YES
We show the termination of the TRS R:

  app(app(|:|(),app(app(|:|(),app(app(|:|(),app(app(|:|(),C()),x)),y)),z)),u) -> app(app(|:|(),app(app(|:|(),x),z)),app(app(|:|(),app(app(|:|(),app(app(|:|(),x),y)),z)),u))
  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(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(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
  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#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: app#(|:|_1(x1),y) -> |:|_2#(x1,y)
p5: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
p6: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(x,z)
p7: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(|:|_2(x,y),z),u)
p8: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,y),z)
p9: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(x,y)
p10: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p11: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p12: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p13: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p14: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p15: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p10, p11, p12, p13, p14, p15}
  {p5, p6, p7, p8, p9}


-- 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(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{6, x1 - 4}
        filter_1_A(x1) = x1 + 12
        filter_2#_A(x1,x2) = max{7, x1}
        cons_2_A(x1,x2) = max{17, x1 - 16, x2}
        map_1_A(x1) = x1 + 5
        map_2#_A(x1,x2) = max{6, x1 - 1}
        filter2_3_A(x1,x2,x3) = max{x1 - 41, x2 + 11, x3 - 15}
        filter2_4#_A(x1,x2,x3,x4) = max{7, x2}
        false_0_A = 46
        app_A(x1,x2) = max{39, x1 + 14, x2 + 37}
        true_0_A = 6
        cons_0_A = 27
        cons_1_A(x1) = max{40, x1 - 15}
        filter_0_A = 1
        filter_2_A(x1,x2) = max{x1 + 13, x2 + 16}
        filter2_0_A = 2
        filter2_1_A(x1) = max{1, x1 - 15}
        filter2_2_A(x1,x2) = max{2, x1 - 42, x2 - 1}
        filter2_4_A(x1,x2,x3,x4) = max{33, x1 - 40, x2 + 13, x3 - 15, x4 + 16}
        map_0_A = 1
        map_2_A(x1,x2) = max{x1 - 2, x2 + 37}
        |:|_0_A = 1
        |:|_1_A(x1) = max{38, x1 - 15}
        |:|_2_A(x1,x2) = 0
        C_0_A = 9
        nil_0_A = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 22, x2 - 13}
          filter_1_A(x1) = max{7, x1 + 6}
          filter_2#_A(x1,x2) = max{0, x1 - 16, x2 - 13}
          cons_2_A(x1,x2) = max{12, x1 + 2, x2}
          map_1_A(x1) = x1 + 5
          map_2#_A(x1,x2) = max{0, x1 - 17, x2 - 15}
          filter2_3_A(x1,x2,x3) = max{16, x1 - 2, x2 + 7, x3 + 13}
          filter2_4#_A(x1,x2,x3,x4) = max{0, x1 - 24, x2 - 16, x3 - 11, x4 - 13}
          false_0_A = 15
          app_A(x1,x2) = max{23, x1 - 6, x2 + 13}
          true_0_A = 15
          cons_0_A = 5
          cons_1_A(x1) = x1 + 13
          filter_0_A = 5
          filter_2_A(x1,x2) = max{14, x2 + 1}
          filter2_0_A = 14
          filter2_1_A(x1) = x1 + 10
          filter2_2_A(x1,x2) = max{x1 + 4, x2 + 13}
          filter2_4_A(x1,x2,x3,x4) = max{14, x3 + 2, x4 + 1}
          map_0_A = 5
          map_2_A(x1,x2) = max{17, x1 - 1, x2 + 13}
          |:|_0_A = 5
          |:|_1_A(x1) = max{14, x1 + 5}
          |:|_2_A(x1,x2) = 23
          C_0_A = 10
          nil_0_A = 18
          app#!_A(x1,x2) = max{x1 - 8, x2 + 8}
          filter_1!_A(x1) = 0
          filter_2#!_A(x1,x2) = 0
          cons_2!_A(x1,x2) = max{4, x2 - 34}
          map_1!_A(x1) = max{2, x1 - 17}
          map_2#!_A(x1,x2) = max{x1 - 4, x2 + 7}
          filter2_3!_A(x1,x2,x3) = max{36, x1 - 13, x2 - 3, x3 - 1}
          filter2_4#!_A(x1,x2,x3,x4) = 0
          false_0!_A = 1
          app!_A(x1,x2) = max{x1 - 15, x2 + 36}
          true_0!_A = 3
          cons_0!_A = 1
          cons_1!_A(x1) = max{36, x1 - 1}
          filter_0!_A = 1
          filter_2!_A(x1,x2) = x2 + 35
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{10, x1 - 1}
          filter2_2!_A(x1,x2) = max{35, x1 - 13, x2 - 4}
          filter2_4!_A(x1,x2,x3,x4) = x4 + 35
          map_0!_A = 1
          map_2!_A(x1,x2) = max{35, x1 - 16, x2 - 20}
          |:|_0!_A = 1
          |:|_1!_A(x1) = max{13, x1 - 1}
          |:|_2!_A(x1,x2) = max{12, x1 - 11}
          C_0!_A = 13
          nil_0!_A = 34
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(app) = [1]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_2) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_4) = [2]
      pi(map_0) = []
      pi(map_2) = []
      pi(|:|_0) = []
      pi(|:|_1) = []
      pi(|:|_2) = []
      pi(C_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p3, p5, p6



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
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(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}
  {p2, p3, p4}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{1, x1 - 1}
        cons_2_A(x1,x2) = max{0, x1 - 13, x2 - 16}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x2 - 7}
          cons_2_A(x1,x2) = max{9, x1 - 20, x2 + 6}
          map_2#!_A(x1,x2) = max{1, x2 - 7}
          cons_2!_A(x1,x2) = max{2, x1 - 6, x2 - 33}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(false_0(),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(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r16

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = 1
        false_0_A = 6
        filter_2#_A(x1,x2) = 1
        cons_2_A(x1,x2) = max{3, x2 - 5}
        app_A(x1,x2) = max{6, x1 + 5}
        true_0_A = 6
        filter_2_A(x1,x2) = 4
        nil_0_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 1, x3 - 30, x4 - 4}
          false_0_A = 19
          filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 24}
          cons_2_A(x1,x2) = max{23, x1 - 6, x2 + 20}
          app_A(x1,x2) = max{x1 - 87, x2 + 19}
          true_0_A = 19
          filter_2_A(x1,x2) = max{23, x1 - 108, x2 + 1}
          nil_0_A = 24
          filter2_4#!_A(x1,x2,x3,x4) = max{x3 + 3, x4 + 11}
          false_0!_A = 3
          filter_2#!_A(x1,x2) = x2 + 10
          cons_2!_A(x1,x2) = max{0, x1 - 62, x2 - 24}
          app!_A(x1,x2) = max{0, x1 - 83, x2 - 25}
          true_0!_A = 1
          filter_2!_A(x1,x2) = 0
          nil_0!_A = 0
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [1]
      pi(true_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
p2: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(x,y)
p3: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,y),z)
p4: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(|:|_2(x,y),z),u)
p5: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(x,z)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        |:|_2#_A(x1,x2) = 1
        |:|_2_A(x1,x2) = max{x1 + 13, x2 - 11}
        C_0_A = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          |:|_2#_A(x1,x2) = max{0, x1 - 38, x2 - 44}
          |:|_2_A(x1,x2) = max{25, x1 + 6, x2}
          C_0_A = 0
          |:|_2#!_A(x1,x2) = max{5, x1 - 19, x2 - 57}
          |:|_2!_A(x1,x2) = max{4, x1 - 89, x2 - 95}
          C_0!_A = 13
  
    partial status:
  
      pi(|:|_2#) = []
      pi(|:|_2) = [1]
      pi(C_0) = []

The following rules are removed:

  p2, p3, p5



-- SCC decomposition.

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

p1: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
p2: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(|:|_2(x,y),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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
p2: |:|_2#(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2#(|:|_2(|:|_2(x,y),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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(|:|_0(),y) -> |:|_1(y)
r12: app(|:|_1(x1),y) -> |:|_2(x1,y)
r13: |:|_2(|:|_2(|:|_2(|:|_2(C_0(),x),y),z),u) -> |:|_2(|:|_2(x,z),|:|_2(|:|_2(|:|_2(x,y),z),u))
r14: map_2(f,nil_0()) -> nil_0()
r15: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r16: filter_2(f,nil_0()) -> nil_0()
r17: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r18: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r19: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        |:|_2#_A(x1,x2) = max{x1 + 17, x2 + 15}
        |:|_2_A(x1,x2) = max{12, x1 + 2, x2}
        C_0_A = 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          |:|_2#_A(x1,x2) = max{0, x1 - 5, x2 - 6}
          |:|_2_A(x1,x2) = max{7, x1 + 1, x2}
          C_0_A = 1
          |:|_2#!_A(x1,x2) = max{0, x1 - 10, x2 - 11}
          |:|_2!_A(x1,x2) = 0
          C_0!_A = 1
  
    partial status:
  
      pi(|:|_2#) = [1]
      pi(|:|_2) = [1, 2]
      pi(C_0) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.