YES
We show the termination of the TRS R:

  app(f(),app(f(),x)) -> app(g(),app(f(),x))
  app(g(),app(g(),x)) -> app(f(),x)
  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(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(g_0(),y) -> g_1(y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  f_1(f_1(x)) -> g_1(f_1(x))
  g_1(g_1(x)) -> f_1(x)
  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_0(),y) -> f_1#(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#(g_0(),y) -> g_1#(y)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: f_1#(f_1(x)) -> g_1#(f_1(x))
p7: g_1#(g_1(x)) -> f_1#(x)
p8: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p9: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p10: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p11: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p12: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p13: 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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

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


-- 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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r4, r6, r7, r11, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 8
        cons_2_A(x1,x2) = max{25, x1 - 26, x2 - 28}
        app#_A(x1,x2) = x1 + 6
        map_1_A(x1) = x1 + 9
        filter2_3_A(x1,x2,x3) = max{x1 - 3, x2 + 29, x3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 6
        false_0_A = 26
        filter_2#_A(x1,x2) = x1 + 6
        filter_1_A(x1) = max{25, x1 + 6}
        app_A(x1,x2) = max{x1, x2 + 25}
        true_0_A = 26
        cons_0_A = 0
        cons_1_A(x1) = max{0, x1 - 26}
        filter_0_A = 0
        filter2_0_A = 0
        filter2_1_A(x1) = max{0, x1 - 1}
        filter2_2_A(x1,x2) = max{0, x1 - 2, x2 - 1}
        map_0_A = 0
        filter_2_A(x1,x2) = max{27, x1 - 1, x2 - 12}
        nil_0_A = 11
  
    base order 2:
  
      precedence:
      
        app = filter_0 = filter_2 > cons_2 = map_1 = filter2_3 = false_0 = filter_1 = nil_0 > true_0 = cons_1 = filter2_2 = map_0 > map_2# = filter2_4# = filter_2# = cons_0 = filter2_1 > app# = filter2_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = []
      pi(app) = [1]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_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#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p7: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p8: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r4, r6, r7, r11, r17

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{12, x1 + 4}
        cons_2_A(x1,x2) = max{11, x1 - 17, x2 - 17}
        app#_A(x1,x2) = x1 + 3
        filter2_3_A(x1,x2,x3) = max{9, x1 - 1, x2 + 7, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 4
        true_0_A = 7
        filter_2#_A(x1,x2) = x1 + 4
        app_A(x1,x2) = max{15, x1 - 14, x2 + 6}
        false_0_A = 7
        map_1_A(x1) = max{13, x1 + 5}
        cons_0_A = 9
        cons_1_A(x1) = max{10, x1 - 1}
        filter_0_A = 6
        filter_1_A(x1) = max{7, x1 - 27}
        filter2_0_A = 6
        filter2_1_A(x1) = max{7, x1 - 1}
        filter2_2_A(x1,x2) = max{8, x1 - 16, x2 - 1}
        map_0_A = 12
        filter_2_A(x1,x2) = max{8, x1 - 26, x2 - 19}
        nil_0_A = 7
  
    base order 2:
  
      precedence:
      
        cons_2 > filter2_3 = app = filter2_0 = filter_2 > false_0 = map_1 = cons_1 = filter_1 = filter2_2 = nil_0 > filter2_4# = filter_2# = cons_0 = filter_0 = filter2_1 = map_0 > app# = true_0 > map_2#
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(map_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(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: 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#(map_1(x1),y) -> map_2#(x1,y)
p7: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}
  {p7}


-- 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(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p5: 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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r2, r4, r6, r7, r11, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{0, x1 - 5}
        filter2_3_A(x1,x2,x3) = max{x1 + 6, x2 + 4, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = max{0, x2 - 4}
        false_0_A = 1
        filter_2#_A(x1,x2) = max{0, x1 - 4}
        cons_2_A(x1,x2) = max{2, x1 - 41, x2 - 11}
        app_A(x1,x2) = max{8, x1 - 9, x2 - 9}
        true_0_A = 23
        cons_1_A(x1) = max{12, x1 - 10}
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 10}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 10}
        filter2_2_A(x1,x2) = max{3, x1 - 20, x2 - 10}
        map_0_A = 1
        map_1_A(x1) = max{0, x1 - 10}
        filter_2_A(x1,x2) = max{2, x1 - 7, x2 - 11}
        nil_0_A = 1
  
    base order 2:
  
      precedence:
      
        app > map_1 > filter2_3 = filter2_4# = filter_2# = map_0 = filter_2 > app# = false_0 = filter_1 = filter2_2 = nil_0 > cons_2 = filter_0 = filter2_1 > true_0 = cons_1 = filter2_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(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      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(),|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#(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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,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(),|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#(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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r4, r6, r8, r11, r17

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 + 7, x4 + 5}
        false_0_A = 1
        filter_2#_A(x1,x2) = x2 + 4
        cons_2_A(x1,x2) = max{x1 + 7, x2 + 5}
        app_A(x1,x2) = max{x1 + 11, x2 + 11}
        true_0_A = 12
        cons_0_A = 2
        cons_1_A(x1) = max{12, x1 + 8}
        filter_0_A = 17
        filter_1_A(x1) = max{27, x1 - 12}
        filter2_0_A = 1
        filter2_1_A(x1) = max{12, x1 - 12}
        filter2_2_A(x1,x2) = max{24, x1 + 1, x2 - 23}
        filter2_3_A(x1,x2,x3) = max{35, x1 - 12, x2 - 12, x3 - 12}
        map_0_A = 1
        map_1_A(x1) = max{7, x1 - 12}
        filter_2_A(x1,x2) = max{37, x1 - 13, x2 - 8}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app > filter_2# = cons_0 = filter_2 > false_0 = filter_1 = map_1 = nil_0 > cons_2 = filter_0 = filter2_3 = map_0 > filter2_4# = true_0 = cons_1 = filter2_2 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_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: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: 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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: 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: 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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r6, r7, r8, r11, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x1 + 1, x2 + 2}
        cons_2_A(x1,x2) = max{2, x1 - 1, x2}
        app_A(x1,x2) = max{0, x1 - 2, x2 - 5}
        cons_0_A = 1
        cons_1_A(x1) = max{0, x1 - 5}
        filter_0_A = 1
        filter_1_A(x1) = 0
        filter2_0_A = 1
        filter2_1_A(x1) = max{0, x1 - 6}
        filter2_2_A(x1,x2) = max{0, x1 - 8, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{0, x1 - 10, x2 - 7, x3 - 5}
        map_0_A = 1
        map_1_A(x1) = max{0, x1 - 5}
        filter_2_A(x1,x2) = max{x1 + 3, x2 - 3}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > map_2# = app = filter_0 = nil_0 > cons_2 = filter_1 = filter2_3 = map_1 > cons_1 = filter2_2 = map_0 > cons_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      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(filter_2) = [1]
      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: f_1#(f_1(x)) -> g_1#(f_1(x))
p2: g_1#(g_1(x)) -> f_1#(x)

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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r6, r7, r8, r10, r11, r13, r14, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_1#_A(x1) = max{10, x1 + 4}
        f_1_A(x1) = max{10, x1 + 8}
        g_1#_A(x1) = max{11, x1 + 3}
        g_1_A(x1) = max{10, x1 + 8}
        app_A(x1,x2) = max{x1 - 11, x2 + 10}
        cons_0_A = 10
        cons_1_A(x1) = max{1, x1 - 1}
        cons_2_A(x1,x2) = max{5, x1 - 28, x2 - 30}
        f_0_A = 9
        filter_0_A = 9
        filter_1_A(x1) = max{10, x1 - 36}
        filter2_0_A = 1
        filter2_1_A(x1) = max{6, x1 - 1}
        filter2_2_A(x1,x2) = max{8, x1 - 12, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 24, x2 - 13, x3 - 1}
        g_0_A = 9
        map_0_A = 9
        map_1_A(x1) = max{10, x1 - 1}
        filter_2_A(x1,x2) = max{4, x1 - 48, x2 - 30}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        g_1# = filter_2 = nil_0 > f_1 = app = filter_0 > g_1 = f_0 = filter_1 = filter2_3 > f_1# = cons_2 = filter2_2 = g_0 = map_1 > cons_1 = filter2_1 = map_0 > cons_0 = filter2_0
      
  
    partial status:
  
      pi(f_1#) = []
      pi(f_1) = []
      pi(g_1#) = [1]
      pi(g_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(f_0) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(map_0) = []
      pi(map_1) = []
      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: g_1#(g_1(x)) -> f_1#(x)

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(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: f_1(f_1(x)) -> g_1(f_1(x))
r14: g_1(g_1(x)) -> f_1(x)
r15: map_2(|fun|,nil_0()) -> nil_0()
r16: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r17: filter_2(|fun|,nil_0()) -> nil_0()
r18: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r19: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r20: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)