YES
We show the termination of the TRS R:

  app(rev(),nil()) -> nil()
  app(rev(),app(app(cons(),x),l)) -> app(app(cons(),app(app(rev1(),x),l)),app(app(rev2(),x),l))
  app(app(rev1(),|0|()),nil()) -> |0|()
  app(app(rev1(),app(s(),x)),nil()) -> app(s(),x)
  app(app(rev1(),x),app(app(cons(),y),l)) -> app(app(rev1(),y),l)
  app(app(rev2(),x),nil()) -> nil()
  app(app(rev2(),x),app(app(cons(),y),l)) -> app(rev(),app(app(cons(),x),app(app(rev2(),y),l)))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
  app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
  app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),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(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(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(rev_0(),y) -> rev_1(y)
  app(rev1_0(),y) -> rev1_1(y)
  app(rev1_1(x1),y) -> rev1_2(x1,y)
  app(rev2_0(),y) -> rev2_1(y)
  app(rev2_1(x1),y) -> rev2_2(x1,y)
  app(s_0(),y) -> s_1(y)
  rev_1(nil_0()) -> nil_0()
  rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
  rev1_2(|0|_0(),nil_0()) -> |0|_0()
  rev1_2(s_1(x),nil_0()) -> s_1(x)
  rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
  rev2_2(x,nil_0()) -> nil_0()
  rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
  filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)


-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: app#(rev_0(),y) -> rev_1#(y)
p5: app#(rev1_1(x1),y) -> rev1_2#(x1,y)
p6: app#(rev2_1(x1),y) -> rev2_2#(x1,y)
p7: rev_1#(cons_2(x,l)) -> rev1_2#(x,l)
p8: rev_1#(cons_2(x,l)) -> rev2_2#(x,l)
p9: rev1_2#(x,cons_2(y,l)) -> rev1_2#(y,l)
p10: rev2_2#(x,cons_2(y,l)) -> rev_1#(cons_2(x,rev2_2(y,l)))
p11: rev2_2#(x,cons_2(y,l)) -> rev2_2#(y,l)
p12: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p13: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p14: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p15: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p16: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p17: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p12, p13, p14, p15, p16, p17}
  {p8, p10, p11}
  {p9}


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p7: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p8: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p9: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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:
          app#_A(x1,x2) = x1 + x2 + 5
          filter_1_A(x1) = x1 + 3
          filter_2#_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          map_1_A(x1) = x1 + 2
          map_2#_A(x1,x2) = x1 + x2 + 5
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          false_0_A() = 1
          app_A(x1,x2) = x1 + x2 + 5
          true_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          map_0_A() = 1
          rev1_0_A() = 1
          rev1_1_A(x1) = x1 + 2
          rev2_0_A() = 1
          rev2_1_A(x1) = x1 + 2
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1 + 2
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        map_2# > app# > filter2_4# > filter_2# > app = filter_2 > filter_1 = map_1 = filter2_3 = false_0 = rev_1 > filter_0 = filter2_2 = map_0 = rev1_2 > cons_2 = filter2_1 = rev1_1 = s_1 > true_0 = cons_1 = filter2_0 = rev1_0 = rev2_1 = s_0 > cons_0 = rev2_0 = nil_0 > |0|_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(map_0) = []
      pi(rev1_0) = []
      pi(rev1_1) = [1]
      pi(rev2_0) = []
      pi(rev2_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(rev_1) = [1]
      pi(nil_0) = []
      pi(rev1_2) = []
      pi(|0|_0) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p8: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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 + 2
          cons_2_A(x1,x2) = x1 + x2 + 6
          app#_A(x1,x2) = x1 + x2 + 5
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          true_0_A() = 1
          app_A(x1,x2) = x1 + x2 + 7
          false_0_A() = 1
          map_1_A(x1) = x1 + 3
          map_2#_A(x1,x2) = x1 + x2 + 7
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 3
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          map_0_A() = 1
          rev1_0_A() = 1
          rev1_1_A(x1) = x1 + 2
          rev2_0_A() = 1
          rev2_1_A(x1) = x1 + 2
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1 + 11
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        map_2# > app# > filter2_4# > filter_2# > app = filter_2 > rev_1 > filter_0 = rev1_2 > cons_2 = filter2_3 = false_0 = map_1 = filter_1 = rev1_1 = s_1 > true_0 = cons_1 = filter2_2 = map_0 = rev1_0 = s_0 > cons_0 = filter2_1 = rev2_1 = nil_0 > filter2_0 = rev2_0 = |0|_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(filter2_3) = [3]
      pi(filter2_4#) = [4]
      pi(true_0) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(map_0) = []
      pi(rev1_0) = []
      pi(rev1_1) = [1]
      pi(rev2_0) = []
      pi(rev2_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(rev_1) = []
      pi(nil_0) = []
      pi(rev1_2) = []
      pi(|0|_0) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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 + 7
          cons_2_A(x1,x2) = x1 + x2 + 6
          app#_A(x1,x2) = x1 + x2 + 7
          map_1_A(x1) = x1 + 2
          app_A(x1,x2) = x2 + 4
          cons_0_A() = 1
          cons_1_A(x1) = 2
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 1
          rev1_0_A() = 2
          rev1_1_A(x1) = 1
          rev2_0_A() = 2
          rev2_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 5
  
    base order 2:
  
      precedence:
      
        app# > map_2# > cons_2 = app = cons_0 = filter_0 = filter2_0 = rev1_0 = rev2_0 = rev_1 = rev1_2 = filter_2 > map_1 = cons_1 = filter_1 = filter2_2 = rev1_1 = rev2_1 = s_1 > filter2_1 = filter2_3 = map_0 = s_0 = nil_0 > |0|_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1]
      pi(app#) = [2]
      pi(map_1) = [1]
      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(rev1_0) = []
      pi(rev1_1) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(rev_1) = []
      pi(nil_0) = []
      pi(rev1_2) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]

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#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26, r27, r28, r29

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 + 6
          app_A(x1,x2) = x2 + 4
          cons_0_A() = 4
          cons_1_A(x1) = 3
          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
          rev1_0_A() = 2
          rev1_1_A(x1) = 1
          rev2_0_A() = 2
          rev2_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1 + 2
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + 5
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 5
          filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 11
          true_0_A() = 1
          false_0_A() = 1
  
    base order 2:
  
      precedence:
      
        rev2_0 > rev2_1 = filter_2 > filter2_4 > cons_2 = filter2_0 > filter2_1 = s_0 = rev_1 = rev1_2 > cons_0 = filter2_2 = s_1 > map_2# = app = cons_1 = filter_0 = filter2_3 = map_0 = rev1_0 = nil_0 = true_0 = false_0 > filter_1 = map_1 = rev1_1 = |0|_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app) = [2]
      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(rev1_0) = []
      pi(rev1_1) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(rev_1) = [1]
      pi(nil_0) = []
      pi(rev1_2) = [1]
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(filter2_4) = [3]
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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:
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          true_0_A() = 1
          filter_2#_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          app_A(x1,x2) = x1 + x2 + 5
          false_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          rev1_0_A() = 1
          rev1_1_A(x1) = x1 + 2
          rev2_0_A() = 1
          rev2_1_A(x1) = x1 + 2
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 3
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      precedence:
      
        filter2_4# > filter_2# = filter_2 > cons_2 = app > filter2_3 = rev2_1 = rev1_2 > true_0 = false_0 = cons_1 = filter_1 = filter2_1 = map_1 = rev1_1 = rev2_0 = s_1 = rev_1 > cons_0 = filter_0 = filter2_0 = filter2_2 = map_0 = rev1_0 = s_0 = nil_0 > |0|_0
      
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(true_0) = []
      pi(filter_2#) = [1, 2]
      pi(cons_2) = []
      pi(app) = [1, 2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(rev1_0) = []
      pi(rev1_1) = [1]
      pi(rev2_0) = []
      pi(rev2_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(rev_1) = []
      pi(nil_0) = []
      pi(rev1_2) = [1, 2]
      pi(|0|_0) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,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#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

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) = 4
          cons_2_A(x1,x2) = 0
          filter2_4#_A(x1,x2,x3,x4) = x1
          app_A(x1,x2) = 3
          false_0_A() = 5
          cons_0_A() = 0
          cons_1_A(x1) = 0
          filter_0_A() = 0
          filter_1_A(x1) = 0
          filter_2_A(x1,x2) = 2
          filter2_0_A() = 0
          filter2_1_A(x1) = 0
          filter2_2_A(x1,x2) = 1
          filter2_3_A(x1,x2,x3) = 0
          filter2_4_A(x1,x2,x3,x4) = 2
          map_0_A() = 0
          map_1_A(x1) = 0
          map_2_A(x1,x2) = 2
          rev_0_A() = 0
          rev_1_A(x1) = 1
          rev1_0_A() = 0
          rev1_1_A(x1) = 0
          rev1_2_A(x1,x2) = 0
          rev2_0_A() = 0
          rev2_1_A(x1) = 0
          rev2_2_A(x1,x2) = 2
          s_0_A() = 0
          s_1_A(x1) = 0
          nil_0_A() = 0
          |0|_0_A() = 0
          true_0_A() = 3
  
    base order 2:
  
      precedence:
      
        filter2_4# = false_0 = filter2_3 = true_0 > filter_2# = filter_2 = filter2_4 > app = map_2 > rev_1 > cons_1 = rev1_2 = rev2_2 = s_1 > cons_2 = cons_0 = filter_1 = filter2_2 = map_1 = rev_0 = rev1_1 = rev2_1 = s_0 = nil_0 > filter_0 = filter2_1 = map_0 = rev1_0 = rev2_0 = |0|_0 > filter2_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(filter2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(rev_0) = []
      pi(rev_1) = []
      pi(rev1_0) = []
      pi(rev1_1) = []
      pi(rev1_2) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(rev2_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(nil_0) = []
      pi(|0|_0) = []
      pi(true_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: rev2_2#(x,cons_2(y,l)) -> rev_1#(cons_2(x,rev2_2(y,l)))
p2: rev_1#(cons_2(x,l)) -> rev2_2#(x,l)
p3: rev2_2#(x,cons_2(y,l)) -> rev2_2#(y,l)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r9, r11, r12, r13, r14, r16, r17, r18, r19, r20, r21, r22, r23, r26

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:
          rev2_2#_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          rev_1#_A(x1) = x1 + 1
          rev2_2_A(x1,x2) = x1 + x2
          app_A(x1,x2) = x2 + 2
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          rev_0_A() = 1
          rev_1_A(x1) = x1
          rev1_0_A() = 2
          rev1_1_A(x1) = 1
          rev1_2_A(x1,x2) = 0
          rev2_0_A() = 2
          rev2_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = 0
          nil_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        rev2_2 = filter_2 > rev_1# = filter2_1 = rev_1 > rev2_2# = cons_2 = filter2_0 = filter2_2 = rev_0 > app = cons_0 = filter_0 = map_0 = rev1_0 = rev2_0 = nil_0 > cons_1 = filter_1 = filter2_3 = map_1 = rev1_1 = rev2_1 > rev1_2 > s_1 = |0|_0 > s_0
      
  
    partial status:
  
      pi(rev2_2#) = []
      pi(cons_2) = [1]
      pi(rev_1#) = []
      pi(rev2_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) = [1, 2]
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(rev_0) = []
      pi(rev_1) = [1]
      pi(rev1_0) = []
      pi(rev1_1) = []
      pi(rev1_2) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(nil_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: rev_1#(cons_2(x,l)) -> rev2_2#(x,l)
p2: rev2_2#(x,cons_2(y,l)) -> rev2_2#(y,l)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2}


-- Reduction pair.

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

p1: rev2_2#(x,cons_2(y,l)) -> rev2_2#(y,l)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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:
          rev2_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 6
          app_A(x1,x2) = x2 + 4
          cons_0_A() = 2
          cons_1_A(x1) = 1
          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
          rev1_0_A() = 4
          rev1_1_A(x1) = 3
          rev2_0_A() = 2
          rev2_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 2
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        filter_0 = filter_2 > cons_2 = app = cons_0 = filter_1 = filter2_0 = map_0 = rev1_0 = rev2_0 > rev2_2# = cons_1 = filter2_1 = map_1 = rev2_1 = rev1_2 > filter2_3 = rev1_1 = s_1 = rev_1 > filter2_2 = s_0 = nil_0 > |0|_0
      
  
    partial status:
  
      pi(rev2_2#) = []
      pi(cons_2) = [1]
      pi(app) = [2]
      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(rev1_0) = []
      pi(rev1_1) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(rev_1) = []
      pi(nil_0) = []
      pi(rev1_2) = [2]
      pi(|0|_0) = []
      pi(filter_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: rev1_2#(x,cons_2(y,l)) -> rev1_2#(y,l)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(map_0(),y) -> map_1(y)
r10: app(map_1(x1),y) -> map_2(x1,y)
r11: app(rev_0(),y) -> rev_1(y)
r12: app(rev1_0(),y) -> rev1_1(y)
r13: app(rev1_1(x1),y) -> rev1_2(x1,y)
r14: app(rev2_0(),y) -> rev2_1(y)
r15: app(rev2_1(x1),y) -> rev2_2(x1,y)
r16: app(s_0(),y) -> s_1(y)
r17: rev_1(nil_0()) -> nil_0()
r18: rev_1(cons_2(x,l)) -> cons_2(rev1_2(x,l),rev2_2(x,l))
r19: rev1_2(|0|_0(),nil_0()) -> |0|_0()
r20: rev1_2(s_1(x),nil_0()) -> s_1(x)
r21: rev1_2(x,cons_2(y,l)) -> rev1_2(y,l)
r22: rev2_2(x,nil_0()) -> nil_0()
r23: rev2_2(x,cons_2(y,l)) -> rev_1(cons_2(x,rev2_2(y,l)))
r24: map_2(f,nil_0()) -> nil_0()
r25: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r26: filter_2(f,nil_0()) -> nil_0()
r27: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r28: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r29: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r20, r26

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:
          rev1_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 6
          app_A(x1,x2) = x2 + 4
          cons_0_A() = 2
          cons_1_A(x1) = 1
          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
          rev1_0_A() = 2
          rev1_1_A(x1) = 1
          rev2_0_A() = 2
          rev2_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          rev_1_A(x1) = x1
          nil_0_A() = 2
          rev1_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        cons_2 = rev_1 > rev1_2 > filter_2 > app = cons_0 = filter_0 = filter2_0 = map_0 = rev1_0 = rev2_0 > rev1_2# = cons_1 = filter_1 = filter2_1 = filter2_3 = map_1 = rev1_1 = rev2_1 = s_1 > filter2_2 = s_0 = nil_0 > |0|_0
      
  
    partial status:
  
      pi(rev1_2#) = []
      pi(cons_2) = [1]
      pi(app) = [2]
      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(rev1_0) = []
      pi(rev1_1) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(rev_1) = []
      pi(nil_0) = []
      pi(rev1_2) = [2]
      pi(|0|_0) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.