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, r5, r12, r14, r16, r17, r19, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{5, x1 - 7}
        filter_1_A(x1) = max{13, x1 + 4}
        filter_2#_A(x1,x2) = max{5, x1 - 7}
        cons_2_A(x1,x2) = max{4, x1 - 41, x2 - 41}
        map_1_A(x1) = x1 + 14
        map_2#_A(x1,x2) = max{6, x1 - 1}
        filter2_3_A(x1,x2,x3) = max{18, x1 - 1, x2 + 6, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x2 - 7}
        false_0_A = 1
        app_A(x1,x2) = max{16, x1 - 19, x2 - 19}
        true_0_A = 3
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 20}
        filter2_0_A = 18
        filter2_1_A(x1) = max{15, x1 - 20}
        rev1_0_A = 1
        rev1_1_A(x1) = max{2, x1 - 74}
        rev2_0_A = 2
        rev2_1_A(x1) = max{3, x1 - 20}
        s_0_A = 5
        s_1_A(x1) = max{6, x1 - 20}
        rev_1_A(x1) = max{15, x1 - 52}
        nil_0_A = 5
        rev1_2_A(x1,x2) = max{3, x1 - 94, x2 - 94}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{14, x1 - 41, x2 - 29}
  
    base order 2:
  
      precedence:
      
        cons_1 = filter2_1 > app = cons_0 = filter2_0 > map_1 = filter_2 > map_2# = false_0 = rev_1 = nil_0 > filter_1 = filter_2# = filter2_3 = filter2_4# = rev2_1 = s_1 > app# = rev2_0 = s_0 > cons_2 > true_0 = rev1_2 > rev1_1 = |0|_0 > rev1_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_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) = []
      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: 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, r3, r5, r6, r7, r9, r12, r14, r16, r17, r19, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{3, x2 + 1}
        cons_2_A(x1,x2) = max{26, x1 + 6, x2 + 6}
        app#_A(x1,x2) = x2 + 6
        filter2_3_A(x1,x2,x3) = max{4, x1 - 27, x2 - 27, x3 - 27}
        filter2_4#_A(x1,x2,x3,x4) = max{5, x4 + 2}
        true_0_A = 25
        app_A(x1,x2) = max{25, x1 - 10, x2 + 17}
        false_0_A = 18
        map_1_A(x1) = max{9, x1 - 27}
        map_2#_A(x1,x2) = x2 + 5
        cons_0_A = 9
        cons_1_A(x1) = x1 + 16
        filter_0_A = 1
        filter_1_A(x1) = max{9, x1 - 1}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 1}
        filter2_2_A(x1,x2) = max{3, x1 - 11, x2 - 16}
        map_0_A = 1
        rev1_0_A = 1
        rev1_1_A(x1) = max{9, x1 - 24}
        rev2_0_A = 8
        rev2_1_A(x1) = max{9, x1 - 20}
        s_0_A = 1
        s_1_A(x1) = max{22, x1 - 1}
        rev_1_A(x1) = max{18, x1 - 34}
        nil_0_A = 7
        rev1_2_A(x1,x2) = max{21, x1 - 23, x2 - 27}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{8, x1 - 44}
  
    base order 2:
  
      precedence:
      
        app = filter_0 > filter_1 > cons_2 = filter2_3 = map_1 = s_1 = rev_1 = rev1_2 = filter_2 > true_0 = false_0 = map_2# = cons_1 = filter2_2 = map_0 = rev1_1 = s_0 > filter_2# = cons_0 = filter2_1 = rev1_0 = rev2_1 = nil_0 > app# = filter2_0 = rev2_0 = |0|_0 > filter2_4#
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(app) = []
      pi(false_0) = []
      pi(map_1) = []
      pi(map_2#) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      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) = []
      pi(|0|_0) = []
      pi(filter_2) = []

The following rules are removed:

  p2



-- 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: 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:

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


-- Reduction pair.

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

p1: filter2_4#(false_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#(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

  r2, r3, r5, r6, r7, r9, r13, r14, r16, r17, r19, r20, r21, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 3, x4 + 3}
        false_0_A = 4
        filter_2#_A(x1,x2) = max{2, x2 + 1}
        cons_2_A(x1,x2) = max{x1 + 24, x2 + 4}
        app_A(x1,x2) = max{8, x1 - 30, x2 + 5}
        true_0_A = 15
        cons_1_A(x1) = x1 + 55
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 6}
        filter2_0_A = 29
        filter2_1_A(x1) = max{7, x1 - 1}
        filter2_2_A(x1,x2) = max{6, x1 - 32, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{7, x1 - 63, x2 - 32, x3 - 1}
        map_0_A = 6
        map_1_A(x1) = max{7, x1 - 6}
        rev1_1_A(x1) = x1 + 30
        rev1_2_A(x1,x2) = max{8, x1, x2 - 23}
        rev2_0_A = 1
        rev2_1_A(x1) = max{0, x1 - 1}
        s_0_A = 7
        s_1_A(x1) = max{8, x1 - 1}
        rev_1_A(x1) = max{8, x1}
        nil_0_A = 3
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{16, x1 - 5, x2 - 18}
  
    base order 2:
  
      precedence:
      
        cons_2 > filter_2 > false_0 > filter_2# = app > filter2_4# = rev2_1 = rev_1 > filter_0 = filter2_3 = rev1_2 = s_1 > true_0 = cons_1 = filter_1 = filter2_2 = map_1 = rev1_1 = rev2_0 = s_0 = nil_0 > filter2_1 = map_0 = |0|_0 > filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_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_1) = []
      pi(rev1_2) = []
      pi(rev2_0) = []
      pi(rev2_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(rev_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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: 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}


-- 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#(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, r3, r5, r6, r7, r9, r12, r14, r16, r19, r20

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{4, x2 - 2}
        cons_2_A(x1,x2) = max{37, x1 + 10, x2 + 4}
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 7, x4 - 1}
        app_A(x1,x2) = max{26, x2 + 5}
        true_0_A = 3
        cons_0_A = 0
        cons_1_A(x1) = x1
        filter_0_A = 0
        filter_1_A(x1) = max{26, x1}
        filter2_0_A = 1
        filter2_1_A(x1) = max{10, x1}
        filter2_2_A(x1,x2) = max{11, x2}
        filter2_3_A(x1,x2,x3) = max{12, x3}
        map_0_A = 1
        map_1_A(x1) = max{6, x1}
        rev1_0_A = 1
        rev1_1_A(x1) = max{26, x1}
        rev2_0_A = 1
        rev2_1_A(x1) = max{6, x1}
        s_0_A = 1
        s_1_A(x1) = max{25, x1 - 13}
        rev1_2_A(x1,x2) = max{x1 + 12, x2 - 38}
        |0|_0_A = 12
        nil_0_A = 24
  
    base order 2:
  
      precedence:
      
        filter2_4# = nil_0 > filter_2# = filter_0 = map_0 = rev1_0 = rev1_2 > app = |0|_0 > filter_1 = filter2_3 = rev1_1 = s_1 > cons_2 = filter2_2 = map_1 = s_0 > true_0 = cons_1 = filter2_1 > cons_0 = filter2_0 = rev2_1 > rev2_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      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(rev1_2) = [1]
      pi(|0|_0) = []
      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#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,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(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: app#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p3: map_2#(f,cons_2(x,xs)) -> app#(f,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(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, r12, r14, r16, r17, r19, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{4, x1 - 3}
        map_1_A(x1) = x1 + 8
        map_2#_A(x1,x2) = max{5, x1 - 2}
        cons_2_A(x1,x2) = max{6, x1 - 51, x2 - 53}
        app_A(x1,x2) = max{14, x1 - 11, x2 - 7}
        cons_0_A = 10
        cons_1_A(x1) = max{1, x1 - 8}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 7}
        filter2_0_A = 8
        filter2_1_A(x1) = max{9, x1 - 8}
        filter2_2_A(x1,x2) = max{10, x1 - 20, x2 - 8}
        filter2_3_A(x1,x2,x3) = max{12, x1 - 31, x2 - 20, x3 - 7}
        rev1_0_A = 1
        rev1_1_A(x1) = max{2, x1 - 100}
        rev2_0_A = 9
        rev2_1_A(x1) = max{10, x1 - 8}
        s_0_A = 10
        s_1_A(x1) = max{14, x1 - 8}
        rev_1_A(x1) = max{13, x1 - 60}
        nil_0_A = 1
        rev1_2_A(x1,x2) = max{5, x1 - 112, x2 - 114}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{12, x1 - 55, x2 - 43}
  
    base order 2:
  
      precedence:
      
        nil_0 = filter_2 > app > filter_1 = filter2_3 = s_1 > app# = map_1 = filter_0 = filter2_2 = s_0 > map_2# = filter2_1 > cons_2 = filter2_0 = rev_1 > cons_1 = rev2_1 = rev1_2 > cons_0 = rev1_1 = rev2_0 = |0|_0 > rev1_0
      
  
    partial status:
  
      pi(app#) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(cons_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(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) = []
      pi(|0|_0) = []
      pi(filter_2) = []

The following rules are removed:

  p3



-- 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, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x2 + 31
        cons_2_A(x1,x2) = max{x1 - 29, x2 + 32}
        app_A(x1,x2) = max{x1 - 11, x2 + 28}
        cons_0_A = 10
        cons_1_A(x1) = max{28, x1 - 18}
        filter_0_A = 22
        filter_1_A(x1) = max{10, x1 - 1}
        filter2_0_A = 8
        filter2_1_A(x1) = max{9, x1 - 1}
        filter2_2_A(x1,x2) = max{10, x1 - 13, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{28, x1 - 25, x2 - 12, x3 - 29}
        map_0_A = 9
        map_1_A(x1) = max{10, x1 - 42}
        rev1_0_A = 9
        rev1_1_A(x1) = max{10, x1 - 73}
        rev2_0_A = 8
        rev2_1_A(x1) = max{9, x1 - 73}
        s_0_A = 12
        s_1_A(x1) = max{13, x1 - 1}
        rev_1_A(x1) = max{10, x1 - 55}
        nil_0_A = 9
        rev1_2_A(x1,x2) = max{7, x1 - 85, x2 - 56}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{27, x2 - 50}
  
    base order 2:
  
      precedence:
      
        s_1 > rev1_2 > app = filter2_0 = filter2_3 = rev2_0 = s_0 = filter_2 > cons_2 = cons_1 = filter_1 = filter2_2 = rev1_1 > map_2# = cons_0 = filter_0 = filter2_1 = map_1 = rev1_0 = nil_0 > map_0 = |0|_0 > rev2_1 = rev_1
      
  
    partial status:
  
      pi(map_2#) = [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) = []
      pi(nil_0) = []
      pi(rev1_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: 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, r2, r3, r5, r6, r7, r9, r12, r14, r16, r17, r18, r19, r20, r21, r22, r23, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        rev2_2#_A(x1,x2) = max{9, x2 + 4}
        cons_2_A(x1,x2) = max{14, x2 + 11}
        rev_1#_A(x1) = max{10, x1 - 6}
        rev2_2_A(x1,x2) = max{3, x2}
        app_A(x1,x2) = max{45, x1 - 15, x2 + 31}
        cons_0_A = 1
        cons_1_A(x1) = x1 + 30
        filter_0_A = 14
        filter_1_A(x1) = max{13, x1 - 49}
        filter2_0_A = 11
        filter2_1_A(x1) = max{12, x1 - 1}
        filter2_2_A(x1,x2) = max{13, x1 - 17, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{14, x1 - 33, x2 - 17, x3 - 1}
        map_0_A = 14
        map_1_A(x1) = max{32, x1 - 16}
        rev1_0_A = 1
        rev1_1_A(x1) = max{0, x1 - 1}
        rev2_0_A = 1
        rev2_1_A(x1) = max{2, x1 - 1}
        s_0_A = 1
        s_1_A(x1) = 0
        rev_1_A(x1) = max{2, x1}
        nil_0_A = 0
        rev1_2_A(x1,x2) = max{10, x2}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{32, x1 - 65, x2 - 15}
  
    base order 2:
  
      precedence:
      
        s_1 = nil_0 > filter_2 > rev2_2# = app > cons_2 = filter_1 = filter2_3 = map_1 = rev2_0 = s_0 = rev_1 > rev_1# = rev2_2 = cons_1 = filter_0 = filter2_2 = map_0 = rev1_2 > cons_0 = filter2_1 = rev1_1 = rev2_1 = |0|_0 > filter2_0 = rev1_0
      
  
    partial status:
  
      pi(rev2_2#) = [2]
      pi(cons_2) = []
      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) = []
      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



-- 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, r9, r12, r14, r16, r17, r19, r22, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        rev2_2#_A(x1,x2) = max{x1 + 8, x2 + 3}
        cons_2_A(x1,x2) = max{19, x1 + 8, x2 + 3}
        app_A(x1,x2) = max{23, x2 - 9}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 14}
        filter_0_A = 1
        filter_1_A(x1) = max{22, x1 - 14}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 14}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 14}
        rev1_0_A = 1
        rev1_1_A(x1) = max{2, x1 - 14}
        rev2_0_A = 1
        rev2_1_A(x1) = max{2, x1 - 14}
        s_0_A = 1
        s_1_A(x1) = max{16, x1 - 10}
        rev_1_A(x1) = max{17, x1 - 22}
        nil_0_A = 15
        rev1_2_A(x1,x2) = max{16, x1 - 22, x2 - 30}
        |0|_0_A = 0
        rev2_2_A(x1,x2) = max{18, x1 - 13, x2 - 21}
        filter_2_A(x1,x2) = max{21, x1 + 14, x2 - 18}
  
    base order 2:
  
      precedence:
      
        app = filter_2 > filter_1 = map_1 = s_1 = rev_1 > cons_2 = filter_0 = map_0 = s_0 > rev2_2# = cons_1 = rev1_2 = rev2_2 > cons_0 = rev1_1 = rev2_1 = nil_0 > filter2_1 = rev1_0 = rev2_0 = |0|_0 > filter2_0
      
  
    partial status:
  
      pi(rev2_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(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) = []
      pi(|0|_0) = []
      pi(rev2_2) = []
      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, r9, r12, r14, r16, r17, r19, r26

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        rev1_2#_A(x1,x2) = max{x1 + 6, x2 + 1}
        cons_2_A(x1,x2) = max{22, x1 + 7, x2 + 6}
        app_A(x1,x2) = max{12, x2 - 14}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 15}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 15}
        filter2_0_A = 2
        filter2_1_A(x1) = max{1, x1 - 15}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 15}
        rev1_0_A = 1
        rev1_1_A(x1) = max{2, x1 - 15}
        rev2_0_A = 1
        rev2_1_A(x1) = max{12, x1 - 15}
        s_0_A = 1
        s_1_A(x1) = max{11, x1 - 15}
        rev_1_A(x1) = max{23, x1}
        nil_0_A = 8
        rev1_2_A(x1,x2) = max{10, x1 - 16, x2 - 23}
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{9, x1 - 14, x2 - 9}
  
    base order 2:
  
      precedence:
      
        rev_1 > cons_2 > rev1_2# = app = nil_0 = filter_2 > cons_1 = filter_1 = filter2_1 = map_1 = rev2_1 = s_1 > cons_0 = filter_0 = filter2_0 = map_0 = rev2_0 = s_0 = rev1_2 > rev1_1 = |0|_0 > rev1_0
      
  
    partial status:
  
      pi(rev1_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(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) = []
      pi(|0|_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.