YES
We show the termination of the TRS R:

  app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
  app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),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(g_1(x)) -> g_1(g_1(f_1(x)))
  f_1(g_1(x)) -> g_1(g_1(g_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#(map_1(x1),y) -> map_2#(x1,y)
p5: f_1#(g_1(x)) -> f_1#(x)
p6: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p7: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p8: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p9: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p10: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p11: 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, p6, p7, p8, p9, p10, p11}
  {p5}


-- 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      model-based reduction triple (=A, >=A!, >A!):
      
        finite algebra:
        
          carrier: {0,1}
          interpretations:
            map_2#_A(0,0) = 0
            map_2#_A(0,1) = 0
            map_2#_A(1,0) = 0
            map_2#_A(1,1) = 0
            cons_2_A(0,0) = 0
            cons_2_A(0,1) = 0
            cons_2_A(1,0) = 0
            cons_2_A(1,1) = 0
            app#_A(0,0) = 0
            app#_A(0,1) = 0
            app#_A(1,0) = 0
            app#_A(1,1) = 0
            map_1_A(0) = 1
            map_1_A(1) = 1
            filter2_3_A(0,0,0) = 1
            filter2_3_A(0,0,1) = 1
            filter2_3_A(0,1,0) = 1
            filter2_3_A(0,1,1) = 1
            filter2_3_A(1,0,0) = 1
            filter2_3_A(1,0,1) = 1
            filter2_3_A(1,1,0) = 1
            filter2_3_A(1,1,1) = 1
            filter2_4#_A(0,0,0,0) = 0
            filter2_4#_A(0,0,0,1) = 0
            filter2_4#_A(0,0,1,0) = 0
            filter2_4#_A(0,0,1,1) = 0
            filter2_4#_A(0,1,0,0) = 0
            filter2_4#_A(0,1,0,1) = 0
            filter2_4#_A(0,1,1,0) = 0
            filter2_4#_A(0,1,1,1) = 0
            filter2_4#_A(1,0,0,0) = 0
            filter2_4#_A(1,0,0,1) = 0
            filter2_4#_A(1,0,1,0) = 0
            filter2_4#_A(1,0,1,1) = 0
            filter2_4#_A(1,1,0,0) = 0
            filter2_4#_A(1,1,0,1) = 0
            filter2_4#_A(1,1,1,0) = 0
            filter2_4#_A(1,1,1,1) = 0
            false_0_A = 1
            filter_2#_A(0,0) = 0
            filter_2#_A(0,1) = 0
            filter_2#_A(1,0) = 0
            filter_2#_A(1,1) = 0
            filter_1_A(0) = 1
            filter_1_A(1) = 1
            app_A(0,0) = 1
            app_A(0,1) = 1
            app_A(1,0) = 0
            app_A(1,1) = 0
            true_0_A = 1
            cons_0_A = 0
            cons_1_A(0) = 1
            cons_1_A(1) = 1
            f_0_A = 1
            f_1_A(0) = 0
            f_1_A(1) = 0
            filter_0_A = 0
            filter_2_A(0,0) = 0
            filter_2_A(0,1) = 0
            filter_2_A(1,0) = 0
            filter_2_A(1,1) = 0
            filter2_0_A = 0
            filter2_1_A(0) = 1
            filter2_1_A(1) = 1
            filter2_2_A(0,0) = 0
            filter2_2_A(0,1) = 0
            filter2_2_A(1,0) = 0
            filter2_2_A(1,1) = 0
            filter2_4_A(0,0,0,0) = 0
            filter2_4_A(0,0,0,1) = 0
            filter2_4_A(0,0,1,0) = 0
            filter2_4_A(0,0,1,1) = 0
            filter2_4_A(0,1,0,0) = 0
            filter2_4_A(0,1,0,1) = 0
            filter2_4_A(0,1,1,0) = 0
            filter2_4_A(0,1,1,1) = 0
            filter2_4_A(1,0,0,0) = 0
            filter2_4_A(1,0,0,1) = 0
            filter2_4_A(1,0,1,0) = 0
            filter2_4_A(1,0,1,1) = 0
            filter2_4_A(1,1,0,0) = 0
            filter2_4_A(1,1,0,1) = 0
            filter2_4_A(1,1,1,0) = 0
            filter2_4_A(1,1,1,1) = 0
            g_0_A = 1
            g_1_A(0) = 0
            g_1_A(1) = 0
            map_0_A = 0
            map_2_A(0,0) = 0
            map_2_A(0,1) = 0
            map_2_A(1,0) = 0
            map_2_A(1,1) = 0
            nil_0_A = 0
            map_2#!_A(0,0) = 0
            map_2#!_A(0,1) = 0
            map_2#!_A(1,0) = 1
            map_2#!_A(1,1) = 0
            cons_2!_A(0,0) = 0
            cons_2!_A(0,1) = 1
            cons_2!_A(1,0) = 0
            cons_2!_A(1,1) = 0
            app#!_A(0,0) = 0
            app#!_A(0,1) = 0
            app#!_A(1,0) = 1
            app#!_A(1,1) = 0
            map_1!_A(0) = 0
            map_1!_A(1) = 0
            filter2_3!_A(0,0,0) = 1
            filter2_3!_A(0,0,1) = 1
            filter2_3!_A(0,1,0) = 1
            filter2_3!_A(0,1,1) = 1
            filter2_3!_A(1,0,0) = 1
            filter2_3!_A(1,0,1) = 1
            filter2_3!_A(1,1,0) = 1
            filter2_3!_A(1,1,1) = 1
            filter2_4#!_A(0,0,0,0) = 0
            filter2_4#!_A(0,0,0,1) = 0
            filter2_4#!_A(0,0,1,0) = 0
            filter2_4#!_A(0,0,1,1) = 0
            filter2_4#!_A(0,1,0,0) = 0
            filter2_4#!_A(0,1,0,1) = 0
            filter2_4#!_A(0,1,1,0) = 1
            filter2_4#!_A(0,1,1,1) = 1
            filter2_4#!_A(1,0,0,0) = 1
            filter2_4#!_A(1,0,0,1) = 0
            filter2_4#!_A(1,0,1,0) = 1
            filter2_4#!_A(1,0,1,1) = 0
            filter2_4#!_A(1,1,0,0) = 1
            filter2_4#!_A(1,1,0,1) = 0
            filter2_4#!_A(1,1,1,0) = 1
            filter2_4#!_A(1,1,1,1) = 0
            false_0!_A = 1
            filter_2#!_A(0,0) = 1
            filter_2#!_A(0,1) = 0
            filter_2#!_A(1,0) = 1
            filter_2#!_A(1,1) = 0
            filter_1!_A(0) = 0
            filter_1!_A(1) = 0
            app!_A(0,0) = 1
            app!_A(0,1) = 1
            app!_A(1,0) = 1
            app!_A(1,1) = 1
            true_0!_A = 1
            cons_0!_A = 1
            cons_1!_A(0) = 1
            cons_1!_A(1) = 0
            f_0!_A = 1
            f_1!_A(0) = 1
            f_1!_A(1) = 0
            filter_0!_A = 0
            filter_2!_A(0,0) = 0
            filter_2!_A(0,1) = 0
            filter_2!_A(1,0) = 0
            filter_2!_A(1,1) = 0
            filter2_0!_A = 1
            filter2_1!_A(0) = 0
            filter2_1!_A(1) = 0
            filter2_2!_A(0,0) = 1
            filter2_2!_A(0,1) = 1
            filter2_2!_A(1,0) = 1
            filter2_2!_A(1,1) = 0
            filter2_4!_A(0,0,0,0) = 0
            filter2_4!_A(0,0,0,1) = 1
            filter2_4!_A(0,0,1,0) = 1
            filter2_4!_A(0,0,1,1) = 0
            filter2_4!_A(0,1,0,0) = 0
            filter2_4!_A(0,1,0,1) = 0
            filter2_4!_A(0,1,1,0) = 0
            filter2_4!_A(0,1,1,1) = 0
            filter2_4!_A(1,0,0,0) = 0
            filter2_4!_A(1,0,0,1) = 0
            filter2_4!_A(1,0,1,0) = 0
            filter2_4!_A(1,0,1,1) = 0
            filter2_4!_A(1,1,0,0) = 1
            filter2_4!_A(1,1,0,1) = 0
            filter2_4!_A(1,1,1,0) = 1
            filter2_4!_A(1,1,1,1) = 1
            g_0!_A = 1
            g_1!_A(0) = 1
            g_1!_A(1) = 1
            map_0!_A = 0
            map_2!_A(0,0) = 0
            map_2!_A(0,1) = 1
            map_2!_A(1,0) = 1
            map_2!_A(1,1) = 1
            nil_0!_A = 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) = [1]
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_4) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(map_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p4



-- 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: 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(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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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#(filter_1(x1),y) -> filter_2#(x1,y)
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)
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)
p8: 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      model-based reduction triple (=A, >=A!, >A!):
      
        finite algebra:
        
          carrier: {0,1}
          interpretations:
            map_2#_A(0,0) = 0
            map_2#_A(0,1) = 0
            map_2#_A(1,0) = 0
            map_2#_A(1,1) = 0
            cons_2_A(0,0) = 0
            cons_2_A(0,1) = 0
            cons_2_A(1,0) = 0
            cons_2_A(1,1) = 0
            app#_A(0,0) = 0
            app#_A(0,1) = 0
            app#_A(1,0) = 0
            app#_A(1,1) = 0
            filter_1_A(0) = 0
            filter_1_A(1) = 0
            filter_2#_A(0,0) = 0
            filter_2#_A(0,1) = 0
            filter_2#_A(1,0) = 0
            filter_2#_A(1,1) = 0
            filter2_4#_A(0,0,0,0) = 0
            filter2_4#_A(0,0,0,1) = 0
            filter2_4#_A(0,0,1,0) = 0
            filter2_4#_A(0,0,1,1) = 0
            filter2_4#_A(0,1,0,0) = 0
            filter2_4#_A(0,1,0,1) = 0
            filter2_4#_A(0,1,1,0) = 0
            filter2_4#_A(0,1,1,1) = 0
            filter2_4#_A(1,0,0,0) = 0
            filter2_4#_A(1,0,0,1) = 0
            filter2_4#_A(1,0,1,0) = 0
            filter2_4#_A(1,0,1,1) = 0
            filter2_4#_A(1,1,0,0) = 0
            filter2_4#_A(1,1,0,1) = 0
            filter2_4#_A(1,1,1,0) = 0
            filter2_4#_A(1,1,1,1) = 0
            app_A(0,0) = 0
            app_A(0,1) = 0
            app_A(1,0) = 0
            app_A(1,1) = 0
            true_0_A = 1
            map_1_A(0) = 0
            map_1_A(1) = 0
            false_0_A = 0
            cons_0_A = 0
            cons_1_A(0) = 0
            cons_1_A(1) = 0
            f_0_A = 0
            f_1_A(0) = 0
            f_1_A(1) = 0
            filter_0_A = 0
            filter_2_A(0,0) = 0
            filter_2_A(0,1) = 0
            filter_2_A(1,0) = 0
            filter_2_A(1,1) = 0
            filter2_0_A = 1
            filter2_1_A(0) = 0
            filter2_1_A(1) = 0
            filter2_2_A(0,0) = 0
            filter2_2_A(0,1) = 0
            filter2_2_A(1,0) = 0
            filter2_2_A(1,1) = 0
            filter2_3_A(0,0,0) = 0
            filter2_3_A(0,0,1) = 0
            filter2_3_A(0,1,0) = 0
            filter2_3_A(0,1,1) = 0
            filter2_3_A(1,0,0) = 0
            filter2_3_A(1,0,1) = 0
            filter2_3_A(1,1,0) = 0
            filter2_3_A(1,1,1) = 0
            filter2_4_A(0,0,0,0) = 0
            filter2_4_A(0,0,0,1) = 0
            filter2_4_A(0,0,1,0) = 0
            filter2_4_A(0,0,1,1) = 0
            filter2_4_A(0,1,0,0) = 0
            filter2_4_A(0,1,0,1) = 0
            filter2_4_A(0,1,1,0) = 0
            filter2_4_A(0,1,1,1) = 0
            filter2_4_A(1,0,0,0) = 0
            filter2_4_A(1,0,0,1) = 0
            filter2_4_A(1,0,1,0) = 0
            filter2_4_A(1,0,1,1) = 0
            filter2_4_A(1,1,0,0) = 0
            filter2_4_A(1,1,0,1) = 0
            filter2_4_A(1,1,1,0) = 0
            filter2_4_A(1,1,1,1) = 0
            g_0_A = 0
            g_1_A(0) = 0
            g_1_A(1) = 0
            map_0_A = 1
            map_2_A(0,0) = 0
            map_2_A(0,1) = 0
            map_2_A(1,0) = 0
            map_2_A(1,1) = 0
            nil_0_A = 0
            map_2#!_A(0,0) = 0
            map_2#!_A(0,1) = 0
            map_2#!_A(1,0) = 0
            map_2#!_A(1,1) = 0
            cons_2!_A(0,0) = 0
            cons_2!_A(0,1) = 0
            cons_2!_A(1,0) = 0
            cons_2!_A(1,1) = 0
            app#!_A(0,0) = 0
            app#!_A(0,1) = 0
            app#!_A(1,0) = 0
            app#!_A(1,1) = 0
            filter_1!_A(0) = 0
            filter_1!_A(1) = 0
            filter_2#!_A(0,0) = 0
            filter_2#!_A(0,1) = 0
            filter_2#!_A(1,0) = 0
            filter_2#!_A(1,1) = 0
            filter2_4#!_A(0,0,0,0) = 0
            filter2_4#!_A(0,0,0,1) = 0
            filter2_4#!_A(0,0,1,0) = 0
            filter2_4#!_A(0,0,1,1) = 0
            filter2_4#!_A(0,1,0,0) = 0
            filter2_4#!_A(0,1,0,1) = 0
            filter2_4#!_A(0,1,1,0) = 0
            filter2_4#!_A(0,1,1,1) = 0
            filter2_4#!_A(1,0,0,0) = 1
            filter2_4#!_A(1,0,0,1) = 1
            filter2_4#!_A(1,0,1,0) = 1
            filter2_4#!_A(1,0,1,1) = 1
            filter2_4#!_A(1,1,0,0) = 1
            filter2_4#!_A(1,1,0,1) = 1
            filter2_4#!_A(1,1,1,0) = 1
            filter2_4#!_A(1,1,1,1) = 1
            app!_A(0,0) = 1
            app!_A(0,1) = 1
            app!_A(1,0) = 1
            app!_A(1,1) = 1
            true_0!_A = 1
            map_1!_A(0) = 0
            map_1!_A(1) = 0
            false_0!_A = 1
            cons_0!_A = 0
            cons_1!_A(0) = 0
            cons_1!_A(1) = 0
            f_0!_A = 1
            f_1!_A(0) = 1
            f_1!_A(1) = 1
            filter_0!_A = 0
            filter_2!_A(0,0) = 0
            filter_2!_A(0,1) = 0
            filter_2!_A(1,0) = 0
            filter_2!_A(1,1) = 0
            filter2_0!_A = 0
            filter2_1!_A(0) = 0
            filter2_1!_A(1) = 0
            filter2_2!_A(0,0) = 1
            filter2_2!_A(0,1) = 0
            filter2_2!_A(1,0) = 0
            filter2_2!_A(1,1) = 0
            filter2_3!_A(0,0,0) = 0
            filter2_3!_A(0,0,1) = 1
            filter2_3!_A(0,1,0) = 1
            filter2_3!_A(0,1,1) = 0
            filter2_3!_A(1,0,0) = 1
            filter2_3!_A(1,0,1) = 1
            filter2_3!_A(1,1,0) = 1
            filter2_3!_A(1,1,1) = 1
            filter2_4!_A(0,0,0,0) = 0
            filter2_4!_A(0,0,0,1) = 0
            filter2_4!_A(0,0,1,0) = 0
            filter2_4!_A(0,0,1,1) = 0
            filter2_4!_A(0,1,0,0) = 0
            filter2_4!_A(0,1,0,1) = 0
            filter2_4!_A(0,1,1,0) = 0
            filter2_4!_A(0,1,1,1) = 0
            filter2_4!_A(1,0,0,0) = 1
            filter2_4!_A(1,0,0,1) = 1
            filter2_4!_A(1,0,1,0) = 1
            filter2_4!_A(1,0,1,1) = 1
            filter2_4!_A(1,1,0,0) = 1
            filter2_4!_A(1,1,0,1) = 1
            filter2_4!_A(1,1,1,0) = 1
            filter2_4!_A(1,1,1,1) = 1
            g_0!_A = 0
            g_1!_A(0) = 0
            g_1!_A(1) = 0
            map_0!_A = 0
            map_2!_A(0,0) = 0
            map_2!_A(0,1) = 0
            map_2!_A(1,0) = 0
            map_2!_A(1,1) = 0
            nil_0!_A = 0
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = []
      pi(filter2_4#) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(map_1) = [1]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_2) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [2]
      pi(filter2_4) = [2]
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(map_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p4



-- 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#(filter_1(x1),y) -> filter_2#(x1,y)
p3: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p4: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p7: 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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}


-- 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#(filter_1(x1),y) -> filter_2#(x1,y)
p5: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p6: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p7: 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      model-based reduction triple (=A, >=A!, >A!):
      
        finite algebra:
        
          carrier: {0,1}
          interpretations:
            map_2#_A(0,0) = 0
            map_2#_A(0,1) = 0
            map_2#_A(1,0) = 0
            map_2#_A(1,1) = 0
            cons_2_A(0,0) = 0
            cons_2_A(0,1) = 0
            cons_2_A(1,0) = 0
            cons_2_A(1,1) = 0
            app#_A(0,0) = 0
            app#_A(0,1) = 0
            app#_A(1,0) = 0
            app#_A(1,1) = 0
            map_1_A(0) = 0
            map_1_A(1) = 0
            filter_1_A(0) = 0
            filter_1_A(1) = 0
            filter_2#_A(0,0) = 0
            filter_2#_A(0,1) = 0
            filter_2#_A(1,0) = 0
            filter_2#_A(1,1) = 0
            filter2_4#_A(0,0,0,0) = 0
            filter2_4#_A(0,0,0,1) = 0
            filter2_4#_A(0,0,1,0) = 0
            filter2_4#_A(0,0,1,1) = 0
            filter2_4#_A(0,1,0,0) = 0
            filter2_4#_A(0,1,0,1) = 0
            filter2_4#_A(0,1,1,0) = 0
            filter2_4#_A(0,1,1,1) = 0
            filter2_4#_A(1,0,0,0) = 0
            filter2_4#_A(1,0,0,1) = 0
            filter2_4#_A(1,0,1,0) = 0
            filter2_4#_A(1,0,1,1) = 0
            filter2_4#_A(1,1,0,0) = 0
            filter2_4#_A(1,1,0,1) = 0
            filter2_4#_A(1,1,1,0) = 0
            filter2_4#_A(1,1,1,1) = 0
            app_A(0,0) = 0
            app_A(0,1) = 0
            app_A(1,0) = 0
            app_A(1,1) = 0
            false_0_A = 1
            cons_0_A = 1
            cons_1_A(0) = 0
            cons_1_A(1) = 0
            f_0_A = 1
            f_1_A(0) = 0
            f_1_A(1) = 0
            filter_0_A = 0
            filter_2_A(0,0) = 0
            filter_2_A(0,1) = 0
            filter_2_A(1,0) = 0
            filter_2_A(1,1) = 0
            filter2_0_A = 1
            filter2_1_A(0) = 0
            filter2_1_A(1) = 0
            filter2_2_A(0,0) = 0
            filter2_2_A(0,1) = 0
            filter2_2_A(1,0) = 0
            filter2_2_A(1,1) = 0
            filter2_3_A(0,0,0) = 0
            filter2_3_A(0,0,1) = 0
            filter2_3_A(0,1,0) = 0
            filter2_3_A(0,1,1) = 0
            filter2_3_A(1,0,0) = 0
            filter2_3_A(1,0,1) = 0
            filter2_3_A(1,1,0) = 0
            filter2_3_A(1,1,1) = 0
            filter2_4_A(0,0,0,0) = 0
            filter2_4_A(0,0,0,1) = 0
            filter2_4_A(0,0,1,0) = 0
            filter2_4_A(0,0,1,1) = 0
            filter2_4_A(0,1,0,0) = 0
            filter2_4_A(0,1,0,1) = 0
            filter2_4_A(0,1,1,0) = 0
            filter2_4_A(0,1,1,1) = 0
            filter2_4_A(1,0,0,0) = 0
            filter2_4_A(1,0,0,1) = 0
            filter2_4_A(1,0,1,0) = 0
            filter2_4_A(1,0,1,1) = 0
            filter2_4_A(1,1,0,0) = 0
            filter2_4_A(1,1,0,1) = 0
            filter2_4_A(1,1,1,0) = 0
            filter2_4_A(1,1,1,1) = 0
            g_0_A = 1
            g_1_A(0) = 0
            g_1_A(1) = 0
            map_0_A = 0
            map_2_A(0,0) = 0
            map_2_A(0,1) = 0
            map_2_A(1,0) = 0
            map_2_A(1,1) = 0
            nil_0_A = 0
            true_0_A = 1
            map_2#!_A(0,0) = 0
            map_2#!_A(0,1) = 0
            map_2#!_A(1,0) = 0
            map_2#!_A(1,1) = 0
            cons_2!_A(0,0) = 0
            cons_2!_A(0,1) = 1
            cons_2!_A(1,0) = 0
            cons_2!_A(1,1) = 1
            app#!_A(0,0) = 0
            app#!_A(0,1) = 0
            app#!_A(1,0) = 0
            app#!_A(1,1) = 0
            map_1!_A(0) = 0
            map_1!_A(1) = 0
            filter_1!_A(0) = 0
            filter_1!_A(1) = 1
            filter_2#!_A(0,0) = 0
            filter_2#!_A(0,1) = 0
            filter_2#!_A(1,0) = 0
            filter_2#!_A(1,1) = 0
            filter2_4#!_A(0,0,0,0) = 0
            filter2_4#!_A(0,0,0,1) = 0
            filter2_4#!_A(0,0,1,0) = 0
            filter2_4#!_A(0,0,1,1) = 0
            filter2_4#!_A(0,1,0,0) = 0
            filter2_4#!_A(0,1,0,1) = 0
            filter2_4#!_A(0,1,1,0) = 0
            filter2_4#!_A(0,1,1,1) = 0
            filter2_4#!_A(1,0,0,0) = 1
            filter2_4#!_A(1,0,0,1) = 1
            filter2_4#!_A(1,0,1,0) = 1
            filter2_4#!_A(1,0,1,1) = 1
            filter2_4#!_A(1,1,0,0) = 1
            filter2_4#!_A(1,1,0,1) = 1
            filter2_4#!_A(1,1,1,0) = 1
            filter2_4#!_A(1,1,1,1) = 1
            app!_A(0,0) = 1
            app!_A(0,1) = 1
            app!_A(1,0) = 1
            app!_A(1,1) = 1
            false_0!_A = 0
            cons_0!_A = 1
            cons_1!_A(0) = 0
            cons_1!_A(1) = 0
            f_0!_A = 1
            f_1!_A(0) = 1
            f_1!_A(1) = 0
            filter_0!_A = 1
            filter_2!_A(0,0) = 0
            filter_2!_A(0,1) = 0
            filter_2!_A(1,0) = 0
            filter_2!_A(1,1) = 0
            filter2_0!_A = 0
            filter2_1!_A(0) = 0
            filter2_1!_A(1) = 0
            filter2_2!_A(0,0) = 0
            filter2_2!_A(0,1) = 0
            filter2_2!_A(1,0) = 0
            filter2_2!_A(1,1) = 0
            filter2_3!_A(0,0,0) = 0
            filter2_3!_A(0,0,1) = 1
            filter2_3!_A(0,1,0) = 1
            filter2_3!_A(0,1,1) = 0
            filter2_3!_A(1,0,0) = 1
            filter2_3!_A(1,0,1) = 1
            filter2_3!_A(1,1,0) = 1
            filter2_3!_A(1,1,1) = 1
            filter2_4!_A(0,0,0,0) = 0
            filter2_4!_A(0,0,0,1) = 0
            filter2_4!_A(0,0,1,0) = 0
            filter2_4!_A(0,0,1,1) = 0
            filter2_4!_A(0,1,0,0) = 0
            filter2_4!_A(0,1,0,1) = 0
            filter2_4!_A(0,1,1,0) = 0
            filter2_4!_A(0,1,1,1) = 0
            filter2_4!_A(1,0,0,0) = 1
            filter2_4!_A(1,0,0,1) = 1
            filter2_4!_A(1,0,1,0) = 1
            filter2_4!_A(1,0,1,1) = 1
            filter2_4!_A(1,1,0,0) = 1
            filter2_4!_A(1,1,0,1) = 1
            filter2_4!_A(1,1,1,0) = 1
            filter2_4!_A(1,1,1,1) = 1
            g_0!_A = 1
            g_1!_A(0) = 1
            g_1!_A(1) = 1
            map_0!_A = 0
            map_2!_A(0,0) = 1
            map_2!_A(0,1) = 1
            map_2!_A(1,0) = 1
            map_2!_A(1,1) = 1
            nil_0!_A = 0
            true_0!_A = 0
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app#) = [2]
      pi(map_1) = [1]
      pi(filter_1) = []
      pi(filter_2#) = [2]
      pi(filter2_4#) = [4]
      pi(app) = [1, 2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_2) = [2]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1]
      pi(filter2_3) = [1, 3]
      pi(filter2_4) = [3, 4]
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(map_2) = [1, 2]
      pi(nil_0) = []
      pi(true_0) = []

The following rules are removed:

  p1, p5, p7



-- SCC decomposition.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p3: app#(filter_1(x1),y) -> filter_2#(x1,y)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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}


-- 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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      model-based reduction triple (=A, >=A!, >A!):
      
        finite algebra:
        
          carrier: {0,1}
          interpretations:
            map_2#_A(0,0) = 0
            map_2#_A(0,1) = 0
            map_2#_A(1,0) = 0
            map_2#_A(1,1) = 0
            cons_2_A(0,0) = 0
            cons_2_A(0,1) = 0
            cons_2_A(1,0) = 0
            cons_2_A(1,1) = 0
            app_A(0,0) = 0
            app_A(0,1) = 0
            app_A(1,0) = 0
            app_A(1,1) = 0
            cons_0_A = 1
            cons_1_A(0) = 0
            cons_1_A(1) = 0
            f_0_A = 1
            f_1_A(0) = 0
            f_1_A(1) = 0
            filter_0_A = 1
            filter_1_A(0) = 0
            filter_1_A(1) = 0
            filter_2_A(0,0) = 0
            filter_2_A(0,1) = 0
            filter_2_A(1,0) = 0
            filter_2_A(1,1) = 0
            filter2_0_A = 1
            filter2_1_A(0) = 0
            filter2_1_A(1) = 0
            filter2_2_A(0,0) = 0
            filter2_2_A(0,1) = 0
            filter2_2_A(1,0) = 0
            filter2_2_A(1,1) = 0
            filter2_3_A(0,0,0) = 0
            filter2_3_A(0,0,1) = 0
            filter2_3_A(0,1,0) = 0
            filter2_3_A(0,1,1) = 0
            filter2_3_A(1,0,0) = 0
            filter2_3_A(1,0,1) = 0
            filter2_3_A(1,1,0) = 0
            filter2_3_A(1,1,1) = 0
            filter2_4_A(0,0,0,0) = 0
            filter2_4_A(0,0,0,1) = 0
            filter2_4_A(0,0,1,0) = 0
            filter2_4_A(0,0,1,1) = 0
            filter2_4_A(0,1,0,0) = 0
            filter2_4_A(0,1,0,1) = 0
            filter2_4_A(0,1,1,0) = 0
            filter2_4_A(0,1,1,1) = 0
            filter2_4_A(1,0,0,0) = 0
            filter2_4_A(1,0,0,1) = 0
            filter2_4_A(1,0,1,0) = 0
            filter2_4_A(1,0,1,1) = 0
            filter2_4_A(1,1,0,0) = 0
            filter2_4_A(1,1,0,1) = 0
            filter2_4_A(1,1,1,0) = 0
            filter2_4_A(1,1,1,1) = 0
            g_0_A = 1
            g_1_A(0) = 0
            g_1_A(1) = 0
            map_0_A = 0
            map_1_A(0) = 0
            map_1_A(1) = 0
            map_2_A(0,0) = 0
            map_2_A(0,1) = 0
            map_2_A(1,0) = 0
            map_2_A(1,1) = 0
            nil_0_A = 0
            true_0_A = 1
            false_0_A = 0
            map_2#!_A(0,0) = 0
            map_2#!_A(0,1) = 0
            map_2#!_A(1,0) = 0
            map_2#!_A(1,1) = 0
            cons_2!_A(0,0) = 0
            cons_2!_A(0,1) = 0
            cons_2!_A(1,0) = 0
            cons_2!_A(1,1) = 0
            app!_A(0,0) = 1
            app!_A(0,1) = 1
            app!_A(1,0) = 1
            app!_A(1,1) = 1
            cons_0!_A = 0
            cons_1!_A(0) = 1
            cons_1!_A(1) = 1
            f_0!_A = 1
            f_1!_A(0) = 0
            f_1!_A(1) = 1
            filter_0!_A = 1
            filter_1!_A(0) = 1
            filter_1!_A(1) = 1
            filter_2!_A(0,0) = 0
            filter_2!_A(0,1) = 0
            filter_2!_A(1,0) = 0
            filter_2!_A(1,1) = 0
            filter2_0!_A = 1
            filter2_1!_A(0) = 0
            filter2_1!_A(1) = 0
            filter2_2!_A(0,0) = 1
            filter2_2!_A(0,1) = 1
            filter2_2!_A(1,0) = 1
            filter2_2!_A(1,1) = 1
            filter2_3!_A(0,0,0) = 0
            filter2_3!_A(0,0,1) = 0
            filter2_3!_A(0,1,0) = 0
            filter2_3!_A(0,1,1) = 0
            filter2_3!_A(1,0,0) = 0
            filter2_3!_A(1,0,1) = 0
            filter2_3!_A(1,1,0) = 0
            filter2_3!_A(1,1,1) = 0
            filter2_4!_A(0,0,0,0) = 0
            filter2_4!_A(0,0,0,1) = 0
            filter2_4!_A(0,0,1,0) = 0
            filter2_4!_A(0,0,1,1) = 0
            filter2_4!_A(0,1,0,0) = 0
            filter2_4!_A(0,1,0,1) = 0
            filter2_4!_A(0,1,1,0) = 0
            filter2_4!_A(0,1,1,1) = 0
            filter2_4!_A(1,0,0,0) = 1
            filter2_4!_A(1,0,0,1) = 1
            filter2_4!_A(1,0,1,0) = 1
            filter2_4!_A(1,0,1,1) = 1
            filter2_4!_A(1,1,0,0) = 1
            filter2_4!_A(1,1,0,1) = 1
            filter2_4!_A(1,1,1,0) = 1
            filter2_4!_A(1,1,1,1) = 1
            g_0!_A = 1
            g_1!_A(0) = 0
            g_1!_A(1) = 1
            map_0!_A = 1
            map_1!_A(0) = 0
            map_1!_A(1) = 0
            map_2!_A(0,0) = 0
            map_2!_A(0,1) = 0
            map_2!_A(1,0) = 0
            map_2!_A(1,1) = 0
            nil_0!_A = 0
            true_0!_A = 1
            false_0!_A = 0
  
    partial status:
  
      pi(map_2#) = [1, 2]
      pi(cons_2) = [2]
      pi(app) = [1, 2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [1, 2]
      pi(filter2_4) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = [2]
      pi(nil_0) = []
      pi(true_0) = []
      pi(false_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#(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(g_1(x)) -> g_1(g_1(f_1(x)))
r14: f_1(g_1(x)) -> g_1(g_1(g_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, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      model-based reduction triple (=A, >=A!, >A!):
      
        finite algebra:
        
          carrier: {0,1}
          interpretations:
            f_1#_A(0) = 0
            f_1#_A(1) = 0
            g_1_A(0) = 0
            g_1_A(1) = 0
            app_A(0,0) = 1
            app_A(0,1) = 1
            app_A(1,0) = 0
            app_A(1,1) = 0
            cons_0_A = 0
            cons_1_A(0) = 1
            cons_1_A(1) = 1
            cons_2_A(0,0) = 0
            cons_2_A(0,1) = 0
            cons_2_A(1,0) = 0
            cons_2_A(1,1) = 0
            f_0_A = 1
            f_1_A(0) = 0
            f_1_A(1) = 0
            filter_0_A = 0
            filter_1_A(0) = 1
            filter_1_A(1) = 1
            filter_2_A(0,0) = 0
            filter_2_A(0,1) = 0
            filter_2_A(1,0) = 0
            filter_2_A(1,1) = 0
            filter2_0_A = 0
            filter2_1_A(0) = 1
            filter2_1_A(1) = 1
            filter2_2_A(0,0) = 0
            filter2_2_A(0,1) = 0
            filter2_2_A(1,0) = 0
            filter2_2_A(1,1) = 0
            filter2_3_A(0,0,0) = 1
            filter2_3_A(0,0,1) = 1
            filter2_3_A(0,1,0) = 1
            filter2_3_A(0,1,1) = 1
            filter2_3_A(1,0,0) = 1
            filter2_3_A(1,0,1) = 1
            filter2_3_A(1,1,0) = 1
            filter2_3_A(1,1,1) = 1
            filter2_4_A(0,0,0,0) = 0
            filter2_4_A(0,0,0,1) = 0
            filter2_4_A(0,0,1,0) = 0
            filter2_4_A(0,0,1,1) = 0
            filter2_4_A(0,1,0,0) = 0
            filter2_4_A(0,1,0,1) = 0
            filter2_4_A(0,1,1,0) = 0
            filter2_4_A(0,1,1,1) = 0
            filter2_4_A(1,0,0,0) = 0
            filter2_4_A(1,0,0,1) = 0
            filter2_4_A(1,0,1,0) = 0
            filter2_4_A(1,0,1,1) = 0
            filter2_4_A(1,1,0,0) = 0
            filter2_4_A(1,1,0,1) = 0
            filter2_4_A(1,1,1,0) = 0
            filter2_4_A(1,1,1,1) = 0
            g_0_A = 1
            map_0_A = 0
            map_1_A(0) = 1
            map_1_A(1) = 1
            map_2_A(0,0) = 0
            map_2_A(0,1) = 0
            map_2_A(1,0) = 0
            map_2_A(1,1) = 0
            nil_0_A = 0
            true_0_A = 1
            false_0_A = 0
            f_1#!_A(0) = 0
            f_1#!_A(1) = 0
            g_1!_A(0) = 0
            g_1!_A(1) = 1
            app!_A(0,0) = 1
            app!_A(0,1) = 1
            app!_A(1,0) = 1
            app!_A(1,1) = 1
            cons_0!_A = 1
            cons_1!_A(0) = 0
            cons_1!_A(1) = 1
            cons_2!_A(0,0) = 0
            cons_2!_A(0,1) = 0
            cons_2!_A(1,0) = 0
            cons_2!_A(1,1) = 1
            f_0!_A = 1
            f_1!_A(0) = 1
            f_1!_A(1) = 1
            filter_0!_A = 0
            filter_1!_A(0) = 0
            filter_1!_A(1) = 0
            filter_2!_A(0,0) = 0
            filter_2!_A(0,1) = 1
            filter_2!_A(1,0) = 0
            filter_2!_A(1,1) = 0
            filter2_0!_A = 0
            filter2_1!_A(0) = 0
            filter2_1!_A(1) = 0
            filter2_2!_A(0,0) = 0
            filter2_2!_A(0,1) = 1
            filter2_2!_A(1,0) = 1
            filter2_2!_A(1,1) = 0
            filter2_3!_A(0,0,0) = 0
            filter2_3!_A(0,0,1) = 0
            filter2_3!_A(0,1,0) = 0
            filter2_3!_A(0,1,1) = 0
            filter2_3!_A(1,0,0) = 0
            filter2_3!_A(1,0,1) = 0
            filter2_3!_A(1,1,0) = 0
            filter2_3!_A(1,1,1) = 0
            filter2_4!_A(0,0,0,0) = 1
            filter2_4!_A(0,0,0,1) = 1
            filter2_4!_A(0,0,1,0) = 1
            filter2_4!_A(0,0,1,1) = 1
            filter2_4!_A(0,1,0,0) = 0
            filter2_4!_A(0,1,0,1) = 0
            filter2_4!_A(0,1,1,0) = 0
            filter2_4!_A(0,1,1,1) = 0
            filter2_4!_A(1,0,0,0) = 0
            filter2_4!_A(1,0,0,1) = 0
            filter2_4!_A(1,0,1,0) = 0
            filter2_4!_A(1,0,1,1) = 0
            filter2_4!_A(1,1,0,0) = 1
            filter2_4!_A(1,1,0,1) = 1
            filter2_4!_A(1,1,1,0) = 1
            filter2_4!_A(1,1,1,1) = 1
            g_0!_A = 1
            map_0!_A = 1
            map_1!_A(0) = 1
            map_1!_A(1) = 1
            map_2!_A(0,0) = 1
            map_2!_A(0,1) = 0
            map_2!_A(1,0) = 1
            map_2!_A(1,1) = 0
            nil_0!_A = 0
            true_0!_A = 1
            false_0!_A = 1
  
    partial status:
  
      pi(f_1#) = [1]
      pi(g_1) = [1]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(f_0) = []
      pi(f_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(filter2_4) = []
      pi(g_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(nil_0) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.