YES
We show the termination of the TRS R:

  app(app(le(),|0|()),y) -> true()
  app(app(le(),app(s(),x)),|0|()) -> false()
  app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
  app(pred(),app(s(),x)) -> x
  app(app(minus(),x),|0|()) -> x
  app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
  app(app(gcd(),|0|()),y) -> y
  app(app(gcd(),app(s(),x)),|0|()) -> app(s(),x)
  app(app(gcd(),app(s(),x)),app(s(),y)) -> app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
  app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
  app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
  app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
  app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filter2_0(),y) -> filter2_1(y)
  app(filter2_1(x1),y) -> filter2_2(x1,y)
  app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
  app(gcd_0(),y) -> gcd_1(y)
  app(gcd_1(x1),y) -> gcd_2(x1,y)
  app(if_gcd_0(),y) -> if_gcd_1(y)
  app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
  app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
  app(le_0(),y) -> le_1(y)
  app(le_1(x1),y) -> le_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(minus_0(),y) -> minus_1(y)
  app(minus_1(x1),y) -> minus_2(x1,y)
  app(pred_0(),y) -> pred_1(y)
  app(s_0(),y) -> s_1(y)
  le_2(|0|_0(),y) -> true_0()
  le_2(s_1(x),|0|_0()) -> false_0()
  le_2(s_1(x),s_1(y)) -> le_2(x,y)
  pred_1(s_1(x)) -> x
  minus_2(x,|0|_0()) -> x
  minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
  gcd_2(|0|_0(),y) -> y
  gcd_2(s_1(x),|0|_0()) -> s_1(x)
  gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
  if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
  if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
  filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
  filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)


-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: app#(gcd_1(x1),y) -> gcd_2#(x1,y)
p4: app#(if_gcd_2(x1,x2),y) -> if_gcd_3#(x1,x2,y)
p5: app#(le_1(x1),y) -> le_2#(x1,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: app#(minus_1(x1),y) -> minus_2#(x1,y)
p8: app#(pred_0(),y) -> pred_1#(y)
p9: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)
p10: minus_2#(x,s_1(y)) -> pred_1#(minus_2(x,y))
p11: minus_2#(x,s_1(y)) -> minus_2#(x,y)
p12: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p13: gcd_2#(s_1(x),s_1(y)) -> le_2#(y,x)
p14: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))
p15: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> minus_2#(x,y)
p16: if_gcd_3#(false_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(y,x),s_1(x))
p17: if_gcd_3#(false_0(),s_1(x),s_1(y)) -> minus_2#(y,x)
p18: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p19: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p20: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p21: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p22: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p23: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p6, p18, p19, p20, p21, p22, p23}
  {p12, p14, p16}
  {p9}
  {p11}


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p7: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p8: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p9: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r7, r9, r11, r12, r14, r16, r18, r21, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x2 + 10
        filter_1_A(x1) = max{3, x1 - 44}
        filter_2#_A(x1,x2) = max{2, x2 + 1}
        cons_2_A(x1,x2) = max{x1 + 10, x2 + 7}
        map_1_A(x1) = max{19, x1 - 44}
        map_2#_A(x1,x2) = max{9, x2 + 4}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 44, x2 - 44, x3 - 44}
        filter2_4#_A(x1,x2,x3,x4) = max{9, x4 + 6}
        false_0_A = 1
        app_A(x1,x2) = max{19, x1 - 20, x2 - 3}
        true_0_A = 3
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 4}
        filter_0_A = 1
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 3}
        filter2_2_A(x1,x2) = max{3, x1 - 23, x2 - 3}
        gcd_0_A = 18
        gcd_1_A(x1) = max{19, x1 - 3}
        if_gcd_0_A = 1
        if_gcd_1_A(x1) = max{17, x1 - 4}
        if_gcd_2_A(x1,x2) = max{18, x1 - 25, x2 - 4}
        le_0_A = 18
        le_1_A(x1) = max{19, x1 - 4}
        map_0_A = 18
        minus_0_A = 19
        minus_1_A(x1) = max{17, x1 - 4}
        s_0_A = 18
        s_1_A(x1) = max{19, x1 - 67}
        le_2_A(x1,x2) = max{17, x1 - 122, x2 - 20}
        |0|_0_A = 4
        filter_2_A(x1,x2) = max{4, x1 - 84, x2 - 76}
        nil_0_A = 3
  
    base order 2:
  
      precedence:
      
        app > map_1 > cons_2 = filter2_3 = filter_0 = map_0 = minus_1 = le_2 = |0|_0 = filter_2 > cons_1 = filter2_2 = le_1 = minus_0 = s_1 = nil_0 > false_0 = true_0 = cons_0 = filter2_1 = gcd_1 = if_gcd_2 = le_0 = s_0 > filter_2# = map_2# = filter2_0 = gcd_0 = if_gcd_1 > app# = filter_1 = if_gcd_0 > filter2_4#
      
  
    partial status:
  
      pi(app#) = [2]
      pi(filter_1) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(map_1) = []
      pi(map_2#) = [2]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      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: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
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)
p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p7: app#(map_1(x1),y) -> map_2#(x1,y)
p8: map_2#(f,cons_2(x,xs)) -> app#(f,x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r5, r7, r9, r11, r12, r14, r18, r21, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 38
        filter_1_A(x1) = x1 + 40
        filter_2#_A(x1,x2) = x1 + 39
        cons_2_A(x1,x2) = max{9, x1 - 43, x2 - 43}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 39
        app_A(x1,x2) = max{25, x1 - 41, x2}
        true_0_A = 6
        filter2_3_A(x1,x2,x3) = max{x1 - 27, x2 + 40, x3 - 27}
        false_0_A = 1
        map_1_A(x1) = x1 + 40
        map_2#_A(x1,x2) = x1 + 39
        cons_0_A = 7
        cons_1_A(x1) = max{8, x1 - 1}
        filter2_0_A = 24
        filter2_1_A(x1) = max{25, x1}
        filter2_2_A(x1,x2) = max{x1 + 24, x2 + 82}
        gcd_0_A = 1
        gcd_1_A(x1) = max{2, x1 - 1}
        if_gcd_0_A = 1
        if_gcd_1_A(x1) = max{2, x1 - 1}
        if_gcd_2_A(x1,x2) = max{3, x1 - 43, x2 - 1}
        le_0_A = 23
        le_1_A(x1) = max{24, x1 - 1}
        minus_0_A = 22
        minus_1_A(x1) = max{23, x1 - 1}
        s_0_A = 4
        s_1_A(x1) = max{5, x1 - 46}
        le_2_A(x1,x2) = max{25, x1 - 43, x2 - 8}
        |0|_0_A = 7
        filter_2_A(x1,x2) = max{24, x1 - 68, x2 - 21}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app > filter_1 = filter2_3 = map_1 > cons_2 = cons_1 = filter2_2 = le_2 = |0|_0 = nil_0 > true_0 = cons_0 = filter2_1 = le_1 > filter2_0 = le_0 > minus_1 > map_2# = gcd_1 = minus_0 = s_1 > app# = false_0 = gcd_0 = if_gcd_2 = s_0 > filter_2# = filter2_4# = if_gcd_1 > if_gcd_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(filter2_3) = []
      pi(false_0) = []
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
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: app#(map_1(x1),y) -> map_2#(x1,y)
p7: map_2#(f,cons_2(x,xs)) -> app#(f,x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: app#(map_1(x1),y) -> map_2#(x1,y)
p5: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p7: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r9, r11, r12, r14, r18, r21, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x1 + 4
        cons_2_A(x1,x2) = max{0, x1 - 31, x2 - 31}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 4
        app_A(x1,x2) = max{20, x1 - 9, x2 - 9}
        false_0_A = 3
        app#_A(x1,x2) = max{1, x1}
        map_1_A(x1) = x1 + 3
        map_2#_A(x1,x2) = max{3, x1 + 2}
        filter2_3_A(x1,x2,x3) = max{8, x1 + 3, x2 + 5, x3 + 3}
        true_0_A = 1
        cons_0_A = 7
        cons_1_A(x1) = max{8, x1 - 9}
        filter_0_A = 7
        filter_1_A(x1) = max{8, x1 - 10}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 10}
        filter2_2_A(x1,x2) = max{3, x1 - 19, x2 - 9}
        gcd_0_A = 8
        gcd_1_A(x1) = max{1, x1 - 10}
        if_gcd_0_A = 3
        if_gcd_1_A(x1) = max{4, x1 - 10}
        if_gcd_2_A(x1,x2) = max{5, x1 - 20, x2 - 10}
        le_0_A = 2
        le_1_A(x1) = max{3, x1 - 10}
        minus_0_A = 7
        minus_1_A(x1) = max{8, x1 - 10}
        s_0_A = 6
        s_1_A(x1) = max{7, x1 - 28}
        le_2_A(x1,x2) = max{4, x1 - 39, x2 - 49}
        |0|_0_A = 4
        filter_2_A(x1,x2) = max{12, x1 - 31, x2 - 20}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > app = filter_0 > filter2_3 = filter_1 > map_1 = nil_0 > filter2_1 = filter2_2 = gcd_0 = if_gcd_2 = minus_1 = s_1 = le_2 > filter2_0 = gcd_1 = if_gcd_1 = le_1 = minus_0 = s_0 = |0|_0 > false_0 = if_gcd_0 = le_0 > filter_2# = filter2_4# > app# > cons_2 = map_2# > true_0 = cons_1 > cons_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(app#) = [1]
      pi(map_1) = []
      pi(map_2#) = []
      pi(filter2_3) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p5, p6}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x1 + 7
        cons_2_A(x1,x2) = max{3, x1 - 30, x2 - 30}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 7
        app_A(x1,x2) = max{24, x1 - 27}
        true_0_A = 2
        app#_A(x1,x2) = x1 + 6
        filter2_3_A(x1,x2,x3) = max{23, x1 - 1, x2 + 7, x3 - 1}
        false_0_A = 8
        le_2_A(x1,x2) = max{22, x1 - 36, x2 - 28}
        |0|_0_A = 9
        s_1_A(x1) = x1 + 21
        filter_2_A(x1,x2) = max{23, x1 - 38, x2 - 9}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        le_2 > s_1 = filter_2 = nil_0 > app = |0|_0 > app# = filter2_3 = false_0 > filter_2# = filter2_4# > cons_2 > true_0
      
  
    partial status:
  
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(filter2_4#) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(false_0) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(s_1) = [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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
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)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r11, r12, r14, r18, r21, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x2 + 40
        cons_2_A(x1,x2) = max{61, x1 + 38, x2 + 42}
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 37, x4 + 41}
        app_A(x1,x2) = max{x1 - 9, x2 + 69}
        false_0_A = 5
        true_0_A = 31
        cons_0_A = 7
        cons_1_A(x1) = x1 + 69
        filter_0_A = 1
        filter_1_A(x1) = max{4, x1 - 33}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 1}
        filter2_2_A(x1,x2) = max{3, x1 - 11, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 20, x2 - 10, x3 - 1}
        gcd_0_A = 14
        gcd_1_A(x1) = max{4, x1 - 62}
        if_gcd_0_A = 5
        if_gcd_1_A(x1) = max{6, x1 - 1}
        if_gcd_2_A(x1,x2) = max{7, x1 - 11, x2 - 306}
        le_0_A = 1
        le_1_A(x1) = max{6, x1 - 61}
        minus_0_A = 7
        minus_1_A(x1) = max{8, x1 - 312}
        s_0_A = 1
        s_1_A(x1) = max{69, x1 - 141}
        le_2_A(x1,x2) = max{69, x1 - 70, x2 - 70}
        |0|_0_A = 6
        filter_2_A(x1,x2) = max{32, x1 - 43, x2 - 62}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app > filter_1 = filter2_3 > cons_2 = filter_0 = filter2_1 = filter2_2 = minus_1 = le_2 > cons_1 = filter2_0 = le_1 = minus_0 = s_1 > true_0 = cons_0 = gcd_1 = if_gcd_2 = le_0 = s_0 > gcd_0 = if_gcd_1 = |0|_0 > false_0 = if_gcd_0 > filter_2# = filter2_4#
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r16, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{8, x2 + 1}
        cons_2_A(x1,x2) = max{x1 + 16, x2 + 18}
        filter2_4#_A(x1,x2,x3,x4) = max{15, x3 + 9, x4 + 9}
        app_A(x1,x2) = max{7, x1 + 4}
        true_0_A = 5
        map_0_A = 1
        map_1_A(x1) = 6
        le_2_A(x1,x2) = max{16, x1 - 15, x2 - 52}
        |0|_0_A = 6
        s_1_A(x1) = max{14, x1 - 52}
        false_0_A = 3
        filter_2_A(x1,x2) = max{7, x1 - 19, x2 - 17}
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        cons_2 > filter_2 = nil_0 > filter2_4# = app = map_0 > filter_2# = map_1 > le_2 > s_1 > |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(filter2_4#) = []
      pi(app) = [1]
      pi(true_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(s_1) = []
      pi(false_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: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: 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: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p2: if_gcd_3#(false_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(y,x),s_1(x))
p3: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r2, r3, r7, r9, r11, r12, r14, r16, r18, r20, r22, r23, r24, r25, r26, r27, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gcd_2#_A(x1,x2) = max{68, x1 + 54, x2 + 37}
        s_1_A(x1) = max{29, x1 + 23}
        if_gcd_3#_A(x1,x2,x3) = max{68, x1 - 2, x2 + 38, x3 + 37}
        le_2_A(x1,x2) = max{82, x1 - 73, x2 - 73}
        false_0_A = 71
        minus_2_A(x1,x2) = max{14, x1 + 3, x2 - 30}
        true_0_A = 1
        app_A(x1,x2) = max{5, x1 - 8, x2 - 21}
        cons_1_A(x1) = max{6, x1 - 1}
        cons_2_A(x1,x2) = max{4, x1 - 10, x2 - 73}
        filter_0_A = 3
        filter_1_A(x1) = max{4, x1 - 22}
        filter2_2_A(x1,x2) = max{17, x1 - 29, x2 - 21}
        filter2_3_A(x1,x2,x3) = max{9, x1 - 80, x2 - 29, x3 - 22}
        gcd_0_A = 1
        gcd_1_A(x1) = max{2, x1 - 88}
        if_gcd_0_A = 7
        if_gcd_1_A(x1) = max{4, x1 - 138}
        if_gcd_2_A(x1,x2) = max{5, x1 - 146, x2 - 21}
        le_0_A = 18
        le_1_A(x1) = max{9, x1 - 74}
        map_0_A = 1
        map_1_A(x1) = max{4, x1 - 22}
        minus_0_A = 3
        minus_1_A(x1) = max{4, x1 - 22}
        pred_0_A = 3
        pred_1_A(x1) = max{4, x1 - 22}
        |0|_0_A = 72
        filter_2_A(x1,x2) = max{72, x1 - 1, x2 - 5}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        minus_2 > pred_1 > gcd_2# = s_1 = if_gcd_3# = filter_0 = map_0 = pred_0 = filter_2 > app = nil_0 > le_2 = cons_1 = cons_2 = filter_1 = filter2_3 = gcd_1 = if_gcd_2 = map_1 = minus_1 = |0|_0 > false_0 = true_0 = filter2_2 = gcd_0 = if_gcd_1 = le_1 = minus_0 > if_gcd_0 = le_0
      
  
    partial status:
  
      pi(gcd_2#) = [2]
      pi(s_1) = []
      pi(if_gcd_3#) = [3]
      pi(le_2) = []
      pi(false_0) = []
      pi(minus_2) = [1]
      pi(true_0) = []
      pi(app) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pred_0) = []
      pi(pred_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: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p2: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: 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: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p2: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r11, r12, r14, r16, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gcd_2#_A(x1,x2) = max{21, x1 + 16}
        s_1_A(x1) = x1 + 40
        if_gcd_3#_A(x1,x2,x3) = max{41, x1 - 1, x2 + 2}
        le_2_A(x1,x2) = x2 + 57
        true_0_A = 23
        minus_2_A(x1,x2) = max{3, x1}
        app_A(x1,x2) = max{x1 + 23, x2 + 160}
        cons_0_A = 138
        cons_1_A(x1) = max{161, x1 + 24}
        cons_2_A(x1,x2) = max{52, x1 + 25, x2 - 27}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 113}
        filter2_0_A = 139
        filter2_1_A(x1) = max{161, x1 - 155}
        filter2_2_A(x1,x2) = max{184, x1 - 132, x2 - 134}
        filter2_3_A(x1,x2,x3) = max{28, x1 - 133, x2 - 133, x3 - 24}
        gcd_0_A = 162
        gcd_1_A(x1) = max{161, x1 - 23}
        if_gcd_0_A = 1
        if_gcd_1_A(x1) = max{54, x1 - 21}
        if_gcd_2_A(x1,x2) = max{x1 - 22, x2 + 159}
        le_0_A = 1
        le_1_A(x1) = max{54, x1 - 24}
        map_0_A = 5
        map_1_A(x1) = max{27, x1}
        minus_0_A = 1
        minus_1_A(x1) = max{0, x1 - 22}
        pred_0_A = 1
        pred_1_A(x1) = max{2, x1 - 2}
        s_0_A = 1
        |0|_0_A = 24
        false_0_A = 21
        gcd_2_A(x1,x2) = max{x1, x2 + 160}
        filter_2_A(x1,x2) = max{26, x1 - 112, x2 - 53}
        nil_0_A = 24
  
    base order 2:
  
      precedence:
      
        minus_2 = if_gcd_2 > app = pred_1 > cons_1 = cons_2 = filter2_3 = if_gcd_0 = gcd_2 > s_1 = cons_0 = filter2_2 = gcd_1 = if_gcd_1 = map_1 = minus_1 = pred_0 = filter_2 = nil_0 > filter_1 = filter2_1 = gcd_0 = map_0 = minus_0 = s_0 = |0|_0 > true_0 = filter_0 = filter2_0 > le_2 > gcd_2# = le_1 = false_0 > if_gcd_3# = le_0
      
  
    partial status:
  
      pi(gcd_2#) = []
      pi(s_1) = []
      pi(if_gcd_3#) = [2]
      pi(le_2) = []
      pi(true_0) = []
      pi(minus_2) = [1]
      pi(app) = []
      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(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pred_0) = []
      pi(pred_1) = []
      pi(s_0) = []
      pi(|0|_0) = []
      pi(false_0) = []
      pi(gcd_2) = [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: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: 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: le_2#(s_1(x),s_1(y)) -> le_2#(x,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r11, r12, r14, r16, r18, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = max{1, x1 - 23, x2 - 26}
        s_1_A(x1) = max{25, x1 + 22}
        app_A(x1,x2) = max{24, x1 - 13, x2 - 4}
        cons_0_A = 1
        cons_1_A(x1) = max{3, x1 - 38}
        cons_2_A(x1,x2) = max{2, x1 - 51, x2 - 24}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 46}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 4}
        filter2_2_A(x1,x2) = max{3, x1 - 18, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 32, x2 - 18, x3 - 4}
        gcd_0_A = 11
        gcd_1_A(x1) = max{12, x1 - 35}
        if_gcd_0_A = 12
        if_gcd_1_A(x1) = max{10, x1 - 5}
        if_gcd_2_A(x1,x2) = max{11, x1 - 19, x2 - 41}
        le_0_A = 8
        le_1_A(x1) = max{9, x1 - 5}
        map_0_A = 11
        map_1_A(x1) = max{12, x1 - 51}
        minus_0_A = 11
        minus_1_A(x1) = max{12, x1 - 5}
        le_2_A(x1,x2) = max{10, x2 - 28}
        |0|_0_A = 24
        true_0_A = 1
        false_0_A = 1
        filter_2_A(x1,x2) = max{3, x1 - 60, x2 - 25}
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        le_2# = le_2 > s_1 = filter2_0 = minus_1 = filter_2 > app = minus_0 = |0|_0 > filter_1 = filter2_3 = gcd_1 = if_gcd_2 = le_1 = false_0 > cons_2 = filter_0 = filter2_2 = gcd_0 = if_gcd_1 = le_0 = map_1 = nil_0 > cons_1 = filter2_1 = if_gcd_0 = map_0 = true_0 > cons_0
      
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]
      pi(app) = []
      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(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_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: minus_2#(x,s_1(y)) -> minus_2#(x,y)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(gcd_0(),y) -> gcd_1(y)
r10: app(gcd_1(x1),y) -> gcd_2(x1,y)
r11: app(if_gcd_0(),y) -> if_gcd_1(y)
r12: app(if_gcd_1(x1),y) -> if_gcd_2(x1,y)
r13: app(if_gcd_2(x1,x2),y) -> if_gcd_3(x1,x2,y)
r14: app(le_0(),y) -> le_1(y)
r15: app(le_1(x1),y) -> le_2(x1,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: app(minus_0(),y) -> minus_1(y)
r19: app(minus_1(x1),y) -> minus_2(x1,y)
r20: app(pred_0(),y) -> pred_1(y)
r21: app(s_0(),y) -> s_1(y)
r22: le_2(|0|_0(),y) -> true_0()
r23: le_2(s_1(x),|0|_0()) -> false_0()
r24: le_2(s_1(x),s_1(y)) -> le_2(x,y)
r25: pred_1(s_1(x)) -> x
r26: minus_2(x,|0|_0()) -> x
r27: minus_2(x,s_1(y)) -> pred_1(minus_2(x,y))
r28: gcd_2(|0|_0(),y) -> y
r29: gcd_2(s_1(x),|0|_0()) -> s_1(x)
r30: gcd_2(s_1(x),s_1(y)) -> if_gcd_3(le_2(y,x),s_1(x),s_1(y))
r31: if_gcd_3(true_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(x,y),s_1(y))
r32: if_gcd_3(false_0(),s_1(x),s_1(y)) -> gcd_2(minus_2(y,x),s_1(x))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r5, r6, r7, r9, r11, r12, r14, r16, r18, r22, r23, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = max{x1 + 11, x2 + 9}
        s_1_A(x1) = max{22, x1 + 9}
        app_A(x1,x2) = max{25, x1 - 4, x2 - 7}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 22}
        cons_2_A(x1,x2) = max{4, x1 - 26, x2 - 7}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 8}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 8}
        filter2_2_A(x1,x2) = max{3, x1 - 12, x2 - 7}
        filter2_3_A(x1,x2,x3) = max{5, x1 - 34, x2 - 12, x3 - 8}
        gcd_0_A = 1
        gcd_1_A(x1) = max{2, x1 - 8}
        if_gcd_0_A = 1
        if_gcd_1_A(x1) = max{2, x1 - 7}
        if_gcd_2_A(x1,x2) = max{3, x1 - 20, x2 - 52}
        le_0_A = 2
        le_1_A(x1) = max{3, x1 - 32}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 8}
        minus_0_A = 1
        minus_1_A(x1) = max{3, x1 - 8}
        le_2_A(x1,x2) = max{23, x1 - 36, x2 - 23}
        |0|_0_A = 2
        true_0_A = 1
        false_0_A = 1
        filter_2_A(x1,x2) = max{3, x1 - 38, x2 - 13}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        minus_2# = filter_2 > s_1 = app = le_2 > gcd_1 = if_gcd_2 = minus_0 = |0|_0 > cons_0 = cons_2 = filter_1 = filter2_3 = gcd_0 = if_gcd_1 = le_1 = map_1 = minus_1 = false_0 = nil_0 > cons_1 = filter_0 = filter2_2 = if_gcd_0 = le_0 = map_0 = true_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = []
      pi(app) = []
      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(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.