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

  r1, r2, r3, r4, r6, r7, r8, r10, r11, r13, r14, r15, r16, r17, r18, r19, r20, r21, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 4
          cons_2_A(x1,x2) = x1 + x2 + 3
          app#_A(x1,x2) = x1 + x2 + 2
          map_1_A(x1) = x1 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          false_0_A() = 4
          filter_2#_A(x1,x2) = x1 + x2 + 3
          filter_1_A(x1) = x1 + 4
          app_A(x1,x2) = x1 + x2 + 4
          true_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          f_0_A() = 1
          f_1_A(x1) = x1 + 4
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          g_0_A() = 1
          g_1_A(x1) = x1 + 2
          map_0_A() = 1
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + 1
          s_0_A() = 4
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        map_2# = filter2_4# > cons_2 = app# = filter_2 > false_0 = true_0 > map_1 = filter_2# > app = map_0 = nil_0 > filter2_3 = filter_1 = cons_1 = f_1 = minus_1 > cons_0 = f_0 = filter_0 = filter2_1 = filter2_2 = minus_0 = minus_2 > filter2_0 = s_1 > g_1 = s_0 > g_0 = |0|_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(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:

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


-- Reduction pair.

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

p1: filter_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: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,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

  r1, r2, r3, r4, r6, r7, r8, r10, r11, r13, r14, r15, r16, r17, r18, r19, r20, r21, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 5
          cons_2_A(x1,x2) = x1 + x2 + 3
          app#_A(x1,x2) = x1 + x2 + 4
          filter_1_A(x1) = x1 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 6
          app_A(x1,x2) = x1 + x2 + 7
          true_0_A() = 1
          false_0_A() = 1
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          f_0_A() = 1
          f_1_A(x1) = x1 + 4
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          g_0_A() = 1
          g_1_A(x1) = x1 + 2
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 3
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > filter2_4# > filter_2# = f_1 > cons_2 = app = f_0 = filter_0 > cons_1 = map_1 = minus_1 > filter2_3 = cons_0 = map_0 = minus_0 = minus_2 = s_1 = filter_2 > filter_1 = true_0 = false_0 = filter2_2 = g_1 = s_0 = nil_0 > filter2_1 = g_0 = |0|_0 > filter2_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [3]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(g_0) = []
      pi(g_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)
p4: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,xs)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,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 estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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)
p2: filter2_4#(false_0(),|fun|,x,xs) -> filter_2#(|fun|,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

  r1, r2, r4, r7, r8, r11, r13, r15, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          filter2_4#_A(x1,x2,x3,x4) = x4 + 3
          app_A(x1,x2) = x1 + x2 + 5
          false_0_A() = 3
          true_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 5
          filter2_1_A(x1) = x1 + 7
          filter2_2_A(x1,x2) = x1 + x2 + 8
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 9
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 3
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        cons_2 = filter2_3 > filter2_4# = minus_0 = s_0 > minus_1 = s_1 = filter_2 > app = nil_0 > filter_2# = cons_1 = filter_1 = filter2_1 = filter2_2 = map_1 > false_0 = true_0 = cons_0 = filter_0 = map_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(filter_2) = [1, 2]
      pi(nil_0) = []

The following rules are removed:

  p3



-- 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)
p2: 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:

  {p1, p2}


-- Reduction pair.

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)
p2: 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 set of usable rules consists of

  r1, r2, r3, r4, r6, r8, r10, r11, r13, r14, r15, r16, r17, r18, r19, r20, r21, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2 + 4
          cons_2_A(x1,x2) = x1 + x2 + 8
          filter2_4#_A(x1,x2,x3,x4) = x4 + 5
          app_A(x1,x2) = x1 + x2 + 3
          false_0_A() = 3
          cons_0_A() = 4
          cons_1_A(x1) = x1 + 6
          f_0_A() = 1
          f_1_A(x1) = x1 + 2
          filter_0_A() = 4
          filter_1_A(x1) = x1 + 3
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 5
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 6
          g_0_A() = 1
          g_1_A(x1) = x1 + 3
          map_0_A() = 8
          map_1_A(x1) = x1 + 10
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 1
          filter_2_A(x1,x2) = 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        g_1 > f_1 = minus_1 > cons_2 = minus_2 > map_1 = s_1 > filter2_4# = app = false_0 = |0|_0 > filter_2# = cons_1 = f_0 = filter_1 = g_0 = map_0 = minus_0 = s_0 > cons_0 = filter_0 = filter2_3 = filter_2 > filter2_0 = filter2_2 = nil_0 > filter2_1
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = [4]
      pi(app) = [2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(g_0) = []
      pi(g_1) = [1]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: 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: 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:

  (no SCCs)

-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(f_0(),y) -> f_1(y)
r4: app(filter_0(),y) -> filter_1(y)
r5: app(filter_1(x1),y) -> filter_2(x1,y)
r6: app(filter2_0(),y) -> filter2_1(y)
r7: app(filter2_1(x1),y) -> filter2_2(x1,y)
r8: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r9: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r10: app(g_0(),y) -> g_1(y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: 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

  r1, r4, r6, r7, r8, r11, r13, r15, r16, r17, r21, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 10
          cons_2_A(x1,x2) = x1 + x2 + 9
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 8
          cons_1_A(x1) = 7
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 7
          minus_2_A(x1,x2) = x1 + 3
          |0|_0_A() = 1
          g_1_A(x1) = x1 + 4
          f_1_A(x1) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > map_2# = nil_0 > cons_2 > filter2_3 > app = cons_0 = filter_0 = filter2_0 = map_0 = minus_0 > cons_1 = filter_1 = filter2_2 = map_1 = s_1 > filter2_1 = s_0 = minus_2 > minus_1 = |0|_0 = f_1 > g_1
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(g_1) = [1]
      pi(f_1) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r1, r3, r4, r7, r8, r10, r11, r13, r15, r16, r17, r18, r19, r20, r21, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    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 + 6
          f_1#_A(x1) = x1 + 2
          g_1_A(x1) = x1 + 3
          f_1_A(x1) = x1 + 7
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 2
          cons_1_A(x1) = 1
          f_0_A() = 1
          filter_0_A() = 8
          filter_1_A(x1) = 7
          filter2_1_A(x1) = 9
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          g_0_A() = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 7
          minus_2_A(x1,x2) = x1 + 2
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        s_1 > g_1# = s_0 = filter_2 > f_1# = app = cons_0 = filter_0 = map_0 = minus_0 = minus_2 = nil_0 > g_1 = f_1 = cons_1 = filter_1 = filter2_1 = map_1 = minus_1 > f_0 = filter2_2 = g_0 = |0|_0 > filter2_3
      
  
    partial status:
  
      pi(g_1#) = [1]
      pi(s_1) = [1]
      pi(f_1#) = [1]
      pi(g_1) = [1]
      pi(f_1) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: f_1#(s_1(x)) -> f_1#(x)
p2: f_1#(s_1(x)) -> g_1#(f_1(x))
p3: 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 estimated dependency graph contains the following SCCs:

  {p1}
  {p3}


-- 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(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

  r1, r4, r6, r7, r8, r11, r13, r15, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          f_1#_A(x1) = x1 + 6
          s_1_A(x1) = x1 + 5
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = x3 + 2
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        s_1 > f_1# = filter_2 > filter2_3 = nil_0 > app = filter_0 = filter2_0 = map_0 = minus_0 = s_0 > cons_1 = filter_1 = filter2_1 = map_1 = minus_1 > cons_0 = filter2_2
      
  
    partial status:
  
      pi(f_1#) = [1]
      pi(s_1) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r1, r4, r6, r7, r8, r11, r13, r15, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          g_1#_A(x1) = x1 + 6
          s_1_A(x1) = x1 + 5
          app_A(x1,x2) = x2 + 8
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = x3 + 2
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        s_1 > g_1# = filter_2 > filter2_3 = nil_0 > app = filter_0 = filter2_0 = map_0 = minus_0 = s_0 > cons_1 = filter_1 = filter2_1 = map_1 = minus_1 > cons_0 = filter2_2
      
  
    partial status:
  
      pi(g_1#) = [1]
      pi(s_1) = []
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r1, r4, r6, r7, r8, r11, r13, r15, r24

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 5
          app_A(x1,x2) = x2 + 10
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          map_0_A() = 10
          map_1_A(x1) = 9
          minus_0_A() = 10
          minus_1_A(x1) = 9
          s_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 11
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_0 > minus_2# = filter_0 = filter2_1 = minus_0 > filter2_2 = nil_0 > filter2_3 = filter_2 > app = map_0 > cons_1 = filter_1 = map_1 > cons_0 = minus_1 > s_1 > s_0
      
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.