YES
We show the termination of the TRS R:

  app(app(|:|(),x),x) -> e()
  app(app(|:|(),e()),x) -> x
  app(app(|:|(),x),app(app(.(),x),y)) -> y
  app(app(|:|(),app(app(/(),x),y)),x) -> y
  app(app(/(),x),x) -> e()
  app(app(/(),x),e()) -> x
  app(app(/(),app(app(.(),y),x)),x) -> y
  app(app(/(),x),app(app(|:|(),y),x)) -> y
  app(app(.(),e()),x) -> x
  app(app(.(),x),e()) -> x
  app(app(.(),x),app(app(|:|(),x),y)) -> y
  app(app(.(),app(app(/(),y),x)),x) -> y
  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(/_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(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(|:|_0(),y) -> |:|_1(y)
  app(|:|_1(x1),y) -> |:|_2(x1,y)
  |:|_2(x,x) -> e_0()
  |:|_2(e_0(),x) -> x
  |:|_2(x,._2(x,y)) -> y
  |:|_2(/_2(x,y),x) -> y
  /_2(x,x) -> e_0()
  /_2(x,e_0()) -> x
  /_2(._2(y,x),x) -> y
  /_2(x,|:|_2(y,x)) -> y
  ._2(e_0(),x) -> x
  ._2(x,e_0()) -> x
  ._2(x,|:|_2(x,y)) -> y
  ._2(/_2(y,x),x) -> y
  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#(/_1(x1),y) -> /_2#(x1,y)
p3: app#(filter_1(x1),y) -> filter_2#(x1,y)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: app#(|:|_1(x1),y) -> |:|_2#(x1,y)
p7: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p8: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p9: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p10: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p11: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p12: 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p3, p4, p5, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p7: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r5, r9, r10, r15, r17, r21, r31

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 5
        filter_1_A(x1) = x1 + 6
        filter_2#_A(x1,x2) = x1 + 5
        cons_2_A(x1,x2) = max{9, x1 - 67, x2 - 11}
        map_1_A(x1) = x1 + 20
        map_2#_A(x1,x2) = max{8, x1 + 6}
        filter2_3_A(x1,x2,x3) = max{7, x1 + 2, x2 + 6, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 5
        false_0_A = 1
        app_A(x1,x2) = max{22, x1 - 25, x2 - 25}
        true_0_A = 1
        ._0_A = 20
        ._1_A(x1) = max{21, x1 - 26}
        cons_0_A = 1
        cons_1_A(x1) = max{22, x1 - 26}
        filter2_0_A = 1
        filter2_1_A(x1) = max{5, x1 - 26}
        filter2_2_A(x1,x2) = max{6, x1 - 52, x2 - 26}
        |:|_0_A = 1
        |:|_1_A(x1) = max{2, x1 - 26}
        |:|_2_A(x1,x2) = max{23, x1 - 1, x2 - 1}
        e_0_A = 0
        /_2_A(x1,x2) = max{23, x1 - 51, x2 - 26}
        filter_2_A(x1,x2) = max{7, x1 - 67, x2 - 27}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_2# = true_0 = /_2 > cons_2 = false_0 = e_0 = nil_0 > |:|_2 = filter_2 > app > filter_1 = map_1 = filter2_3 = ._1 = cons_1 = |:|_1 > filter_2# = filter2_4# = ._0 = cons_0 = filter2_2 = |:|_0 > app# = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(map_1) = []
      pi(map_2#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(false_0) = []
      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(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(e_0) = []
      pi(/_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: 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: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)
p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p8: 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r9, r10, r11, r15, r17, r21, r31

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 2
        filter_1_A(x1) = max{13, x1 + 11}
        filter_2#_A(x1,x2) = x1 + 10
        cons_2_A(x1,x2) = max{8, x1 - 28, x2 - 28}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 10
        app_A(x1,x2) = max{11, x1 - 14, x2}
        true_0_A = 3
        map_1_A(x1) = max{11, x1 + 4}
        map_2#_A(x1,x2) = max{5, x1 + 3}
        false_0_A = 1
        ._0_A = 8
        ._1_A(x1) = max{9, x1 - 1}
        /_0_A = 9
        /_1_A(x1) = max{10, x1 - 1}
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 1}
        filter2_0_A = 1
        filter2_1_A(x1) = max{0, x1 - 1}
        filter2_2_A(x1,x2) = max{0, x1 - 16, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{0, x1 - 31, x2 - 50, x3 - 1}
        |:|_0_A = 1
        |:|_1_A(x1) = max{2, x1 - 1}
        |:|_2_A(x1,x2) = max{3, x1 - 16, x2 - 1}
        e_0_A = 0
        /_2_A(x1,x2) = max{1, x1 - 16, x2 - 1}
        filter_2_A(x1,x2) = max{12, x1 - 35, x2 - 7}
        nil_0_A = 6
  
    base order 2:
  
      precedence:
      
        filter_2 > app > filter_1 = false_0 = /_0 = /_2 = nil_0 > filter_2# = filter2_4# = /_1 > cons_2 = map_1 = ._1 = |:|_2 > app# = true_0 = ._0 = cons_1 = filter2_3 = |:|_1 = e_0 > map_2# = cons_0 = filter2_2 = |:|_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = [1]
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(filter2_4#) = [2]
      pi(app) = [2]
      pi(true_0) = []
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(false_0) = []
      pi(._0) = []
      pi(._1) = []
      pi(/_0) = []
      pi(/_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(e_0) = []
      pi(/_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)
p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p7: 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p6}
  {p1, p2, p3, p4, p7}


-- 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r10, r13, r15, r17, r21, r31

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x2 - 4}
        cons_2_A(x1,x2) = max{x1 - 11, x2 + 9}
        app_A(x1,x2) = max{10, x1 - 12, x2 - 25}
        ._0_A = 11
        ._1_A(x1) = max{1, x1 - 26}
        /_0_A = 9
        /_1_A(x1) = max{10, x1 - 25}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 26}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 26}
        filter2_0_A = 1
        filter2_1_A(x1) = max{0, x1 - 26}
        filter2_2_A(x1,x2) = max{10, x2 - 25}
        map_0_A = 8
        map_1_A(x1) = max{9, x1 - 26}
        |:|_0_A = 7
        |:|_1_A(x1) = max{8, x1 - 26}
        |:|_2_A(x1,x2) = max{9, x1 - 39, x2 - 26}
        e_0_A = 0
        /_2_A(x1,x2) = max{9, x1 - 37, x2 - 25}
        filter_2_A(x1,x2) = max{11, x1 - 11, x2 - 13}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app = map_0 = nil_0 > cons_2 = ._0 = filter_1 = filter2_2 = map_1 = |:|_2 = /_2 > map_2# = ._1 = /_1 = cons_1 = filter_0 = filter2_1 = |:|_1 = e_0 > /_0 = cons_0 = filter2_0 = |:|_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(._0) = []
      pi(._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(map_0) = []
      pi(map_1) = []
      pi(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(e_0) = []
      pi(/_2) = []
      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#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r10, r11, r13, r15, r17, r21, r31

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{2, x1 - 2}
        filter_1_A(x1) = x1 + 5
        filter_2#_A(x1,x2) = max{2, x1 - 1}
        cons_2_A(x1,x2) = max{10, x1 - 15, x2 - 15}
        filter2_4#_A(x1,x2,x3,x4) = max{2, x2 - 1}
        app_A(x1,x2) = max{12, x1 - 13, x2 + 6}
        false_0_A = 7
        true_0_A = 7
        ._0_A = 1
        ._1_A(x1) = max{2, x1 - 1}
        /_0_A = 11
        /_1_A(x1) = max{12, x1 + 6}
        cons_0_A = 12
        cons_1_A(x1) = max{9, x1 - 1}
        filter_0_A = 12
        filter2_0_A = 6
        filter2_1_A(x1) = max{7, x1 - 1}
        filter2_2_A(x1,x2) = max{8, x1 - 14, x2 + 6}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 28, x2 - 10, x3 - 33}
        map_0_A = 54
        map_1_A(x1) = max{40, x1 - 1}
        |:|_0_A = 6
        |:|_1_A(x1) = max{7, x1 - 1}
        |:|_2_A(x1,x2) = max{8, x1 - 15, x2 - 1}
        e_0_A = 0
        /_2_A(x1,x2) = max{12, x1 - 8, x2 - 1}
        filter_2_A(x1,x2) = max{10, x1 - 9, x2 - 16}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_1 > app = filter_0 = /_2 > filter_2# = filter2_4# = ._1 = /_1 = filter2_3 = |:|_2 > app# = ._0 = /_0 = filter2_2 = |:|_1 = e_0 > cons_2 = filter2_1 = |:|_0 > cons_1 = filter2_0 = map_1 = filter_2 = nil_0 > false_0 = true_0 = cons_0 = map_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(true_0) = []
      pi(._0) = []
      pi(._1) = []
      pi(/_0) = []
      pi(/_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(e_0) = []
      pi(/_2) = []
      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: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: 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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r10, r11, r13, r15, r17, r21, r31

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 + 5
        cons_2_A(x1,x2) = max{19, x1 + 7, x2 + 7}
        filter2_4#_A(x1,x2,x3,x4) = max{8, x3 + 6, x4 + 6}
        app_A(x1,x2) = max{x1 - 8, x2 + 25}
        true_0_A = 28
        false_0_A = 26
        ._0_A = 1
        ._1_A(x1) = max{7, x1 - 1}
        /_0_A = 1
        /_1_A(x1) = max{7, x1 - 1}
        cons_0_A = 1
        cons_1_A(x1) = x1 + 16
        filter_0_A = 1
        filter_1_A(x1) = max{7, x1 - 1}
        filter2_0_A = 5
        filter2_1_A(x1) = max{6, x1 - 1}
        filter2_2_A(x1,x2) = max{7, x1 - 9, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{5, x1 - 18, x2 - 10, x3 - 1}
        map_0_A = 1
        map_1_A(x1) = max{7, x1 - 1}
        |:|_0_A = 6
        |:|_1_A(x1) = max{7, x1 - 1}
        |:|_2_A(x1,x2) = max{24, x1 - 10, x2 - 1}
        e_0_A = 0
        /_2_A(x1,x2) = max{26, x1 - 8, x2 - 1}
        filter_2_A(x1,x2) = max{22, x1 - 35, x2 - 20}
        nil_0_A = 5
  
    base order 2:
  
      precedence:
      
        false_0 > map_1 = filter_2 = nil_0 > cons_2 > filter2_4# > filter_2# > app = ._0 = cons_0 > true_0 = ._1 = /_0 = cons_1 = filter_1 = filter2_0 = filter2_3 = map_0 = |:|_2 = /_2 > /_1 = filter_0 = filter2_1 = filter2_2 = |:|_1 = e_0 > |:|_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(._0) = []
      pi(._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(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(e_0) = []
      pi(/_2) = []
      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(),f,x,xs) -> filter_2#(f,xs)
p2: 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(/_0(),y) -> /_1(y)
r4: app(/_1(x1),y) -> /_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,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(map_0(),y) -> map_1(y)
r14: app(map_1(x1),y) -> map_2(x1,y)
r15: app(|:|_0(),y) -> |:|_1(y)
r16: app(|:|_1(x1),y) -> |:|_2(x1,y)
r17: |:|_2(x,x) -> e_0()
r18: |:|_2(e_0(),x) -> x
r19: |:|_2(x,._2(x,y)) -> y
r20: |:|_2(/_2(x,y),x) -> y
r21: /_2(x,x) -> e_0()
r22: /_2(x,e_0()) -> x
r23: /_2(._2(y,x),x) -> y
r24: /_2(x,|:|_2(y,x)) -> y
r25: ._2(e_0(),x) -> x
r26: ._2(x,e_0()) -> x
r27: ._2(x,|:|_2(x,y)) -> y
r28: ._2(/_2(y,x),x) -> y
r29: map_2(f,nil_0()) -> nil_0()
r30: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r31: filter_2(f,nil_0()) -> nil_0()
r32: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r33: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r34: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)