YES
We show the termination of the TRS R:

  app(app(ack(),|0|()),y) -> app(succ(),y)
  app(app(ack(),app(succ(),x)),y) -> app(app(ack(),x),app(succ(),|0|()))
  app(app(ack(),app(succ(),x)),app(succ(),y)) -> app(app(ack(),x),app(app(ack(),app(succ(),x)),y))
  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(ack_0(),y) -> ack_1(y)
  app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
  ack_2(|0|_0(),y) -> succ_1(y)
  ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
  ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
  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#(ack_1(x1),y) -> ack_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: ack_2#(succ_1(x),y) -> ack_2#(x,succ_1(|0|_0()))
p6: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y))
p7: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)
p8: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p9: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p10: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p11: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p12: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p13: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p8, p9, p10, p11, p12, p13}
  {p5, p6, p7}


-- 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(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r7, r8, r14, r19

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 + 6
        cons_2_A(x1,x2) = max{45, x1 + 23, x2 - 23}
        app#_A(x1,x2) = max{4, x1 - 1}
        map_1_A(x1) = x1 + 8
        filter2_3_A(x1,x2,x3) = max{x1 + 8, x2 + 7, x3 + 8}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 5
        false_0_A = 3
        filter_2#_A(x1,x2) = x1 + 5
        filter_1_A(x1) = max{11, x1 + 7}
        app_A(x1,x2) = max{10, x1 - 12, x2 - 24}
        true_0_A = 1
        ack_0_A = 8
        ack_1_A(x1) = max{9, x1 - 25}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 25}
        filter2_0_A = 13
        filter2_1_A(x1) = max{6, x1 - 24}
        filter2_2_A(x1,x2) = max{7, x1 - 37, x2 - 25}
        ack_2_A(x1,x2) = max{12, x1 - 26, x2 - 13}
        |0|_0_A = 1
        succ_1_A(x1) = max{11, x1 - 27}
        filter_2_A(x1,x2) = max{8, x1 - 47, x2 - 81}
        nil_0_A = 7
  
    base order 2:
  
      precedence:
      
        ack_2 = succ_1 > filter2_4# = filter_2# = app = |0|_0 > cons_2 = map_1 = ack_0 = cons_0 > map_2# = filter2_3 = filter_1 = ack_1 = cons_1 = filter_2 > app# = false_0 = true_0 = filter2_1 = filter2_2 = nil_0 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app#) = []
      pi(map_1) = []
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(ack_0) = []
      pi(ack_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      pi(succ_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: 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(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r8, r9, r11, r13, r14, r19

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 + 17, x2 + 40}
        cons_2_A(x1,x2) = max{15, x1 - 23, x2 + 3}
        app_A(x1,x2) = max{5, x1 - 5, x2 - 16}
        ack_0_A = 3
        ack_1_A(x1) = max{4, x1 - 17}
        cons_0_A = 1
        cons_1_A(x1) = max{4, x1 - 24}
        filter_0_A = 4
        filter_1_A(x1) = max{3, x1 - 17}
        filter2_0_A = 4
        filter2_1_A(x1) = max{2, x1 - 16}
        filter2_2_A(x1,x2) = max{3, x1 - 21, x2 - 16}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 27, x2 - 22, x3 - 17}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 17}
        succ_0_A = 1
        succ_1_A(x1) = max{2, x1 - 35}
        ack_2_A(x1,x2) = max{3, x1 - 23, x2 - 17}
        |0|_0_A = 1
        filter_2_A(x1,x2) = max{4, x1 - 23, x2 - 17}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > map_2# = app > cons_2 = ack_1 = filter_1 = filter2_3 = succ_1 = ack_2 > ack_0 = cons_1 = filter_0 = filter2_2 = map_1 = succ_0 = |0|_0 = nil_0 > cons_0 = filter2_1 = map_0 > filter2_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(ack_0) = []
      pi(ack_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(succ_0) = []
      pi(succ_1) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      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(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r7, r11, r13, r14, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 15
        filter2_3_A(x1,x2,x3) = max{x1 - 11, x2 + 17, x3 - 11}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 16
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 16
        cons_2_A(x1,x2) = max{5, x1 - 15, x2 - 6}
        app_A(x1,x2) = max{10, x2 - 11}
        false_0_A = 3
        filter_1_A(x1) = x1 + 17
        ack_0_A = 1
        ack_1_A(x1) = max{10, x1 - 11}
        cons_0_A = 1
        cons_1_A(x1) = max{4, x1 - 11}
        filter2_0_A = 1
        filter2_1_A(x1) = max{10, x1 - 11}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 11}
        succ_0_A = 1
        succ_1_A(x1) = max{2, x1 - 27}
        ack_2_A(x1,x2) = max{11, x1 - 11, x2 - 13}
        |0|_0_A = 1
        filter_2_A(x1,x2) = max{5, x1 - 3, x2 - 8}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > false_0 > filter2_4# = filter_2# > filter2_3 > cons_2 > app = filter2_0 = nil_0 > app# = true_0 = filter_1 = cons_1 = filter2_1 = map_1 = succ_1 = ack_2 > ack_1 = cons_0 = map_0 = succ_0 = |0|_0 > ack_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = [2]
      pi(true_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(ack_0) = []
      pi(ack_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(succ_0) = []
      pi(succ_1) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> app#(f,x)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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, r13, r14, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{1, x1 - 4}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 22, x2 + 3, x3 - 18}
        filter2_4#_A(x1,x2,x3,x4) = max{2, x2 - 2}
        false_0_A = 3
        filter_2#_A(x1,x2) = max{2, x1 - 2}
        cons_2_A(x1,x2) = max{0, x1 - 26, x2 - 7}
        app_A(x1,x2) = max{25, x1 - 10, x2 + 4}
        true_0_A = 1
        ack_0_A = 8
        ack_1_A(x1) = max{9, x1 - 55}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 1}
        filter_0_A = 4
        filter_1_A(x1) = max{5, x1 - 9}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 1}
        filter2_2_A(x1,x2) = max{3, x1 - 11, x2 - 1}
        map_0_A = 1
        map_1_A(x1) = max{5, x1 - 1}
        succ_0_A = 4
        succ_1_A(x1) = max{5, x1 - 55}
        ack_2_A(x1,x2) = max{26, x1 - 54, x2 - 27}
        |0|_0_A = 4
        filter_2_A(x1,x2) = max{6, x1 - 9, x2 - 2}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = ack_1 = succ_1 = ack_2 > ack_0 = succ_0 = |0|_0 = filter_2 > false_0 = filter_1 > filter2_4# = filter_2# = filter_0 > app# > cons_2 = cons_1 > filter2_3 = true_0 = cons_0 = map_1 = nil_0 > filter2_2 = map_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter2_3) = [2]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(ack_0) = []
      pi(ack_1) = []
      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(map_1) = []
      pi(succ_0) = []
      pi(succ_1) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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#(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(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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, r13, r14, r19

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{33, x3 + 3, x4 + 29}
        false_0_A = 3
        filter_2#_A(x1,x2) = x2 + 28
        cons_2_A(x1,x2) = max{40, x1 + 32, x2 + 30}
        app_A(x1,x2) = max{59, x1 - 5, x2 + 39}
        true_0_A = 1
        ack_0_A = 1
        ack_1_A(x1) = max{4, x1 - 1}
        cons_0_A = 4
        cons_1_A(x1) = x1 + 38
        filter_0_A = 4
        filter_1_A(x1) = max{0, x1 - 36}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 1}
        filter2_2_A(x1,x2) = max{3, x1 - 6, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 12, x2 - 7, x3 - 1}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 1}
        succ_0_A = 1
        succ_1_A(x1) = max{59, x1 - 123}
        ack_2_A(x1,x2) = max{59, x1 - 122, x2 - 61}
        |0|_0_A = 1
        filter_2_A(x1,x2) = max{42, x1 - 42}
        nil_0_A = 32
  
    base order 2:
  
      precedence:
      
        app = ack_1 = cons_0 = ack_2 > ack_0 = succ_1 > cons_2 = filter2_3 = succ_0 = |0|_0 = filter_2 > filter2_4# = filter_2# = cons_1 = filter_1 = filter2_2 = map_1 = nil_0 > false_0 = true_0 = filter_0 = filter2_1 = map_0 > filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app) = [2]
      pi(true_0) = []
      pi(ack_0) = []
      pi(ack_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(succ_0) = []
      pi(succ_1) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      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#(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)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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#(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)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r4, r5, r7, r8, r11, r13, r14, r19

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 + 10, x4 - 2}
        false_0_A = 1
        filter_2#_A(x1,x2) = max{2, x2 - 7}
        cons_2_A(x1,x2) = max{x1 + 18, x2 + 6}
        app_A(x1,x2) = max{21, x1 - 23, x2 + 7}
        ack_0_A = 1
        ack_1_A(x1) = max{8, x1 - 1}
        cons_1_A(x1) = x1 + 42
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 1}
        filter2_0_A = 9
        filter2_1_A(x1) = max{8, x1 - 1}
        filter2_2_A(x1,x2) = max{20, x1 - 25, x2 - 2}
        map_0_A = 7
        map_1_A(x1) = max{8, x1 - 1}
        succ_0_A = 1
        succ_1_A(x1) = max{8, x1 - 26}
        ack_2_A(x1,x2) = max{9, x1 - 25, x2 - 10}
        |0|_0_A = 1
        filter_2_A(x1,x2) = max{9, x1 - 25, x2 - 16}
        nil_0_A = 1
  
    base order 2:
  
      precedence:
      
        ack_0 = filter_0 > filter_2 = nil_0 > app > filter_2# = ack_1 = filter_1 = filter2_2 = succ_1 = ack_2 > filter2_4# = false_0 = cons_2 = filter2_1 = map_1 = succ_0 = |0|_0 > cons_1 = filter2_0 = map_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(ack_0) = []
      pi(ack_1) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(succ_0) = []
      pi(succ_1) = []
      pi(ack_2) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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: ack_2#(succ_1(x),y) -> ack_2#(x,succ_1(|0|_0()))
p2: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)
p3: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y))

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack_2#_A(x1,x2) = x1 + 12
        succ_1_A(x1) = x1 + 14
        |0|_0_A = 11
        ack_2_A(x1,x2) = max{13, x1 - 40, x2 - 26}
        app_A(x1,x2) = max{12, x1 - 3, x2 - 8}
        ack_0_A = 1
        ack_1_A(x1) = max{2, x1 - 41}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 9}
        cons_2_A(x1,x2) = max{3, x1 - 13, x2 - 15}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 9}
        filter2_0_A = 1
        filter2_1_A(x1) = max{0, x1 - 8}
        filter2_2_A(x1,x2) = max{0, x1 - 12, x2 - 9}
        filter2_3_A(x1,x2,x3) = max{0, x1 - 27, x2 - 13}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 13}
        filter_2_A(x1,x2) = max{9, x1 - 30, x2 - 21}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        ack_2 > ack_2# > succ_1 = |0|_0 = filter_2 = nil_0 > app > ack_1 = filter_1 = filter2_1 = filter2_2 > ack_0 = cons_2 = filter_0 = filter2_0 = filter2_3 = map_1 > cons_1 = map_0 > cons_0
      
  
    partial status:
  
      pi(ack_2#) = [1]
      pi(succ_1) = [1]
      pi(|0|_0) = []
      pi(ack_2) = []
      pi(app) = []
      pi(ack_0) = []
      pi(ack_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:

  p1



-- SCC decomposition.

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

p1: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)
p2: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y))

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)
p2: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y))

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack_2#_A(x1,x2) = max{1, x1 - 25}
        succ_1_A(x1) = max{70, x1 + 46}
        ack_2_A(x1,x2) = max{115, x1 - 24, x2 + 47}
        app_A(x1,x2) = max{69, x1 - 5, x2 - 1}
        ack_0_A = 1
        ack_1_A(x1) = max{2, x1 - 24}
        cons_0_A = 3
        cons_1_A(x1) = max{4, x1 - 2}
        cons_2_A(x1,x2) = max{36, x1 - 8, x2 - 23}
        filter_0_A = 4
        filter_1_A(x1) = max{3, x1 - 30}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 2}
        filter2_2_A(x1,x2) = max{3, x1 - 8, x2 - 2}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 14, x2 - 8, x3 - 27}
        map_0_A = 4
        map_1_A(x1) = max{6, x1 - 2}
        |0|_0_A = 23
        filter_2_A(x1,x2) = max{34, x1 - 36, x2 - 25}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        ack_2# = ack_2 > succ_1 > app = |0|_0 > ack_1 = cons_2 = filter2_2 = map_1 = nil_0 > ack_0 = cons_1 = filter2_1 = filter2_3 = map_0 = filter_2 > cons_0 = filter_1 = filter2_0 > filter_0
      
  
    partial status:
  
      pi(ack_2#) = []
      pi(succ_1) = [1]
      pi(ack_2) = []
      pi(app) = []
      pi(ack_0) = []
      pi(ack_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(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: 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: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y)

and R consists of:

r1: app(ack_0(),y) -> ack_1(y)
r2: app(ack_1(x1),y) -> ack_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(succ_0(),y) -> succ_1(y)
r14: ack_2(|0|_0(),y) -> succ_1(y)
r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0()))
r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y))
r17: map_2(f,nil_0()) -> nil_0()
r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r19: filter_2(f,nil_0()) -> nil_0()
r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r7, r8, r9, r11, r13, r19

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        ack_2#_A(x1,x2) = max{1, x2 - 1}
        succ_1_A(x1) = x1 + 23
        app_A(x1,x2) = max{x1, x2 + 23}
        ack_0_A = 1
        ack_1_A(x1) = x1
        cons_0_A = 0
        cons_1_A(x1) = max{0, x1 - 25}
        filter_0_A = 0
        filter_1_A(x1) = max{23, x1 - 24}
        filter2_0_A = 0
        filter2_1_A(x1) = max{0, x1 - 1}
        filter2_2_A(x1,x2) = max{0, x1 - 2, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{0, x2 - 12, x3 - 38}
        map_0_A = 17
        map_1_A(x1) = max{16, x1 - 24}
        succ_0_A = 1
        filter_2_A(x1,x2) = max{2, x1 - 25, x2 - 14}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = cons_1 = map_1 > cons_0 = filter2_3 = map_0 = filter_2 = nil_0 > ack_2# = succ_1 = ack_1 = filter_1 = filter2_2 > ack_0 = filter_0 = filter2_1 = succ_0 > filter2_0
      
  
    partial status:
  
      pi(ack_2#) = []
      pi(succ_1) = [1]
      pi(app) = [2]
      pi(ack_0) = []
      pi(ack_1) = [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(succ_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.