YES
We show the termination of the TRS R:

  app(not(),app(not(),x)) -> x
  app(not(),app(app(or(),x),y)) -> app(app(and(),app(not(),x)),app(not(),y))
  app(not(),app(app(and(),x),y)) -> app(app(or(),app(not(),x)),app(not(),y))
  app(app(and(),x),app(app(or(),y),z)) -> app(app(or(),app(app(and(),x),y)),app(app(and(),x),z))
  app(app(and(),app(app(or(),y),z)),x) -> app(app(or(),app(app(and(),x),y)),app(app(and(),x),z))
  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(and_0(),y) -> and_1(y)
  app(and_1(x1),y) -> and_2(x1,y)
  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(not_0(),y) -> not_1(y)
  app(or_0(),y) -> or_1(y)
  app(or_1(x1),y) -> or_2(x1,y)
  not_1(not_1(x)) -> x
  not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
  not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
  and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
  and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
  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#(and_1(x1),y) -> and_2#(x1,y)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: app#(map_1(x1),y) -> map_2#(x1,y)
p5: app#(not_0(),y) -> not_1#(y)
p6: not_1#(or_2(x,y)) -> and_2#(not_1(x),not_1(y))
p7: not_1#(or_2(x,y)) -> not_1#(x)
p8: not_1#(or_2(x,y)) -> not_1#(y)
p9: not_1#(and_2(x,y)) -> not_1#(x)
p10: not_1#(and_2(x,y)) -> not_1#(y)
p11: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p12: and_2#(x,or_2(y,z)) -> and_2#(x,z)
p13: and_2#(or_2(y,z),x) -> and_2#(x,y)
p14: and_2#(or_2(y,z),x) -> and_2#(x,z)
p15: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p16: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p17: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p18: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p19: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p20: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p15, p16, p17, p18, p19, p20}
  {p7, p8, p9, p10}
  {p11, p12, p13, p14}


-- 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)
p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p7: app#(filter_1(x1),y) -> filter_2#(x1,y)
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(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r14, r15, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 23
        cons_2_A(x1,x2) = max{21, x2 - 19}
        app#_A(x1,x2) = x1 + 7
        map_1_A(x1) = x1 + 24
        filter2_3_A(x1,x2,x3) = max{x1 + 40, x2 + 11, x3 + 10}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 11
        false_0_A = 33
        filter_2#_A(x1,x2) = x1 + 11
        filter_1_A(x1) = x1 + 12
        app_A(x1,x2) = max{x1 - 17, x2 + 19}
        true_0_A = 20
        and_0_A = 15
        and_1_A(x1) = max{16, x1 - 1}
        cons_0_A = 16
        cons_1_A(x1) = max{0, x1 - 1}
        filter_0_A = 1
        filter2_1_A(x1) = max{20, x1 - 1}
        filter2_2_A(x1,x2) = max{x1 - 19, x2 + 18}
        or_0_A = 14
        or_1_A(x1) = max{15, x1 - 1}
        or_2_A(x1,x2) = max{16, x1 - 38, x2 - 40}
        filter_2_A(x1,x2) = max{13, x1 - 52, x2 - 42}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        and_1 = filter_2 = nil_0 > app > filter_0 > cons_2 = map_1 = filter2_3 = false_0 = filter_1 = and_0 = or_2 > app# = true_0 = cons_1 = filter2_1 = filter2_2 = or_1 > map_2# = filter2_4# = filter_2# = cons_0 = or_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)
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(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: 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, p8}


-- 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(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r16, r17, r18, r19, r20, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 10, x2 + 11}
        cons_2_A(x1,x2) = max{x1 + 8, x2 + 10}
        not_1_A(x1) = max{20, x1 + 15}
        or_2_A(x1,x2) = max{6, x1 - 22, x2 - 23}
        and_2_A(x1,x2) = max{21, x1 - 22, x2 - 23}
        filter_2_A(x1,x2) = max{3, x1 - 14, x2 - 12}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        not_1 > map_2# = and_2 > or_2 > cons_2 > filter_2 > nil_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(not_1) = [1]
      pi(or_2) = []
      pi(and_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter_1(x1),y) -> filter_2#(x1,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r11, r14, r15, r23

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 - 3}
        filter2_3_A(x1,x2,x3) = max{13, x1 + 10, x2 + 10, x3 + 10}
        filter2_4#_A(x1,x2,x3,x4) = max{6, x2 - 2}
        true_0_A = 1
        filter_2#_A(x1,x2) = max{6, x1 - 2}
        cons_2_A(x1,x2) = max{4, x1 - 26, x2 - 26}
        app_A(x1,x2) = max{12, x1 - 14, x2 + 3}
        false_0_A = 5
        filter_1_A(x1) = max{10, x1 + 2}
        and_0_A = 13
        and_1_A(x1) = max{11, x1 - 1}
        cons_0_A = 11
        cons_1_A(x1) = max{12, x1 - 1}
        filter_0_A = 9
        filter2_0_A = 1
        filter2_1_A(x1) = max{4, x1 - 1}
        filter2_2_A(x1,x2) = max{5, x1 - 16, x2 - 1}
        map_0_A = 1
        map_1_A(x1) = max{4, x1 - 16}
        or_0_A = 1
        or_1_A(x1) = max{4, x1 - 1}
        or_2_A(x1,x2) = max{5, x1 - 48, x2 - 18}
        filter_2_A(x1,x2) = max{11, x1 - 60, x2 - 16}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app = nil_0 > filter2_3 = cons_2 = false_0 = filter_1 = cons_0 = map_1 > app# = and_1 = cons_1 = filter_0 = filter2_0 = filter2_2 = map_0 > filter2_4# = filter_2# = and_0 = filter2_1 = or_2 > true_0 = or_1 > or_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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)
p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter_1(x1),y) -> filter_2#(x1,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- 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)) -> app#(f,x)
p3: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r7, r8, r9, r11, r14, r15, r23

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) = x2 + 5
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 5
        cons_2_A(x1,x2) = max{3, x1 - 33, x2 - 33}
        app#_A(x1,x2) = x1 + 5
        filter_1_A(x1) = x1 + 5
        app_A(x1,x2) = max{10, x1 - 6, x2 - 6}
        false_0_A = 2
        and_0_A = 4
        and_1_A(x1) = max{5, x1 - 7}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 7}
        filter2_0_A = 4
        filter2_1_A(x1) = max{5, x1 - 7}
        filter2_2_A(x1,x2) = max{4, x1 - 14, x2 - 7}
        filter2_3_A(x1,x2,x3) = max{5, x1 - 21, x2 - 14, x3 - 54}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 33}
        or_0_A = 4
        or_1_A(x1) = max{5, x1 - 9}
        or_2_A(x1,x2) = max{0, x1 - 15}
        filter_2_A(x1,x2) = max{9, x1 - 32, x2 - 20}
        nil_0_A = 5
  
    base order 2:
  
      precedence:
      
        false_0 > or_0 = filter_2 > app > filter_1 = or_1 = nil_0 > filter2_4# = filter_2# = map_1 = or_2 > app# = map_0 > cons_2 = and_1 = filter2_3 > true_0 = and_0 = cons_1 = filter2_2 > cons_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filter_1) = []
      pi(app) = []
      pi(false_0) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p2: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4}


-- 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(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r23

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{10, x3 + 3, x4 + 8}
        true_0_A = 1
        filter_2#_A(x1,x2) = x2 + 7
        cons_2_A(x1,x2) = max{14, x1 + 4, x2 + 9}
        app_A(x1,x2) = max{22, x1 - 5, x2 + 11}
        false_0_A = 12
        and_0_A = 3
        and_1_A(x1) = max{4, x1 - 43}
        cons_0_A = 3
        cons_1_A(x1) = x1 + 10
        filter_0_A = 4
        filter_1_A(x1) = max{3, x1 - 34}
        filter2_0_A = 6
        filter2_1_A(x1) = max{4, x1 - 1}
        filter2_2_A(x1,x2) = max{1, x1 - 7, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 13, x2 - 7, x3 - 6}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 5}
        or_0_A = 3
        or_1_A(x1) = max{4, x1 - 1}
        or_2_A(x1,x2) = max{12, x1 - 16, x2 - 51}
        filter_2_A(x1,x2) = max{13, x1 - 40, x2 - 15}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app = map_0 > cons_0 = filter_1 = filter2_3 = map_1 > cons_2 = and_1 = filter_0 = filter2_2 > filter2_4# = filter_2# = and_0 = cons_1 = filter2_1 = or_2 > true_0 = false_0 = filter2_0 = or_1 > or_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(false_0) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: 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: 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)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r14, r15, r23

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{10, x3 + 8, x4 + 8}
        true_0_A = 6
        filter_2#_A(x1,x2) = max{7, x2 + 2}
        cons_2_A(x1,x2) = max{65, x1 + 9, x2 + 9}
        app_A(x1,x2) = max{68, x1 + 15, x2 + 15}
        and_0_A = 1
        and_1_A(x1) = max{17, x1 - 16}
        cons_0_A = 1
        cons_1_A(x1) = max{66, x1 + 1}
        filter_0_A = 2
        filter_1_A(x1) = max{16, x1 - 42}
        filter2_0_A = 1
        filter2_1_A(x1) = max{17, x1 - 16}
        filter2_2_A(x1,x2) = max{33, x1 - 17, x2 - 16}
        filter2_3_A(x1,x2,x3) = max{49, x1 - 21, x2 - 17, x3 - 16}
        or_0_A = 1
        or_1_A(x1) = max{17, x1 - 16}
        or_2_A(x1,x2) = max{33, x1 - 37, x2 - 37}
        filter_2_A(x1,x2) = max{64, x1 - 28, x2 - 66}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app > filter_1 = filter2_3 = or_1 = or_2 > cons_2 = and_1 = filter_0 = filter2_2 = or_0 > filter2_4# = and_0 = cons_1 = filter2_1 > filter_2# = cons_0 = filter2_0 > true_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app) = [1, 2]
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: 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: not_1#(and_2(x,y)) -> not_1#(y)
p2: not_1#(and_2(x,y)) -> not_1#(x)
p3: not_1#(or_2(x,y)) -> not_1#(y)
p4: not_1#(or_2(x,y)) -> not_1#(x)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r14, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_1#_A(x1) = max{28, x1 + 14}
        and_2_A(x1,x2) = max{35, x1 + 15, x2 + 31}
        or_2_A(x1,x2) = max{x1 + 33, x2 + 15}
        app_A(x1,x2) = max{34, x1 - 7, x2 - 4}
        and_0_A = 1
        and_1_A(x1) = max{2, x1 - 5}
        cons_0_A = 6
        cons_1_A(x1) = max{1, x1 - 5}
        cons_2_A(x1,x2) = max{2, x1 - 41, x2 - 6}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 39}
        filter2_0_A = 4
        filter2_1_A(x1) = max{5, x1 - 5}
        filter2_2_A(x1,x2) = max{6, x1 - 14, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{1, x1 - 22, x2 - 13, x3 - 5}
        map_0_A = 5
        map_1_A(x1) = max{6, x1 - 41}
        or_0_A = 5
        or_1_A(x1) = max{6, x1 - 5}
        filter_2_A(x1,x2) = max{3, x1 - 38, x2 - 6}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        or_2 > and_2 = filter_2 > app = and_0 > not_1# = and_1 = filter_1 = filter2_3 = or_1 > cons_2 = filter_0 = filter2_2 = map_1 = or_0 = nil_0 > cons_1 = filter2_1 = map_0 > cons_0 = filter2_0
      
  
    partial status:
  
      pi(not_1#) = [1]
      pi(and_2) = [1, 2]
      pi(or_2) = []
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(or_0) = []
      pi(or_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: not_1#(and_2(x,y)) -> not_1#(y)
p2: not_1#(and_2(x,y)) -> not_1#(x)
p3: not_1#(or_2(x,y)) -> not_1#(y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: not_1#(and_2(x,y)) -> not_1#(y)
p2: not_1#(or_2(x,y)) -> not_1#(y)
p3: not_1#(and_2(x,y)) -> not_1#(x)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_1#_A(x1) = x1 + 33
        and_2_A(x1,x2) = max{22, x1 + 8, x2 + 10}
        or_2_A(x1,x2) = max{16, x1 - 23, x2 + 5}
        app_A(x1,x2) = max{17, x1 - 18, x2 + 9}
        and_0_A = 15
        and_1_A(x1) = max{16, x1 - 1}
        cons_0_A = 16
        cons_1_A(x1) = max{17, x1 - 30}
        cons_2_A(x1,x2) = max{11, x1 - 49, x2 - 95}
        filter_0_A = 15
        filter_1_A(x1) = max{16, x1 - 77}
        filter2_0_A = 16
        filter2_1_A(x1) = max{17, x1 - 1}
        filter2_2_A(x1,x2) = max{17, x1 - 19, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{11, x1 - 38, x2 - 20, x3 - 1}
        map_0_A = 17
        map_1_A(x1) = max{10, x1 - 17}
        or_0_A = 17
        or_1_A(x1) = max{15, x1 - 1}
        filter_2_A(x1,x2) = max{48, x1 - 76, x2 - 52}
        nil_0_A = 13
  
    base order 2:
  
      precedence:
      
        not_1# > and_2 = or_2 = app = filter_0 = map_0 = or_0 > and_1 = map_1 = or_1 > and_0 = cons_1 = cons_2 = filter2_3 = filter_2 > cons_0 = filter_1 = filter2_2 = nil_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(not_1#) = []
      pi(and_2) = [1, 2]
      pi(or_2) = []
      pi(app) = [2]
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(or_0) = []
      pi(or_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: not_1#(and_2(x,y)) -> not_1#(y)
p2: not_1#(or_2(x,y)) -> not_1#(y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: 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: not_1#(and_2(x,y)) -> not_1#(y)
p2: not_1#(or_2(x,y)) -> not_1#(y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r14

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_1#_A(x1) = max{14, x1}
        and_2_A(x1,x2) = max{x1 + 17, x2 + 27}
        or_2_A(x1,x2) = max{x1 + 14, x2 + 11}
        app_A(x1,x2) = max{7, x1 - 8, x2 - 29}
        and_0_A = 6
        and_1_A(x1) = max{7, x1 - 29}
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 29}
        cons_2_A(x1,x2) = max{6, x1 - 87, x2 - 53}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 57}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 30}
        filter2_2_A(x1,x2) = max{3, x1 - 39, x2 - 30}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 48, x2 - 39, x3 - 30}
        map_0_A = 5
        map_1_A(x1) = max{6, x1 - 96}
        or_0_A = 5
        or_1_A(x1) = max{6, x1 - 30}
  
    base order 2:
  
      precedence:
      
        and_2 = or_2 > not_1# > app = and_0 = cons_0 = filter_0 > and_1 = filter_1 = map_1 = or_1 > cons_1 = cons_2 = filter2_2 = filter2_3 = map_0 = or_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(not_1#) = [1]
      pi(and_2) = [1, 2]
      pi(or_2) = []
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(or_0) = []
      pi(or_1) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: not_1#(and_2(x,y)) -> not_1#(y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: not_1#(and_2(x,y)) -> not_1#(y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r11, r14, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        not_1#_A(x1) = x1 + 17
        and_2_A(x1,x2) = max{27, x1 - 15, x2 + 22}
        app_A(x1,x2) = max{29, x2 + 3}
        and_0_A = 1
        and_1_A(x1) = max{28, x1 - 71}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 71}
        filter_0_A = 1
        filter_1_A(x1) = max{4, x1 - 71}
        filter2_0_A = 1
        filter2_1_A(x1) = max{29, x1 - 71}
        map_0_A = 1
        map_1_A(x1) = max{4, x1 - 71}
        or_0_A = 1
        or_1_A(x1) = max{4, x1 - 71}
        filter_2_A(x1,x2) = max{x1 - 41, x2 + 23}
        nil_0_A = 8
  
    base order 2:
  
      precedence:
      
        and_2 > not_1# = app = filter_2 > and_1 = cons_1 = filter_1 = filter2_1 = map_1 = or_1 = nil_0 > and_0 = cons_0 = filter_0 = filter2_0 = map_0 = or_0
      
  
    partial status:
  
      pi(not_1#) = []
      pi(and_2) = []
      pi(app) = [2]
      pi(and_0) = []
      pi(and_1) = []
      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(or_0) = []
      pi(or_1) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p2: and_2#(or_2(y,z),x) -> and_2#(x,z)
p3: and_2#(or_2(y,z),x) -> and_2#(x,y)
p4: and_2#(x,or_2(y,z)) -> and_2#(x,z)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r14, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        and_2#_A(x1,x2) = max{x1 + 2, x2 + 3}
        or_2_A(x1,x2) = max{15, x1 + 9, x2 + 3}
        app_A(x1,x2) = max{25, x1 - 7, x2 - 7}
        and_0_A = 5
        and_1_A(x1) = max{6, x1 - 8}
        cons_0_A = 1
        cons_1_A(x1) = max{0, x1 - 7}
        cons_2_A(x1,x2) = max{4, x2 - 14}
        filter_0_A = 5
        filter_1_A(x1) = max{6, x1 - 7}
        filter2_0_A = 3
        filter2_1_A(x1) = max{4, x1 - 8}
        filter2_2_A(x1,x2) = max{5, x1 - 16, x2 - 8}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 24, x2 - 16, x3 - 8}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 9}
        or_0_A = 5
        or_1_A(x1) = max{6, x1 - 8}
        filter_2_A(x1,x2) = max{16, x1 - 14, x2 - 7}
        nil_0_A = 6
  
    base order 2:
  
      precedence:
      
        and_2# > or_2 > app > and_1 = filter2_3 = or_1 = filter_2 > and_0 = filter_1 = filter2_2 = or_0 = nil_0 > filter_0 = filter2_1 > cons_2 = filter2_0 = map_1 > cons_1 = map_0 > cons_0
      
  
    partial status:
  
      pi(and_2#) = [2]
      pi(or_2) = [2]
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(or_0) = []
      pi(or_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p2: and_2#(or_2(y,z),x) -> and_2#(x,z)
p3: and_2#(or_2(y,z),x) -> and_2#(x,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p2: and_2#(or_2(y,z),x) -> and_2#(x,y)
p3: and_2#(or_2(y,z),x) -> and_2#(x,z)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r7, r8, r9, r11, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        and_2#_A(x1,x2) = max{x1 + 5, x2 + 6}
        or_2_A(x1,x2) = max{10, x1 + 2, x2 + 7}
        app_A(x1,x2) = max{12, x1 - 13, x2 - 5}
        and_0_A = 9
        and_1_A(x1) = max{10, x1 - 5}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 5}
        cons_2_A(x1,x2) = max{3, x1 - 20, x2 - 6}
        filter_0_A = 12
        filter_1_A(x1) = max{1, x1 - 46}
        filter2_0_A = 11
        filter2_1_A(x1) = max{12, x1 - 5}
        filter2_2_A(x1,x2) = max{12, x1 - 19, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{11, x1 - 33, x2 - 19, x3 - 6}
        map_0_A = 1
        map_1_A(x1) = max{4, x1 - 6}
        filter_2_A(x1,x2) = max{2, x1 - 60, x2 - 30}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        and_2# > or_2 = cons_0 = filter_0 = filter2_0 > app > and_1 > and_0 = cons_2 = filter2_3 = map_1 = filter_2 > cons_1 = filter_1 = filter2_2 = map_0 = nil_0 > filter2_1
      
  
    partial status:
  
      pi(and_2#) = [1, 2]
      pi(or_2) = [2]
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p2: and_2#(or_2(y,z),x) -> and_2#(x,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: 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: and_2#(x,or_2(y,z)) -> and_2#(x,y)
p2: and_2#(or_2(y,z),x) -> and_2#(x,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r16, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        and_2#_A(x1,x2) = max{x1 + 4, x2 + 5}
        or_2_A(x1,x2) = max{32, x1 + 6, x2 + 21}
        not_1_A(x1) = max{12, x1 + 7}
        filter_2_A(x1,x2) = max{47, x1 - 33, x2 - 82}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        and_2# > or_2 > not_1 = filter_2 > nil_0
      
  
    partial status:
  
      pi(and_2#) = [2]
      pi(or_2) = [1, 2]
      pi(not_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: and_2#(or_2(y,z),x) -> and_2#(x,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: and_2#(or_2(y,z),x) -> and_2#(x,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(filter_0(),y) -> filter_1(y)
r6: app(filter_1(x1),y) -> filter_2(x1,y)
r7: app(filter2_0(),y) -> filter2_1(y)
r8: app(filter2_1(x1),y) -> filter2_2(x1,y)
r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: app(map_0(),y) -> map_1(y)
r12: app(map_1(x1),y) -> map_2(x1,y)
r13: app(not_0(),y) -> not_1(y)
r14: app(or_0(),y) -> or_1(y)
r15: app(or_1(x1),y) -> or_2(x1,y)
r16: not_1(not_1(x)) -> x
r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y))
r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y))
r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z))
r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z))
r21: map_2(f,nil_0()) -> nil_0()
r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r23: filter_2(f,nil_0()) -> nil_0()
r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r21, r22, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        and_2#_A(x1,x2) = max{x1 + 22, x2 + 23}
        or_2_A(x1,x2) = max{8, x1 + 2, x2 - 12}
        map_2_A(x1,x2) = max{8, x1 - 2, x2 - 8}
        nil_0_A = 7
        cons_2_A(x1,x2) = max{3, x2 - 9}
        app_A(x1,x2) = max{6, x1}
        filter_2_A(x1,x2) = max{6, x1 + 1, x2 + 2}
  
    base order 2:
  
      precedence:
      
        map_2 = cons_2 > and_2# > or_2 = filter_2 > nil_0 = app
      
  
    partial status:
  
      pi(and_2#) = [1]
      pi(or_2) = [1]
      pi(map_2) = []
      pi(nil_0) = []
      pi(cons_2) = []
      pi(app) = [1]
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.