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, r2, r3, r4, r5, r7, r8, r9, r11, r13, r14, r15, r16, r18, r19, r20, r21, r22, r23, r28

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 + 3
          cons_2_A(x1,x2) = x1 + x2 + 2
          app#_A(x1,x2) = x1 + x2 + 3
          map_1_A(x1) = x1 + 2
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 4
          false_0_A() = 4
          filter_2#_A(x1,x2) = x1 + x2 + 3
          filter_1_A(x1) = x1 + 2
          app_A(x1,x2) = x1 + x2 + 4
          true_0_A() = 2
          c_0_A() = 8
          c_1_A(x1) = 7
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 3
          f_0_A() = 6
          f_1_A(x1) = 5
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 1
          g_0_A() = 4
          g_1_A(x1) = x1 + 7
          if_0_A() = 1
          if_1_A(x1) = x1 + 2
          if_2_A(x1,x2) = x1 + x2 + 3
          if_3_A(x1,x2,x3) = x2 + 1
          map_0_A() = 1
          s_0_A() = 2
          s_1_A(x1) = 1
          |0|_0_A() = 3
          |1|_0_A() = 5
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > filter2_4# > filter_2# = filter_2 > filter_1 > map_2# = app > map_1 = g_1 = if_3 > c_0 > c_1 > f_1 = |1|_0 > cons_2 = filter2_3 = false_0 = f_0 = filter_0 = g_0 = if_2 = map_0 = s_0 = |0|_0 = nil_0 > true_0 = cons_1 = filter2_2 = if_1 > cons_0 = filter2_1 = if_0 = s_1 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = [2]
      pi(true_0) = []
      pi(c_0) = []
      pi(c_1) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(g_0) = []
      pi(g_1) = [1]
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = [2]
      pi(if_3) = [2]
      pi(map_0) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(|1|_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(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:

  {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(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, r2, r3, r4, r5, r7, r8, r11, r13, r14, r15, r16, r18, r19, r20, r21, r22, r23, r28

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 + 3
          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 + 4
          app_A(x1,x2) = x1 + x2 + 4
          true_0_A() = 1
          false_0_A() = 1
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
          c_0_A() = 1
          c_1_A(x1) = 3
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          f_0_A() = 3
          f_1_A(x1) = 2
          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
          if_0_A() = 1
          if_1_A(x1) = x1 + 3
          if_2_A(x1,x2) = x1 + x2 + 6
          if_3_A(x1,x2,x3) = 5
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = 4
          |0|_0_A() = 2
          |1|_0_A() = 2
          filter_2_A(x1,x2) = x1 + x2 + 3
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > filter2_4# = filter2_3 > filter_2# = f_0 = |0|_0 = filter_2 > filter_1 > app = filter_0 = nil_0 > cons_2 = filter2_2 = g_1 = if_2 = map_1 > cons_1 = filter2_1 = g_0 = if_1 = if_3 = map_0 > cons_0 = filter2_0 = if_0 = s_1 > c_1 = s_0 = |1|_0 > false_0 = c_0 = f_1 > true_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(filter_1) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [3]
      pi(c_0) = []
      pi(c_1) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(g_0) = []
      pi(g_1) = [1]
      pi(if_0) = []
      pi(if_1) = [1]
      pi(if_2) = [2]
      pi(if_3) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(|1|_0) = []
      pi(filter_2) = [1, 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(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, 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(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, r2, r5, r8, r9, r13, r15, r18, r19, r20, r21, r22, r23, r28

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 + 5
          cons_2_A(x1,x2) = x1 + x2 + 7
          filter2_4#_A(x1,x2,x3,x4) = x4 + 6
          app_A(x1,x2) = 7
          false_0_A() = 6
          true_0_A() = 4
          c_0_A() = 1
          c_1_A(x1) = 2
          cons_0_A() = 1
          cons_1_A(x1) = 7
          filter_0_A() = 7
          filter_1_A(x1) = 6
          filter2_1_A(x1) = 8
          filter2_2_A(x1,x2) = 5
          filter2_3_A(x1,x2,x3) = 6
          if_0_A() = 2
          if_1_A(x1) = 1
          if_2_A(x1,x2) = x1 + x2 + 8
          if_3_A(x1,x2,x3) = 4
          s_0_A() = 1
          s_1_A(x1) = 3
          f_1_A(x1) = 8
          |0|_0_A() = 5
          |1|_0_A() = 7
          filter_2_A(x1,x2) = x1 + 5
          nil_0_A() = 0
  
    base order 2:
  
      precedence:
      
        if_2 = if_3 > s_1 > c_1 = s_0 = filter_2 > cons_2 = c_0 = f_1 = |0|_0 > filter2_4# = true_0 > filter_2# = |1|_0 = nil_0 > app = false_0 > cons_1 = filter_1 = filter2_3 = if_1 > cons_0 = filter_0 = filter2_1 = filter2_2 = if_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(filter2_4#) = [4]
      pi(app) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(c_0) = []
      pi(c_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = [2]
      pi(if_3) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(|1|_0) = []
      pi(filter_2) = [1]
      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(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}


-- 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(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, r11, r12, r13, r14, r24, r25, r28

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 + 8
          cons_2_A(x1,x2) = x1 + x2 + 10
          filter2_4#_A(x1,x2,x3,x4) = x4 + 9
          app_A(x1,x2) = 2
          false_0_A() = 7
          c_0_A() = 0
          c_1_A(x1) = 1
          g_0_A() = 0
          g_1_A(x1) = 0
          g_2_A(x1,x2) = 1
          if_0_A() = 0
          if_1_A(x1) = 0
          if_2_A(x1,x2) = 0
          if_3_A(x1,x2,x3) = 5
          f_1_A(x1) = 3
          s_1_A(x1) = 6
          filter_2_A(x1,x2) = x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > cons_2 > filter_2# = filter2_4# = nil_0 > app = false_0 > c_0 = g_1 = f_1 > c_1 = g_0 = g_2 > if_3 > if_2 = s_1 > if_1 > if_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1]
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(c_0) = []
      pi(c_1) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(g_2) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(if_3) = []
      pi(f_1) = []
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: 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: map_2#(|fun|,cons_2(x,xs)) -> map_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, r2, r4, r5, r7, r8, r9, r11, r13, r14, r16, r18, r19, r20, r21, r22, r28

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) = x2
          cons_2_A(x1,x2) = x1 + x2 + 1
          app_A(x1,x2) = x2 + 7
          c_0_A() = 1
          c_1_A(x1) = x1 + 2
          cons_0_A() = 3
          cons_1_A(x1) = 2
          f_0_A() = 4
          f_1_A(x1) = 3
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          filter2_3_A(x1,x2,x3) = 2
          g_0_A() = 7
          g_1_A(x1) = 6
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = 4
          |0|_0_A() = 2
          true_0_A() = 1
          |1|_0_A() = 2
          false_0_A() = 1
          if_3_A(x1,x2,x3) = x3
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        cons_0 > cons_1 > cons_2 > app = g_0 = filter_2 > g_1 > filter_0 = if_0 = map_0 = s_0 > map_2# = c_1 = filter_1 = filter2_0 = filter2_3 = if_1 = map_1 = s_1 = |0|_0 = |1|_0 = if_3 = nil_0 > c_0 = f_1 = filter2_2 = if_2 > f_0 = filter2_1 = true_0 = false_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app) = []
      pi(c_0) = []
      pi(c_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []
      pi(if_3) = []
      pi(filter_2) = [1]
      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_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

  r1, r2, r4, r5, r7, r8, r9, r11, r13, r14, r16, r18, r19, r20, r21, r23, r28

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_2#_A(x1,x2) = x1 + x2 + 4
          c_1_A(x1) = x1 + 3
          s_1_A(x1) = x1 + 2
          app_A(x1,x2) = x2 + 8
          c_0_A() = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          f_0_A() = 8
          f_1_A(x1) = 7
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x2 + 2
          filter2_3_A(x1,x2,x3) = 3
          g_0_A() = 8
          g_1_A(x1) = 7
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          s_0_A() = 1
          |0|_0_A() = 4
          true_0_A() = 3
          |1|_0_A() = 4
          false_0_A() = 3
          if_3_A(x1,x2,x3) = x3 + 2
          filter_2_A(x1,x2) = x2 + 9
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app = cons_0 = f_0 = filter_0 = g_0 = if_0 = map_0 > g_2# = cons_1 = f_1 = filter_1 = filter2_3 = g_1 = if_1 = map_1 = |0|_0 = |1|_0 > filter2_0 = filter2_2 = true_0 = false_0 > s_1 = filter2_1 > s_0 = if_3 > c_1 = if_2 > c_0
      
  
    partial status:
  
      pi(g_2#) = [1]
      pi(c_1) = [1]
      pi(s_1) = []
      pi(app) = []
      pi(c_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(f_0) = []
      pi(f_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []
      pi(if_3) = []
      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: 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

  r3, r5, r7, r11, r13, r14, r16, r17, r19, r20, r21, r26, r27, r28, r30

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_2#_A(x1,x2) = x1 + x2
          c_1_A(x1) = x1 + 2
          app_A(x1,x2) = 5
          cons_1_A(x1) = x1 + 6
          cons_2_A(x1,x2) = 1
          filter_0_A() = 5
          filter_1_A(x1) = 4
          filter2_0_A() = 1
          filter2_1_A(x1) = 2
          g_0_A() = 2
          g_1_A(x1) = 1
          if_0_A() = 3
          if_1_A(x1) = 2
          if_2_A(x1,x2) = 1
          map_0_A() = 4
          map_1_A(x1) = 3
          map_2_A(x1,x2) = 2
          f_1_A(x1) = x1 + 4
          |0|_0_A() = 8
          true_0_A() = 7
          |1|_0_A() = 8
          false_0_A() = 7
          s_1_A(x1) = x1 + 3
          nil_0_A() = 1
          filter_2_A(x1,x2) = 6
          filter2_4_A(x1,x2,x3,x4) = x3 + 2
  
    base order 2:
  
      precedence:
      
        f_1 = |0|_0 > true_0 > g_2# = |1|_0 = s_1 > c_1 = false_0 > filter_2 > filter2_4 > app = filter_0 = filter2_0 = g_0 = if_0 = map_0 > cons_1 = cons_2 = filter_1 = filter2_1 = g_1 = if_1 = map_1 > if_2 = map_2 = nil_0
      
  
    partial status:
  
      pi(g_2#) = [1]
      pi(c_1) = [1]
      pi(app) = []
      pi(cons_1) = [1]
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(g_0) = []
      pi(g_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []
      pi(s_1) = []
      pi(nil_0) = []
      pi(filter_2) = []
      pi(filter2_4) = [3]

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

  r1, r2, r5, r7, r8, r9, r11, r13, r14, r16, r18, r19, r20, r21, r28

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
          s_1_A(x1) = x1 + 2
          app_A(x1,x2) = x2 + 6
          c_0_A() = 2
          c_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          filter2_3_A(x1,x2,x3) = 2
          g_0_A() = 2
          g_1_A(x1) = 1
          if_0_A() = 6
          if_1_A(x1) = 5
          if_2_A(x1,x2) = 4
          map_0_A() = 2
          map_1_A(x1) = 1
          s_0_A() = 1
          f_1_A(x1) = 7
          |0|_0_A() = 4
          true_0_A() = 3
          |1|_0_A() = 2
          false_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_0 = g_0 = if_0 = |0|_0 > app = cons_0 = filter2_0 = map_0 > cons_1 = filter2_1 = g_1 = if_1 = map_1 = f_1 = filter_2 > s_1 = filter_1 = if_2 = nil_0 > f_1# = filter2_3 = s_0 = true_0 = false_0 > c_1 = filter2_2 = |1|_0 > c_0
      
  
    partial status:
  
      pi(f_1#) = []
      pi(s_1) = [1]
      pi(app) = []
      pi(c_0) = []
      pi(c_1) = []
      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(g_0) = []
      pi(g_1) = []
      pi(if_0) = []
      pi(if_1) = []
      pi(if_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(f_1) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(|1|_0) = []
      pi(false_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.