YES
We show the termination of the TRS R:

  app(app(app(f(),|0|()),|1|()),x) -> app(app(app(f(),app(s(),x)),x),x)
  app(app(app(f(),x),y),app(s(),z)) -> app(s(),app(app(app(f(),|0|()),|1|()),z))
  app(app(map(),|fun|),nil()) -> nil()
  app(app(map(),|fun|),app(app(cons(),x),xs)) -> app(app(cons(),app(|fun|,x)),app(app(map(),|fun|),xs))
  app(app(filter(),|fun|),nil()) -> nil()
  app(app(filter(),|fun|),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(|fun|,x)),|fun|),x),xs)
  app(app(app(app(filter2(),true()),|fun|),x),xs) -> app(app(cons(),x),app(app(filter(),|fun|),xs))
  app(app(app(app(filter2(),false()),|fun|),x),xs) -> app(app(filter(),|fun|),xs)

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(f_0(),y) -> f_1(y)
  app(f_1(x1),y) -> f_2(x1,y)
  app(f_2(x1,x2),y) -> f_3(x1,x2,y)
  app(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(s_0(),y) -> s_1(y)
  f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
  f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
  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_2(x1,x2),y) -> f_3#(x1,x2,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#(map_1(x1),y) -> map_2#(x1,y)
p5: f_3#(|0|_0(),|1|_0(),x) -> f_3#(s_1(x),x,x)
p6: f_3#(x,y,s_1(z)) -> f_3#(|0|_0(),|1|_0(),z)
p7: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p8: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p9: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p10: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p11: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p12: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p7, p8, p9, p10, p11, p12}
  {p5, p6}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r4, r19

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{6, x1 + 4}
        cons_2_A(x1,x2) = max{6, x1 - 22, x2 - 22}
        app#_A(x1,x2) = x1 + 3
        map_1_A(x1) = max{7, x1 + 5}
        filter2_3_A(x1,x2,x3) = max{x1 + 5, x2 + 8, x3 + 5}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 7
        false_0_A = 1
        filter_2#_A(x1,x2) = x1 + 7
        filter_1_A(x1) = max{13, x1 + 8}
        app_A(x1,x2) = max{21, x1 - 11}
        true_0_A = 1
        cons_0_A = 1
        cons_1_A(x1) = 2
        f_1_A(x1) = max{2, x1 - 11}
        f_2_A(x1,x2) = max{12, x1 - 23}
        filter_2_A(x1,x2) = max{20, x1 - 56, x2 - 30}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_1 > map_2# > app# = false_0 > filter2_4# = filter_2# = app > filter_1 = cons_0 = f_2 = filter_2 > f_1 = nil_0 > cons_2 = filter2_3 > true_0 = cons_1
      
  
    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) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_1) = []
      pi(f_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: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)
p7: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p8: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r3, r4, r6, r8, r9, r10, r14, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 8, x2 + 8}
        cons_2_A(x1,x2) = max{6, x1, x2 + 2}
        app_A(x1,x2) = max{x1 + 2, x2}
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 3}
        f_0_A = 6
        f_1_A(x1) = max{7, x1 - 3}
        f_2_A(x1,x2) = max{8, x1 - 4, x2 - 19}
        filter_0_A = 3
        filter_1_A(x1) = max{2, x1 - 7}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 11}
        filter2_2_A(x1,x2) = max{3, x1 - 10, x2 - 4}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 9, x2 - 3, x3 - 3}
        s_0_A = 9
        s_1_A(x1) = max{10, x1 - 13}
        filter_2_A(x1,x2) = max{4, x1 - 5, x2 - 15}
        nil_0_A = 3
  
    base order 2:
  
      precedence:
      
        map_2# = app > filter2_3 = s_1 > cons_2 = filter2_2 = s_0 = filter_2 > cons_1 = f_2 = filter_1 = filter2_1 = nil_0 > cons_0 = f_1 = filter_0 = filter2_0 > f_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(s_0) = []
      pi(s_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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 7
        filter2_3_A(x1,x2,x3) = max{16, x1 - 3, x2 + 8, x3 - 3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 8
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 8
        cons_2_A(x1,x2) = max{7, x1 - 19, x2 - 19}
        app_A(x1,x2) = max{15, x1 - 42}
        false_0_A = 17
        filter_1_A(x1) = max{17, x1 + 8}
        filter_2_A(x1,x2) = max{18, x1 - 32, x2 - 8}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > false_0 = nil_0 > cons_2 = app > app# = filter2_3 = filter_1 > filter2_4# = filter_2# > true_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      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#(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)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: 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(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r6, r8, r12, r14, r19

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 - 14}
        filter2_3_A(x1,x2,x3) = max{x1 - 1, x2 + 20, x3 - 12}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x2 - 13}
        false_0_A = 1
        filter_2#_A(x1,x2) = max{5, x1 - 13}
        cons_2_A(x1,x2) = max{3, x1 - 40, x2 - 56}
        app_A(x1,x2) = max{23, x1 - 25, x2 - 13}
        true_0_A = 1
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 14}
        f_0_A = 21
        f_1_A(x1) = max{22, x1 - 31}
        f_2_A(x1,x2) = max{23, x1 - 56, x2 - 14}
        filter_0_A = 20
        filter_1_A(x1) = max{21, x1 - 14}
        filter2_0_A = 22
        filter2_1_A(x1) = max{23, x1 - 13}
        map_0_A = 15
        map_1_A(x1) = max{16, x1 - 39}
        s_0_A = 21
        s_1_A(x1) = max{22, x1 - 19}
        filter_2_A(x1,x2) = max{22, x1 - 104, x2 - 14}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app# = app = f_0 > filter2_4# = filter_2# = s_1 > cons_2 = map_1 = s_0 = filter_2 > filter2_3 = false_0 = true_0 = cons_1 = f_2 = filter_1 = map_0 = nil_0 > cons_0 = f_1 = filter_0 = filter2_0 > filter2_1
      
  
    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_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_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(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: 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(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r6, r8, r12, r14, r19

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 + 17, x4 + 17}
        false_0_A = 31
        filter_2#_A(x1,x2) = x2 + 16
        cons_2_A(x1,x2) = max{25, x1 + 18, x2 + 18}
        app_A(x1,x2) = max{x1 - 5, x2 + 29}
        true_0_A = 33
        cons_0_A = 4
        cons_1_A(x1) = max{29, x1 + 24}
        f_0_A = 1
        f_1_A(x1) = max{3, x1 - 19}
        f_2_A(x1,x2) = max{4, x1 - 24, x2 - 85}
        filter_0_A = 1
        filter_1_A(x1) = max{4, x1 - 19}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 1}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 1}
        s_0_A = 4
        s_1_A(x1) = max{28, x1 - 28}
        filter_2_A(x1,x2) = 30
        nil_0_A = 16
  
    base order 2:
  
      precedence:
      
        app = cons_0 > cons_2 = s_1 > filter2_4# = true_0 = cons_1 = filter2_1 = map_1 = s_0 > filter_2# = f_2 = filter2_0 = map_0 > false_0 = f_1 > f_0 = filter_0 = filter_2 > filter_1 = nil_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_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(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: f_3#(|0|_0(),|1|_0(),x) -> f_3#(s_1(x),x,x)
p2: f_3#(x,y,s_1(z)) -> f_3#(|0|_0(),|1|_0(),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(f_0(),y) -> f_1(y)
r4: app(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r6, r9, r12, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_3#_A(x1,x2,x3) = max{x1 + 147, x2 + 116, x3 + 180}
        |0|_0_A = 146
        |1|_0_A = 81
        s_1_A(x1) = max{114, x1 + 16}
        app_A(x1,x2) = max{4, x1 - 6, x2 - 18}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 19}
        cons_2_A(x1,x2) = max{3, x1 - 26, x2 - 32}
        f_0_A = 3
        f_1_A(x1) = max{4, x1 - 18}
        f_2_A(x1,x2) = max{3, x1 - 25, x2 - 19}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 19}
        filter2_1_A(x1) = x1 + 8
        filter2_2_A(x1,x2) = max{1, x1 - 7}
        map_0_A = 7
        map_1_A(x1) = 3
        filter_2_A(x1,x2) = max{30, x1 - 5, x2 + 9}
        nil_0_A = 46
  
    base order 2:
  
      precedence:
      
        |1|_0 > f_3# > s_1 = filter2_1 = filter_2 > app = map_0 = nil_0 > |0|_0 = cons_0 = cons_2 = f_2 = filter_1 = filter2_2 = map_1 > cons_1 = f_1 = filter_0 > f_0
      
  
    partial status:
  
      pi(f_3#) = []
      pi(|0|_0) = []
      pi(|1|_0) = []
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(f_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: f_3#(|0|_0(),|1|_0(),x) -> f_3#(s_1(x),x,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(f_1(x1),y) -> f_2(x1,y)
r5: app(f_2(x1,x2),y) -> f_3(x1,x2,y)
r6: app(filter_0(),y) -> filter_1(y)
r7: app(filter_1(x1),y) -> filter_2(x1,y)
r8: app(filter2_0(),y) -> filter2_1(y)
r9: app(filter2_1(x1),y) -> filter2_2(x1,y)
r10: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r11: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r12: app(map_0(),y) -> map_1(y)
r13: app(map_1(x1),y) -> map_2(x1,y)
r14: app(s_0(),y) -> s_1(y)
r15: f_3(|0|_0(),|1|_0(),x) -> f_3(s_1(x),x,x)
r16: f_3(x,y,s_1(z)) -> s_1(f_3(|0|_0(),|1|_0(),z))
r17: map_2(|fun|,nil_0()) -> nil_0()
r18: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r19: filter_2(|fun|,nil_0()) -> nil_0()
r20: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r21: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r22: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  (no SCCs)