YES
We show the termination of the TRS R:

  app(f(),|0|()) -> true()
  app(f(),|1|()) -> false()
  app(f(),app(s(),x)) -> app(f(),x)
  app(app(app(if(),true()),app(s(),x)),app(s(),y)) -> app(s(),x)
  app(app(app(if(),false()),app(s(),x)),app(s(),y)) -> app(s(),y)
  app(app(g(),x),app(c(),y)) -> app(c(),app(app(g(),x),y))
  app(app(g(),x),app(c(),y)) -> app(app(g(),x),app(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y)))
  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(c_0(),y) -> c_1(y)
  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(g_1(x1),y) -> g_2(x1,y)
  app(if_0(),y) -> if_1(y)
  app(if_1(x1),y) -> if_2(x1,y)
  app(if_2(x1,x2),y) -> if_3(x1,x2,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_1(|0|_0()) -> true_0()
  f_1(|1|_0()) -> false_0()
  f_1(s_1(x)) -> f_1(x)
  if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
  if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
  g_2(x,c_1(y)) -> c_1(g_2(x,y))
  g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
  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_1(x1),y) -> g_2#(x1,y)
p5: app#(if_2(x1,x2),y) -> if_3#(x1,x2,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: f_1#(s_1(x)) -> f_1#(x)
p8: g_2#(x,c_1(y)) -> g_2#(x,y)
p9: g_2#(x,c_1(y)) -> g_2#(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
p10: g_2#(x,c_1(y)) -> if_3#(f_1(x),c_1(g_2(s_1(x),y)),c_1(y))
p11: g_2#(x,c_1(y)) -> f_1#(x)
p12: g_2#(x,c_1(y)) -> g_2#(s_1(x),y)
p13: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p14: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p15: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p16: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p17: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p18: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p13, p14, p15, p16, p17, p18}
  {p8, p12}
  {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(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r1, r5, r7, r9, r11, r13, r14, r15, r16, r20, r22, r23, r26, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = 8
        cons_2_A(x1,x2) = max{14, x1 + 9, x2}
        app#_A(x1,x2) = 8
        map_1_A(x1) = max{50, x1 + 48}
        filter2_3_A(x1,x2,x3) = max{x1 - 18, x2 + 24, x3 + 48}
        filter2_4#_A(x1,x2,x3,x4) = 8
        false_0_A = 52
        filter_2#_A(x1,x2) = 8
        filter_1_A(x1) = max{60, x1 + 22}
        app_A(x1,x2) = max{59, x1 + 21, x2 + 49}
        true_0_A = 50
        c_0_A = 34
        c_1_A(x1) = max{54, x1 - 56}
        filter_0_A = 40
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 22}
        filter2_2_A(x1,x2) = max{x1 - 18, x2 + 24}
        g_0_A = 61
        g_1_A(x1) = 60
        if_0_A = 1
        if_1_A(x1) = max{35, x1 + 14}
        if_2_A(x1,x2) = max{x1 + 34, x2 + 36}
        if_3_A(x1,x2,x3) = max{58, x1 - 49, x2 + 56, x3 - 58}
        map_0_A = 1
        f_1_A(x1) = 53
        |1|_0_A = 53
        s_1_A(x1) = 113
        map_2_A(x1,x2) = max{71, x1 + 69, x2 + 49}
        nil_0_A = 22
        filter_2_A(x1,x2) = max{53, x1 + 23, x2 + 46}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{1, x1 - 231, x2 - 208}
          cons_2_A(x1,x2) = max{x1 + 207, x2 + 104}
          app#_A(x1,x2) = max{1, x1 - 247, x2 - 1}
          map_1_A(x1) = x1 + 16
          filter2_3_A(x1,x2,x3) = max{246, x1 + 18, x2 + 116, x3 + 38}
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 239, x3 - 209, x4 - 104}
          false_0_A = 72
          filter_2#_A(x1,x2) = max{1, x1 - 239, x2 - 208}
          filter_1_A(x1) = x1 + 8
          app_A(x1,x2) = max{x1 - 37, x2 + 102}
          true_0_A = 74
          c_0_A = 8
          c_1_A(x1) = max{3, x1 - 4}
          filter_0_A = 8
          filter2_0_A = 8
          filter2_1_A(x1) = x1 + 92
          filter2_2_A(x1,x2) = max{x1 + 55, x2 + 283}
          g_0_A = 10
          g_1_A(x1) = max{8, x1 - 67}
          if_0_A = 8
          if_1_A(x1) = max{8, x1 - 1}
          if_2_A(x1,x2) = max{x1 - 38, x2 + 46}
          if_3_A(x1,x2,x3) = max{10, x1 - 75, x2 + 9, x3}
          map_0_A = 8
          f_1_A(x1) = max{102, x1 - 23}
          |1|_0_A = 0
          s_1_A(x1) = max{103, x1 + 1}
          map_2_A(x1,x2) = max{x1 - 105, x2 + 102}
          nil_0_A = 209
          filter_2_A(x1,x2) = max{209, x1 - 105, x2 - 208}
          map_2#!_A(x1,x2) = max{0, x1 - 1}
          cons_2!_A(x1,x2) = 5
          app#!_A(x1,x2) = max{0, x1 - 17}
          map_1!_A(x1) = max{0, x1 - 1}
          filter2_3!_A(x1,x2,x3) = max{x1 - 1, x2 + 275, x3 - 1}
          filter2_4#!_A(x1,x2,x3,x4) = max{0, x2 - 9}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{0, x1 - 9}
          filter_1!_A(x1) = max{0, x1 - 9}
          app!_A(x1,x2) = max{0, x1 - 9}
          true_0!_A = 4
          c_0!_A = 1
          c_1!_A(x1) = max{2, x1 - 109}
          filter_0!_A = 1
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{2, x1 - 1}
          filter2_2!_A(x1,x2) = max{276, x2 - 1}
          g_0!_A = 1
          g_1!_A(x1) = max{7, x1 - 120}
          if_0!_A = 1
          if_1!_A(x1) = max{2, x1 - 1}
          if_2!_A(x1,x2) = max{3, x2 - 114}
          if_3!_A(x1,x2,x3) = max{38, x1 - 212, x2 - 113, x3 - 104}
          map_0!_A = 1
          f_1!_A(x1) = max{3, x1 - 120}
          |1|_0!_A = 2
          s_1!_A(x1) = max{5, x1 - 119}
          map_2!_A(x1,x2) = x1 + 6
          nil_0!_A = 5
          filter_2!_A(x1,x2) = max{109, x1 + 4, x2 + 1}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(c_0) = []
      pi(c_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(g_0) = []
      pi(g_1) = []
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = [1]
      pi(if_3) = []
      pi(map_0) = []
      pi(f_1) = []
      pi(|1|_0) = []
      pi(s_1) = []
      pi(map_2) = [1, 2]
      pi(nil_0) = []
      pi(filter_2) = [1]

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(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: 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(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r19, r20, r21, r26, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 3
        cons_2_A(x1,x2) = max{8, x2}
        app#_A(x1,x2) = max{2, x1 - 3}
        filter_1_A(x1) = max{7, x1 + 2}
        filter_2#_A(x1,x2) = max{3, x1 - 2}
        filter2_4#_A(x1,x2,x3,x4) = max{3, x2 - 2}
        app_A(x1,x2) = max{48, x1 - 11, x2 + 19}
        true_0_A = 20
        map_1_A(x1) = x1 + 10
        false_0_A = 20
        f_1_A(x1) = max{47, x1 - 22}
        |0|_0_A = 21
        |1|_0_A = 21
        s_1_A(x1) = x1
        map_2_A(x1,x2) = 9
        nil_0_A = 7
        filter_2_A(x1,x2) = max{8, x1 - 10, x2 - 11}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{21, x2 - 26}
          cons_2_A(x1,x2) = max{46, x1, x2 + 26}
          app#_A(x1,x2) = max{21, x2 - 26}
          filter_1_A(x1) = max{32, x1 - 25}
          filter_2#_A(x1,x2) = max{21, x2 - 26}
          filter2_4#_A(x1,x2,x3,x4) = max{21, x3 - 26, x4 - 1}
          app_A(x1,x2) = max{65, x1}
          true_0_A = 0
          map_1_A(x1) = max{40, x1 + 21}
          false_0_A = 11
          f_1_A(x1) = max{36, x1 - 13}
          |0|_0_A = 0
          |1|_0_A = 0
          s_1_A(x1) = x1 + 29
          map_2_A(x1,x2) = max{41, x1 + 21, x2 + 19}
          nil_0_A = 48
          filter_2_A(x1,x2) = max{33, x1 - 25, x2 + 13}
          map_2#!_A(x1,x2) = max{51, x2 + 26}
          cons_2!_A(x1,x2) = max{0, x2 - 42}
          app#!_A(x1,x2) = max{49, x2 + 2}
          filter_1!_A(x1) = max{50, x1 - 1}
          filter_2#!_A(x1,x2) = x2 + 2
          filter2_4#!_A(x1,x2,x3,x4) = max{48, x3 + 2, x4 + 28}
          app!_A(x1,x2) = 49
          true_0!_A = 53
          map_1!_A(x1) = max{50, x1 - 21}
          false_0!_A = 2
          f_1!_A(x1) = max{54, x1 - 30}
          |0|_0!_A = 54
          |1|_0!_A = 3
          s_1!_A(x1) = max{52, x1 - 31}
          map_2!_A(x1,x2) = max{0, x1 - 21, x2 - 49}
          nil_0!_A = 105
          filter_2!_A(x1,x2) = x2 + 56
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = []
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(map_1) = []
      pi(false_0) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(|1|_0) = []
      pi(s_1) = [1]
      pi(map_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p4, p5, p6, p7, p8



-- SCC decomposition.

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

p1: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: 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: g_2#(x,c_1(y)) -> g_2#(s_1(x),y)
p2: g_2#(x,c_1(y)) -> g_2#(x,y)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g_2#_A(x1,x2) = 1
        c_1_A(x1) = max{4, x1 - 7}
        s_1_A(x1) = max{8, x1 + 7}
        f_1_A(x1) = 2
        |0|_0_A = 8
        true_0_A = 1
        |1|_0_A = 2
        false_0_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g_2#_A(x1,x2) = max{0, x1 - 10, x2 - 19}
          c_1_A(x1) = max{17, x1 + 6}
          s_1_A(x1) = max{9, x1 - 1}
          f_1_A(x1) = max{20, x1 - 2}
          |0|_0_A = 1
          true_0_A = 0
          |1|_0_A = 0
          false_0_A = 0
          g_2#!_A(x1,x2) = max{1, x1 - 76, x2 - 15}
          c_1!_A(x1) = max{6, x1 - 14}
          s_1!_A(x1) = max{9, x1 - 75}
          f_1!_A(x1) = max{7, x1 - 77}
          |0|_0!_A = 2
          true_0!_A = 1
          |1|_0!_A = 2
          false_0!_A = 1
  
    partial status:
  
      pi(g_2#) = []
      pi(c_1) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: g_2#(x,c_1(y)) -> g_2#(x,y)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: g_2#(x,c_1(y)) -> g_2#(x,y)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g_2#_A(x1,x2) = max{2, x1 + 1}
        c_1_A(x1) = max{1, x1 - 39}
        f_1_A(x1) = max{9, x1 - 9}
        |0|_0_A = 6
        true_0_A = 5
        |1|_0_A = 8
        false_0_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g_2#_A(x1,x2) = max{0, x2 - 8}
          c_1_A(x1) = max{18, x1 + 7}
          f_1_A(x1) = max{22, x1 - 1}
          |0|_0_A = 0
          true_0_A = 0
          |1|_0_A = 0
          false_0_A = 0
          g_2#!_A(x1,x2) = max{9, x2 - 8}
          c_1!_A(x1) = max{8, x1 - 33}
          f_1!_A(x1) = max{12, x1 - 24}
          |0|_0!_A = 6
          true_0!_A = 5
          |1|_0!_A = 2
          false_0!_A = 1
  
    partial status:
  
      pi(g_2#) = [1]
      pi(c_1) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_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#(s_1(x)) -> f_1#(x)

and R consists of:

r1: app(c_0(),y) -> c_1(y)
r2: app(cons_0(),y) -> cons_1(y)
r3: app(cons_1(x1),y) -> cons_2(x1,y)
r4: app(f_0(),y) -> f_1(y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(g_0(),y) -> g_1(y)
r12: app(g_1(x1),y) -> g_2(x1,y)
r13: app(if_0(),y) -> if_1(y)
r14: app(if_1(x1),y) -> if_2(x1,y)
r15: app(if_2(x1,x2),y) -> if_3(x1,x2,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(s_0(),y) -> s_1(y)
r19: f_1(|0|_0()) -> true_0()
r20: f_1(|1|_0()) -> false_0()
r21: f_1(s_1(x)) -> f_1(x)
r22: if_3(true_0(),s_1(x),s_1(y)) -> s_1(x)
r23: if_3(false_0(),s_1(x),s_1(y)) -> s_1(y)
r24: g_2(x,c_1(y)) -> c_1(g_2(x,y))
r25: g_2(x,c_1(y)) -> g_2(x,if_3(f_1(x),c_1(g_2(s_1(x),y)),c_1(y)))
r26: map_2(|fun|,nil_0()) -> nil_0()
r27: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r28: filter_2(|fun|,nil_0()) -> nil_0()
r29: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r30: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r31: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f_1#_A(x1) = 0
        s_1_A(x1) = max{2, x1 - 5}
        f_1_A(x1) = x1 + 3
        |0|_0_A = 2
        true_0_A = 1
        |1|_0_A = 2
        false_0_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f_1#_A(x1) = max{0, x1 - 2}
          s_1_A(x1) = max{10, x1 + 1}
          f_1_A(x1) = max{22, x1 - 2}
          |0|_0_A = 1
          true_0_A = 0
          |1|_0_A = 1
          false_0_A = 0
          f_1#!_A(x1) = max{7, x1 - 2}
          s_1!_A(x1) = max{6, x1 - 25}
          f_1!_A(x1) = max{7, x1 - 24}
          |0|_0!_A = 2
          true_0!_A = 1
          |1|_0!_A = 2
          false_0!_A = 1
  
    partial status:
  
      pi(f_1#) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.