) -> x
  app(i(),app(app(.(),x),y)) -> app(app(.(),app(i(),y)),app(i(),x))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
  app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
  app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Instead we prove termination of the uncurried system:

  app(._0(),y) -> ._1(y)
  app(._1(x1),y) -> ._2(x1,y)
  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,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(i_0(),y) -> i_1(y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  ._2(|1|_0(),x) -> x
  ._2(x,|1|_0()) -> x
  ._2(i_1(x),x) -> |1|_0()
  ._2(x,i_1(x)) -> |1|_0()
  ._2(i_1(y),._2(y,z)) -> z
  ._2(y,._2(i_1(y),z)) -> z
  ._2(._2(x,y),z) -> ._2(x,._2(y,z))
  i_1(|1|_0()) -> |1|_0()
  i_1(i_1(x)) -> x
  i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
  filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)


-- SCC decomposition.

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

p1: app#(._1(x1),y) -> ._2#(x1,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#(i_0(),y) -> i_1#(y)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: ._2#(._2(x,y),z) -> ._2#(x,._2(y,z))
p7: ._2#(._2(x,y),z) -> ._2#(y,z)
p8: i_1#(._2(x,y)) -> ._2#(i_1(y),i_1(x))
p9: i_1#(._2(x,y)) -> i_1#(y)
p10: i_1#(._2(x,y)) -> i_1#(x)
p11: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p12: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p13: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p14: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p15: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p16: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p5, p11, p12, p13, p14, p15, p16}
  {p9, p10}
  {p6, p7}


-- Reduction pair.

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

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

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r7, r8, r16, r17, r21, r26

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{5, x1 - 1}
        cons_2_A(x1,x2) = max{3, x1 - 31, x2 - 31}
        app#_A(x1,x2) = max{4, x1 - 2}
        map_1_A(x1) = max{15, x1 + 8}
        filter2_3_A(x1,x2,x3) = max{15, x1 + 6, x2 + 8, x3 + 6}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 5
        false_0_A = 3
        filter_2#_A(x1,x2) = x1 + 5
        filter_1_A(x1) = max{14, x1 + 8}
        app_A(x1,x2) = max{17, x1 - 18, x2 - 12}
        true_0_A = 1
        ._0_A = 14
        ._1_A(x1) = max{15, x1 - 13}
        cons_0_A = 16
        cons_1_A(x1) = max{17, x1 - 12}
        filter2_0_A = 12
        filter2_1_A(x1) = max{13, x1 - 13}
        filter2_2_A(x1,x2) = max{14, x1 - 31, x2 - 12}
        ._2_A(x1,x2) = max{16, x1 - 32, x2 - 82}
        i_1_A(x1) = max{15, x1 - 34}
        |1|_0_A = 0
        filter_2_A(x1,x2) = max{15, x1 + 9, x2 - 11}
        nil_0_A = 10
  
    base order 2:
  
      precedence:
      
        app > cons_0 = ._2 = filter_2 > cons_2 = map_1 = filter2_3 = false_0 = filter_1 = ._1 = i_1 = nil_0 > map_2# = ._0 = cons_1 = filter2_2 = |1|_0 > app# = true_0 = filter2_1 > filter2_4# = filter_2# = filter2_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(._2) = []
      pi(i_1) = []
      pi(|1|_0) = []
      pi(filter_2) = [1]
      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#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,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#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: app#(map_1(x1),y) -> map_2#(x1,y)
p8: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r16, r17, r21, r26

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{7, x1 - 2}
        cons_2_A(x1,x2) = max{3, x1 - 24, x2 - 24}
        app#_A(x1,x2) = max{4, 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 = 1
        filter_2#_A(x1,x2) = x1 + 6
        app_A(x1,x2) = max{18, x1 - 11, x2 - 11}
        false_0_A = 1
        map_1_A(x1) = max{10, x1 + 2}
        ._0_A = 9
        ._1_A(x1) = max{10, x1 - 12}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 12}
        filter_0_A = 9
        filter_1_A(x1) = max{10, x1 - 12}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 11}
        filter2_2_A(x1,x2) = max{4, x1 - 23, x2 - 12}
        ._2_A(x1,x2) = max{2, x1 - 24}
        i_1_A(x1) = max{2, x1 - 24}
        |1|_0_A = 1
        filter_2_A(x1,x2) = max{11, x1 - 24, x2 - 27}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        nil_0 > filter_2 > app = false_0 > filter2_4# = filter_2# = map_1 = ._1 > cons_2 = filter2_3 = ._0 = cons_0 = filter_1 = filter2_0 = ._2 > app# = true_0 = cons_1 = filter_0 = filter2_2 = i_1 > map_2# = filter2_1 = |1|_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(app) = []
      pi(false_0) = []
      pi(map_1) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(._2) = []
      pi(i_1) = []
      pi(|1|_0) = []
      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#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p7}
  {p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r12, r16, r17, r21, r26

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 + 5, x2 + 29}
        cons_2_A(x1,x2) = max{3, x1 - 24, x2}
        app_A(x1,x2) = max{x1 - 9, x2 + 8}
        ._0_A = 8
        ._1_A(x1) = max{6, x1 - 1}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 1}
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 31}
        filter2_0_A = 7
        filter2_1_A(x1) = max{8, x1 - 1}
        filter2_2_A(x1,x2) = max{5, x1 - 11, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 21, x2 - 11, x3 - 1}
        map_0_A = 7
        map_1_A(x1) = max{8, x1 - 1}
        ._2_A(x1,x2) = max{7, x1 - 11, x2 - 12}
        i_1_A(x1) = max{6, x1 - 14}
        |1|_0_A = 0
        filter_2_A(x1,x2) = max{2, x1 - 41, x2 - 31}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > map_2# > cons_2 = cons_0 > cons_1 = filter2_3 = map_1 = ._2 = filter_2 = nil_0 > ._1 = filter_1 = filter2_2 = map_0 = i_1 > ._0 = filter_0 = filter2_1 = |1|_0 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(._2) = []
      pi(i_1) = []
      pi(|1|_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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r12, r16, r17, r21, r26

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 - 3}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 21, x2 + 2, x3 - 11}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x2 - 2}
        false_0_A = 6
        filter_2#_A(x1,x2) = max{5, x1 - 2}
        cons_2_A(x1,x2) = max{3, x1 - 43, x2 - 43}
        app_A(x1,x2) = max{15, x1 - 10, x2 - 10}
        true_0_A = 1
        ._0_A = 8
        ._1_A(x1) = max{9, x1 - 11}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 11}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 11}
        filter2_0_A = 6
        filter2_1_A(x1) = max{7, x1 - 11}
        filter2_2_A(x1,x2) = max{8, x1 - 22, x2 - 11}
        map_0_A = 8
        map_1_A(x1) = max{9, x1 - 43}
        ._2_A(x1,x2) = max{14, x1 - 22, x2 - 19}
        i_1_A(x1) = max{2, x1 - 35}
        |1|_0_A = 0
        filter_2_A(x1,x2) = max{13, x1 - 43, x2 - 32}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = ._0 > ._1 = ._2 > filter2_3 = map_1 = i_1 = filter_2 > false_0 = filter_1 = filter2_2 = map_0 = |1|_0 = nil_0 > filter2_4# = filter_2# = filter_0 = filter2_1 > app# = filter2_0 > cons_2 > true_0 = cons_1 > cons_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(true_0) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(._2) = []
      pi(i_1) = []
      pi(|1|_0) = []
      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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r12, r16, r17, r21, r26

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 + 2}
        false_0_A = 10
        filter_2#_A(x1,x2) = max{1, x2 - 1}
        cons_2_A(x1,x2) = max{21, x1 + 4, x2 + 4}
        app_A(x1,x2) = max{41, x1 - 5, x2 + 11}
        true_0_A = 20
        ._0_A = 1
        ._1_A(x1) = max{4, x1 - 37}
        cons_0_A = 4
        cons_1_A(x1) = max{12, x1 + 10}
        filter_0_A = 3
        filter_1_A(x1) = max{4, x1 - 1}
        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}
        map_0_A = 3
        map_1_A(x1) = x1 + 6
        ._2_A(x1,x2) = max{41, x1 - 42, x2 - 126}
        i_1_A(x1) = max{12, x1 - 84}
        |1|_0_A = 0
        filter_2_A(x1,x2) = max{21, x1 - 25, x2 - 22}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > filter_2# = app > false_0 = filter_1 = filter2_3 = ._2 > cons_2 = ._1 = filter_0 = filter2_2 = map_1 = i_1 = nil_0 > filter2_4# = true_0 = ._0 = cons_1 = filter2_1 = map_0 = |1|_0 > cons_0 = filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(true_0) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(._2) = []
      pi(i_1) = []
      pi(|1|_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: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: i_1#(._2(x,y)) -> i_1#(x)
p2: i_1#(._2(x,y)) -> i_1#(y)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r12, r14, r15, r16, r17, r18, r19, r21, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        i_1#_A(x1) = max{8, x1 + 5}
        ._2_A(x1,x2) = max{x1 + 7, x2 + 9}
        app_A(x1,x2) = max{7, x1 - 8, x2 - 4}
        ._0_A = 7
        ._1_A(x1) = max{6, x1 - 5}
        cons_0_A = 7
        cons_1_A(x1) = max{1, x1 - 5}
        cons_2_A(x1,x2) = max{2, x1 - 37, x2 - 7}
        filter_0_A = 6
        filter_1_A(x1) = max{7, x1 - 4}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 5}
        filter2_2_A(x1,x2) = max{3, x1 - 14, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 23, x2 - 14, x3 - 5}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 37}
        |1|_0_A = 2
        i_1_A(x1) = max{3, x1 - 10}
        filter_2_A(x1,x2) = max{4, x1 - 41, x2 - 28}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        ._2 = filter_2 > app > i_1# = ._1 = cons_2 = filter_1 = filter2_3 = map_1 = i_1 = nil_0 > ._0 = cons_1 = filter_0 = filter2_2 = map_0 = |1|_0 > cons_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(i_1#) = [1]
      pi(._2) = [2]
      pi(app) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|1|_0) = []
      pi(i_1) = []
      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: i_1#(._2(x,y)) -> i_1#(x)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: i_1#(._2(x,y)) -> i_1#(x)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r3, r4, r5, r7, r8, r9, r12, r15, r16, r17, r21, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        i_1#_A(x1) = x1 + 27
        ._2_A(x1,x2) = max{x1 + 28, x2 - 14}
        app_A(x1,x2) = max{11, x1 - 14, x2 - 3}
        cons_0_A = 13
        cons_1_A(x1) = max{1, x1 - 4}
        cons_2_A(x1,x2) = max{2, x1 - 52, x2 - 56}
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 49}
        filter2_0_A = 6
        filter2_1_A(x1) = max{7, x1 - 4}
        filter2_2_A(x1,x2) = max{8, x1 - 19, x2 - 4}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 34, x2 - 19, x3 - 4}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 52}
        |1|_0_A = 9
        i_1_A(x1) = max{10, x1 - 41}
        filter_2_A(x1,x2) = max{9, x1 - 64, x2 - 49}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        ._2 = filter_2 = nil_0 > app = filter_0 > i_1# = cons_0 = filter_1 = filter2_3 = i_1 > cons_1 = cons_2 = filter2_2 = map_1 = |1|_0 > filter2_1 = map_0 > filter2_0
      
  
    partial status:
  
      pi(i_1#) = [1]
      pi(._2) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|1|_0) = []
      pi(i_1) = []
      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: ._2#(._2(x,y),z) -> ._2#(x,._2(y,z))
p2: ._2#(._2(x,y),z) -> ._2#(y,z)

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r12, r14, r15, r16, r17, r18, r19, r20, r21, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ._2#_A(x1,x2) = max{12, x1 + 5, x2 + 2}
        ._2_A(x1,x2) = max{7, x1 + 3, x2}
        app_A(x1,x2) = max{6, x1 - 7, x2 - 5}
        ._0_A = 4
        ._1_A(x1) = max{5, x1 - 6}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 5}
        cons_2_A(x1,x2) = max{3, x1 - 19, x2 - 6}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 6}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 6}
        filter2_2_A(x1,x2) = max{4, x1 - 14, x2 - 6}
        filter2_3_A(x1,x2,x3) = max{5, x1 - 22, x2 - 14, x3 - 11}
        map_0_A = 6
        map_1_A(x1) = 1
        |1|_0_A = 2
        i_1_A(x1) = max{5, x1 - 8}
        filter_2_A(x1,x2) = max{3, x1 - 21, x2 - 11}
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        ._2# > ._2 > app = cons_0 = map_0 > ._1 = filter2_3 = i_1 = filter_2 > ._0 = cons_2 = filter_1 = filter2_1 = filter2_2 = map_1 = |1|_0 = nil_0 > cons_1 = filter_0 = filter2_0
      
  
    partial status:
  
      pi(._2#) = [1]
      pi(._2) = [1, 2]
      pi(app) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|1|_0) = []
      pi(i_1) = []
      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: ._2#(._2(x,y),z) -> ._2#(x,._2(y,z))

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: ._2#(._2(x,y),z) -> ._2#(x,._2(y,z))

and R consists of:

r1: app(._0(),y) -> ._1(y)
r2: app(._1(x1),y) -> ._2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(i_0(),y) -> i_1(y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: ._2(|1|_0(),x) -> x
r15: ._2(x,|1|_0()) -> x
r16: ._2(i_1(x),x) -> |1|_0()
r17: ._2(x,i_1(x)) -> |1|_0()
r18: ._2(i_1(y),._2(y,z)) -> z
r19: ._2(y,._2(i_1(y),z)) -> z
r20: ._2(._2(x,y),z) -> ._2(x,._2(y,z))
r21: i_1(|1|_0()) -> |1|_0()
r22: i_1(i_1(x)) -> x
r23: i_1(._2(x,y)) -> ._2(i_1(y),i_1(x))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r12, r15, r16, r17, r21, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ._2#_A(x1,x2) = x1 + 14
        ._2_A(x1,x2) = max{9, x1 + 5, x2 - 10}
        app_A(x1,x2) = max{8, x1 - 9, x2 - 9}
        ._0_A = 1
        ._1_A(x1) = max{2, x1 - 10}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 40}
        cons_2_A(x1,x2) = max{3, x1 - 49, x2 - 40}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 39}
        filter2_0_A = 8
        filter2_1_A(x1) = max{4, x1 - 9}
        filter2_2_A(x1,x2) = max{5, x1 - 19, x2 - 10}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 29, x2 - 20, x3 - 25}
        map_0_A = 7
        map_1_A(x1) = max{8, x1 - 9}
        |1|_0_A = 6
        i_1_A(x1) = max{7, x1 - 13}
        filter_2_A(x1,x2) = max{6, x1 - 49, x2 - 36}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        ._2# > ._2 = filter_2 > app > ._1 = cons_2 = filter_1 = filter2_3 = map_1 = i_1 = nil_0 > ._0 = cons_1 = filter_0 = filter2_2 = map_0 = |1|_0 > cons_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(._2#) = []
      pi(._2) = [1]
      pi(app) = []
      pi(._0) = []
      pi(._1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|1|_0) = []
      pi(i_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.