YES
We show the termination of the TRS R:

  app(app(minus(),x),|0|()) -> x
  app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
  app(f(),|0|()) -> app(s(),|0|())
  app(f(),app(s(),x)) -> app(app(minus(),app(s(),x)),app(g(),app(f(),x)))
  app(g(),|0|()) -> |0|()
  app(g(),app(s(),x)) -> app(app(minus(),app(s(),x)),app(f(),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)
  app(minus_0(),y) -> minus_1(y)
  app(minus_1(x1),y) -> minus_2(x1,y)
  app(s_0(),y) -> s_1(y)
  minus_2(x,|0|_0()) -> x
  minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
  f_1(|0|_0()) -> s_1(|0|_0())
  f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
  g_1(|0|_0()) -> |0|_0()
  g_1(s_1(x)) -> minus_2(s_1(x),f_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#(g_0(),y) -> g_1#(y)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: app#(minus_1(x1),y) -> minus_2#(x1,y)
p7: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p8: f_1#(s_1(x)) -> minus_2#(s_1(x),g_1(f_1(x)))
p9: f_1#(s_1(x)) -> g_1#(f_1(x))
p10: f_1#(s_1(x)) -> f_1#(x)
p11: g_1#(s_1(x)) -> minus_2#(s_1(x),f_1(g_1(x)))
p12: g_1#(s_1(x)) -> f_1#(g_1(x))
p13: g_1#(s_1(x)) -> g_1#(x)
p14: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p15: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p16: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p17: filter_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p18: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p19: 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: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p5, p14, p15, p16, p17, p18, p19}
  {p9, p10, p12, p13}
  {p7}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = 4
          cons_2_A(x1,x2) = 5
          app#_A(x1,x2) = 4
          map_1_A(x1) = x1 + 7
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 6
          filter2_4#_A(x1,x2,x3,x4) = 4
          false_0_A() = 5
          filter_2#_A(x1,x2) = 4
          filter_1_A(x1) = x1 + 5
          app_A(x1,x2) = x1 + x2 + 3
          true_0_A() = 6
          filter_2_A(x1,x2) = x1 + 2
          nil_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{1, x1 - 112, x2 - 27}
          cons_2_A(x1,x2) = max{26, x1 + 23, x2}
          app#_A(x1,x2) = max{1, x1 - 133, x2 - 4}
          map_1_A(x1) = x1 + 21
          filter2_3_A(x1,x2,x3) = max{x1 + 14, x2 + 58, x3 + 132}
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 75, x3 - 4, x4 - 27}
          false_0_A = 2
          filter_2#_A(x1,x2) = max{1, x1 - 75, x2 - 27}
          filter_1_A(x1) = x1 + 58
          app_A(x1,x2) = max{13, x1 - 131, x2 - 131}
          true_0_A = 0
          filter_2_A(x1,x2) = max{26, x1 - 146, x2 + 1}
          nil_0_A = 27
          map_2#!_A(x1,x2) = x1 + 5
          cons_2!_A(x1,x2) = max{4, x1 - 42, x2 - 31}
          app#!_A(x1,x2) = max{5, x1 - 15}
          map_1!_A(x1) = max{2, x1 - 24}
          filter2_3!_A(x1,x2,x3) = max{168, x1 - 9, x2 + 35}
          filter2_4#!_A(x1,x2,x3,x4) = x2 + 34
          false_0!_A = 1
          filter_2#!_A(x1,x2) = x1 + 34
          filter_1!_A(x1) = max{32, x1 - 1}
          app!_A(x1,x2) = max{x1 - 22, x2 + 169}
          true_0!_A = 5
          filter_2!_A(x1,x2) = max{169, x1 + 36, x2 + 6}
          nil_0!_A = 37
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p2, p4, p6, p7



-- SCC decomposition.

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

p1: map_2#(|fun|,cons_2(x,xs)) -> app#(|fun|,x)
p2: map_2#(|fun|,cons_2(x,xs)) -> map_2#(|fun|,xs)
p3: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p4: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p5: filter2_4#(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: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p3, p4, p5}


-- 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: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = 0
          cons_2_A(x1,x2) = max{3, x1 - 28, x2 - 4}
          map_2#!_A(x1,x2) = 0
          cons_2!_A(x1,x2) = max{2, x1 - 2, x2 - 5}
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p2: filter_2#(|fun|,cons_2(x,xs)) -> filter2_4#(app(|fun|,x),|fun|,x,xs)
p3: filter2_4#(true_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 4
          false_0_A() = 3
          filter_2#_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x2 + 3
          app_A(x1,x2) = x1 + x2 + 8
          true_0_A() = 3
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          filter2_4#_A(x1,x2,x3,x4) = max{2, x2 - 1, x3 - 15, x4 + 1}
          false_0_A = 6
          filter_2#_A(x1,x2) = max{2, x1 - 1, x2 - 13}
          cons_2_A(x1,x2) = max{x1 - 2, x2 + 14}
          app_A(x1,x2) = max{x1 - 46, x2 + 10}
          true_0_A = 6
          filter_2_A(x1,x2) = max{12, x1 - 54, x2}
          nil_0_A = 13
          filter2_4#!_A(x1,x2,x3,x4) = max{x2 + 6, x4 - 30}
          false_0!_A = 3
          filter_2#!_A(x1,x2) = max{6, x2 - 45}
          cons_2!_A(x1,x2) = max{4, x1 - 27, x2 - 29}
          app!_A(x1,x2) = max{x1 - 40, x2 + 4}
          true_0!_A = 3
          filter_2!_A(x1,x2) = max{4, x1 - 81, x2 - 27}
          nil_0!_A = 0
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- Reduction pair.

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

p1: g_1#(s_1(x)) -> f_1#(g_1(x))
p2: f_1#(s_1(x)) -> f_1#(x)
p3: f_1#(s_1(x)) -> g_1#(f_1(x))
p4: g_1#(s_1(x)) -> g_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: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  r16, r17, r18, r19, r20, r21

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          g_1#_A(x1) = x1
          s_1_A(x1) = x1 + 7
          f_1#_A(x1) = x1 + 2
          g_1_A(x1) = x1 + 4
          f_1_A(x1) = x1 + 8
          minus_2_A(x1,x2) = x1 + 3
          |0|_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g_1#_A(x1) = max{0, x1 - 46}
          s_1_A(x1) = max{33, x1 + 10}
          f_1#_A(x1) = max{0, x1 - 46}
          g_1_A(x1) = max{34, x1 + 10}
          f_1_A(x1) = max{45, x1 + 2}
          minus_2_A(x1,x2) = max{36, x1 + 2, x2 - 56}
          |0|_0_A = 35
          g_1#!_A(x1) = max{3, x1 - 3}
          s_1!_A(x1) = x1 + 6
          f_1#!_A(x1) = 30
          g_1!_A(x1) = 6
          f_1!_A(x1) = max{42, x1 - 36}
          minus_2!_A(x1,x2) = max{5, x1 - 60, x2 - 59}
          |0|_0!_A = 1
  
    partial status:
  
      pi(g_1#) = [1]
      pi(s_1) = [1]
      pi(f_1#) = []
      pi(g_1) = []
      pi(f_1) = []
      pi(minus_2) = []
      pi(|0|_0) = []

The following rules are removed:

  p1, p2, p3, p4

Then no dependency pair remains.

-- Reduction pair.

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

p1: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,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(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: app(minus_0(),y) -> minus_1(y)
r14: app(minus_1(x1),y) -> minus_2(x1,y)
r15: app(s_0(),y) -> s_1(y)
r16: minus_2(x,|0|_0()) -> x
r17: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r18: f_1(|0|_0()) -> s_1(|0|_0())
r19: f_1(s_1(x)) -> minus_2(s_1(x),g_1(f_1(x)))
r20: g_1(|0|_0()) -> |0|_0()
r21: g_1(s_1(x)) -> minus_2(s_1(x),f_1(g_1(x)))
r22: map_2(|fun|,nil_0()) -> nil_0()
r23: map_2(|fun|,cons_2(x,xs)) -> cons_2(app(|fun|,x),map_2(|fun|,xs))
r24: filter_2(|fun|,nil_0()) -> nil_0()
r25: filter_2(|fun|,cons_2(x,xs)) -> filter2_4(app(|fun|,x),|fun|,x,xs)
r26: filter2_4(true_0(),|fun|,x,xs) -> cons_2(x,filter_2(|fun|,xs))
r27: filter2_4(false_0(),|fun|,x,xs) -> filter_2(|fun|,xs)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2 + 6
          s_1_A(x1) = x1 + 5
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_2#_A(x1,x2) = 0
          s_1_A(x1) = max{14, x1 - 62}
          minus_2#!_A(x1,x2) = max{15, x2 - 24}
          s_1!_A(x1) = max{14, x1 - 23}
  
    partial status:
  
      pi(minus_2#) = [1]
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.