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, r2, r3, r5, r6, r7, r9, r11, r12, r13, r14, r15, r16, r18, r21, r22, r23, r24, r28, r29, r30, r31, r32, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + x2 + 4
          filter_1_A(x1) = x1 + 7
          filter_2#_A(x1,x2) = x1 + x2 + 8
          cons_2_A(x1,x2) = x1 + x2 + 3
          map_1_A(x1) = x1 + 3
          map_2#_A(x1,x2) = x1 + x2 + 4
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 8
          filter2_4#_A(x1,x2,x3,x4) = x2 + x3 + x4 + 9
          false_0_A() = 7
          app_A(x1,x2) = x1 + x2 + 2
          true_0_A() = 7
          cons_0_A() = 3
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 6
          filter2_0_A() = 5
          filter2_1_A(x1) = x1 + 6
          filter2_2_A(x1,x2) = x1 + x2 + 7
          gcd_0_A() = 1
          gcd_1_A(x1) = x1 + 2
          if_gcd_0_A() = 5
          if_gcd_1_A(x1) = x1 + 6
          if_gcd_2_A(x1,x2) = x1 + x2 + 7
          if_gcd_3_A(x1,x2,x3) = 6
          le_0_A() = 8
          le_1_A(x1) = x1 + 9
          le_2_A(x1,x2) = 8
          map_0_A() = 2
          minus_0_A() = 6
          minus_1_A(x1) = x1 + 7
          s_0_A() = 2
          s_1_A(x1) = 1
          |0|_0_A() = 8
          gcd_2_A(x1,x2) = x2 + 5
          minus_2_A(x1,x2) = x2 + 8
          filter_2_A(x1,x2) = x1 + x2 + 8
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_4# > filter_2# = map_2# > app# = app = cons_0 = filter_2 > filter_1 = cons_2 = map_1 = filter2_3 = le_1 = minus_2 = nil_0 > cons_1 = filter_0 = filter2_2 = le_0 = le_2 = map_0 = minus_1 = |0|_0 > false_0 = true_0 = filter2_1 = if_gcd_2 = minus_0 > filter2_0 = if_gcd_1 = if_gcd_3 = s_0 = gcd_2 > gcd_1 = if_gcd_0 = s_1 > gcd_0
      
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = []
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(map_1) = []
      pi(map_2#) = []
      pi(filter2_3) = [3]
      pi(filter2_4#) = [3, 4]
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(gcd_0) = []
      pi(gcd_1) = [1]
      pi(if_gcd_0) = []
      pi(if_gcd_1) = [1]
      pi(if_gcd_2) = [2]
      pi(if_gcd_3) = []
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(s_0) = []
      pi(s_1) = []
      pi(|0|_0) = []
      pi(gcd_2) = []
      pi(minus_2) = []
      pi(filter_2) = [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)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p8: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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, r9, r11, r12, r14, r16, r18, r21, r22, r23, r24, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 3
          cons_2_A(x1,x2) = x1 + x2 + 2
          app#_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 4
          true_0_A() = 1
          app_A(x1,x2) = x2 + 4
          false_0_A() = 1
          map_1_A(x1) = x1 + 2
          map_2#_A(x1,x2) = x1 + x2 + 3
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          gcd_0_A() = 2
          gcd_1_A(x1) = 1
          if_gcd_0_A() = 3
          if_gcd_1_A(x1) = 2
          if_gcd_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 3
          le_2_A(x1,x2) = x1 + x2 + 5
          |0|_0_A() = 2
          gcd_2_A(x1,x2) = x1 + x2 + 5
          filter_2_A(x1,x2) = x1 + x2 + 5
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        map_2# > app# = |0|_0 = gcd_2 = filter_2 > filter2_4# = false_0 > filter_2# = cons_2 = cons_0 > filter2_3 = cons_1 > app = filter_0 = filter2_0 = gcd_0 = if_gcd_0 = le_0 = minus_0 = nil_0 > map_1 = filter_1 = filter2_1 = gcd_1 = if_gcd_1 = minus_1 > filter2_2 = if_gcd_2 = map_0 = s_1 = le_2 > true_0 = le_1 = s_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      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(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(gcd_2) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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:

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


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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

  r3, r5, r6, r7, r9, r11, r12, r14, r18, r22, r23, r24, r28, r29, r35, r37

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + 1
          cons_2_A(x1,x2) = x1 + 5
          app#_A(x1,x2) = x1
          map_1_A(x1) = x1 + 2
          app_A(x1,x2) = x2 + 4
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          gcd_0_A() = 2
          gcd_1_A(x1) = 1
          if_gcd_0_A() = 3
          if_gcd_1_A(x1) = 2
          if_gcd_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          le_2_A(x1,x2) = x1 + x2 + 14
          |0|_0_A() = 7
          true_0_A() = 6
          s_1_A(x1) = x1 + 13
          false_0_A() = 1
          gcd_2_A(x1,x2) = x1 + x2 + 12
          filter_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 1
          filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 13
  
    base order 2:
  
      precedence:
      
        app# > map_2# = gcd_2 > cons_2 = s_1 > map_1 = filter_1 = le_2 > app = filter_0 = if_gcd_0 = le_0 = minus_0 > filter2_1 = gcd_1 = if_gcd_1 = le_1 = minus_1 = |0|_0 = filter_2 = filter2_4 > filter2_0 = filter2_2 = gcd_0 = if_gcd_2 = true_0 = false_0 = nil_0 > filter2_3
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app#) = []
      pi(map_1) = [1]
      pi(app) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = [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(le_2) = [2]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(s_1) = []
      pi(false_0) = []
      pi(gcd_2) = []
      pi(filter_2) = [2]
      pi(nil_0) = []
      pi(filter2_4) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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:

  {p2}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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, r12, r14, r16, r18, r21, r22, r28, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x2 + 6
          app_A(x1,x2) = x2 + 9
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          gcd_0_A() = 1
          gcd_1_A(x1) = x1 + 3
          if_gcd_1_A(x1) = x1 + 10
          if_gcd_2_A(x1,x2) = 1
          le_0_A() = 8
          le_1_A(x1) = 7
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 5
          s_1_A(x1) = 4
          le_2_A(x1,x2) = 6
          |0|_0_A() = 1
          true_0_A() = 5
          gcd_2_A(x1,x2) = x2 + 2
          filter_2_A(x1,x2) = x2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_0 > map_2# = cons_2 = filter_1 > if_gcd_1 = le_0 = s_0 = filter_2 > nil_0 > map_0 > app = cons_0 = map_1 = minus_0 > cons_1 = filter2_1 = if_gcd_2 = le_1 = minus_1 = gcd_2 > filter2_0 = filter2_2 = gcd_1 = le_2 > filter2_3 = gcd_0 = true_0 > |0|_0 > s_1
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(gcd_0) = []
      pi(gcd_1) = [1]
      pi(if_gcd_1) = [1]
      pi(if_gcd_2) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = []
      pi(|0|_0) = []
      pi(true_0) = []
      pi(gcd_2) = [2]
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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, r15, r16, r18, r21, r22, r23, r24, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter2_4#_A(x1,x2,x3,x4) = x4 + 3
          true_0_A() = 1
          filter_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          app_A(x1,x2) = x1 + x2 + 7
          false_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          gcd_0_A() = 1
          gcd_1_A(x1) = x1 + 2
          if_gcd_0_A() = 1
          if_gcd_1_A(x1) = x1 + 2
          if_gcd_2_A(x1,x2) = x1 + x2 + 3
          le_0_A() = 1
          le_1_A(x1) = x1 + 2
          le_2_A(x1,x2) = x1 + x2 + 4
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 2
          s_0_A() = 2
          s_1_A(x1) = x1 + 3
          |0|_0_A() = 2
          gcd_2_A(x1,x2) = x1 + x2 + 5
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app = gcd_0 = filter_2 > cons_2 = filter_1 = filter2_3 = if_gcd_1 = map_1 = gcd_2 = nil_0 > filter2_4# = cons_1 = filter_0 = filter2_2 = gcd_1 = if_gcd_0 = map_0 > filter_2# = cons_0 = filter2_1 = if_gcd_2 = le_2 = s_1 > filter2_0 = le_1 = minus_1 = s_0 = |0|_0 > true_0 = false_0 = le_0 = minus_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = [1]
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = [3]
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = [1]
      pi(if_gcd_2) = [1, 2]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(gcd_2) = [2]
      pi(filter_2) = [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#(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}


-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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, r4, r5, r9, r11, r14, r16, r18, r21, r22, r23, r24, r28, r29, r35, r36, r37, r38

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 5
          cons_2_A(x1,x2) = x1 + x2 + 2
          filter2_4#_A(x1,x2,x3,x4) = x2 + x3 + x4 + 6
          app_A(x1,x2) = x2 + 6
          false_0_A() = 1
          cons_0_A() = 0
          cons_1_A(x1) = x1
          filter_0_A() = 0
          filter_1_A(x1) = x1
          filter_2_A(x1,x2) = x2 + 2
          filter2_0_A() = 0
          filter2_1_A(x1) = x1
          gcd_0_A() = 1
          gcd_1_A(x1) = x1
          if_gcd_0_A() = 0
          if_gcd_1_A(x1) = x1
          le_0_A() = 0
          le_1_A(x1) = x1
          map_0_A() = 0
          map_1_A(x1) = x1
          minus_0_A() = 0
          minus_1_A(x1) = 0
          s_0_A() = 0
          s_1_A(x1) = x1 + 4
          le_2_A(x1,x2) = x1 + x2 + 7
          |0|_0_A() = 2
          true_0_A() = 1
          gcd_2_A(x1,x2) = x1 + x2 + 7
          nil_0_A() = 0
          filter2_4_A(x1,x2,x3,x4) = x3 + x4 + 4
  
    base order 2:
  
      precedence:
      
        filter_2 > app = cons_0 = minus_0 = nil_0 > filter_2# = filter_1 = if_gcd_1 = map_1 = minus_1 > filter2_4# = filter_0 = if_gcd_0 = map_0 = filter2_4 > cons_2 = filter2_1 = s_0 = |0|_0 > cons_1 = filter2_0 = s_1 = le_2 = true_0 > le_1 = gcd_2 > false_0 = gcd_1 = le_0 > gcd_0
      
  
    partial status:
  
      pi(filter_2#) = [1, 2]
      pi(cons_2) = []
      pi(filter2_4#) = [2, 3]
      pi(app) = [2]
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(le_2) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(gcd_2) = [1, 2]
      pi(nil_0) = []
      pi(filter2_4) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gcd_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 6
          if_gcd_3#_A(x1,x2,x3) = x2 + x3
          le_2_A(x1,x2) = x1 + x2 + 11
          false_0_A() = 1
          minus_2_A(x1,x2) = x1
          true_0_A() = 1
          app_A(x1,x2) = x2 + 10
          cons_0_A() = 10
          cons_1_A(x1) = 9
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          gcd_0_A() = 2
          gcd_1_A(x1) = 1
          if_gcd_0_A() = 3
          if_gcd_1_A(x1) = 2
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 7
          |0|_0_A() = 2
          pred_1_A(x1) = x1
          gcd_2_A(x1,x2) = x2 + 9
          filter_2_A(x1,x2) = x1 + x2 + 11
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        gcd_0 > gcd_1 > gcd_2 > minus_2 = |0|_0 > false_0 = s_0 = pred_1 > s_1 > gcd_2# > if_gcd_3# = le_2 = filter_0 > app = cons_0 = filter_1 = filter2_0 = if_gcd_0 = le_0 = map_0 = filter_2 > true_0 = cons_1 = filter2_1 = if_gcd_1 = le_1 = map_1 = minus_1 = nil_0 > filter2_2 = minus_0 > filter2_3
      
  
    partial status:
  
      pi(gcd_2#) = [1]
      pi(s_1) = []
      pi(if_gcd_3#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(minus_2) = [1]
      pi(true_0) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(s_0) = []
      pi(|0|_0) = []
      pi(pred_1) = []
      pi(gcd_2) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p3



-- 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#(false_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(y,x),s_1(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}


-- 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))

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, r7, r9, r11, r12, r14, r15, r16, r18, r20, r21, r22, r23, r24, r25, r26, r27, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gcd_2#_A(x1,x2) = x1 + x2 + 1
          s_1_A(x1) = x1 + 5
          if_gcd_3#_A(x1,x2,x3) = x2 + x3
          le_2_A(x1,x2) = 4
          false_0_A() = 0
          minus_2_A(x1,x2) = x1 + 4
          app_A(x1,x2) = x2 + 9
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 10
          filter2_3_A(x1,x2,x3) = 8
          gcd_0_A() = 2
          gcd_1_A(x1) = 1
          if_gcd_0_A() = 9
          if_gcd_1_A(x1) = 8
          if_gcd_2_A(x1,x2) = 7
          le_0_A() = 6
          le_1_A(x1) = 5
          map_0_A() = 9
          map_1_A(x1) = 8
          minus_0_A() = 2
          minus_1_A(x1) = 1
          pred_0_A() = 1
          pred_1_A(x1) = x1
          s_0_A() = 4
          |0|_0_A() = 0
          true_0_A() = 3
          gcd_2_A(x1,x2) = x1 + 10
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_2 > filter2_3 > filter_0 = le_0 = minus_0 = |0|_0 = gcd_2 = filter_2 > filter_1 = nil_0 > app = cons_0 = filter2_0 = gcd_0 = if_gcd_0 = map_0 > minus_2 = cons_1 = filter2_1 = gcd_1 = if_gcd_1 = map_1 = minus_1 = pred_1 > if_gcd_2 = le_1 = pred_0 = s_0 = true_0 > gcd_2# = s_1 = if_gcd_3# = le_2 = false_0
      
  
    partial status:
  
      pi(gcd_2#) = []
      pi(s_1) = [1]
      pi(if_gcd_3#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(minus_2) = []
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [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(true_0) = []
      pi(gcd_2) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: if_gcd_3#(false_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(y,x),s_1(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:

  (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, r3, r5, r6, r7, r9, r11, r12, r14, r16, r18, r21, r22, r23, r24, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le_2#_A(x1,x2) = x1 + x2 + 4
          s_1_A(x1) = x1 + 3
          app_A(x1,x2) = x2 + 7
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 4
          filter_1_A(x1) = 3
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          gcd_0_A() = 2
          gcd_1_A(x1) = 1
          if_gcd_0_A() = 3
          if_gcd_1_A(x1) = 2
          if_gcd_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          s_0_A() = 1
          le_2_A(x1,x2) = x1 + x2 + 6
          |0|_0_A() = 2
          true_0_A() = 1
          false_0_A() = 1
          gcd_2_A(x1,x2) = x1 + 2
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_0 = le_0 > filter_1 = gcd_0 = nil_0 > gcd_1 = le_2 = filter_2 > app = cons_0 = filter2_0 = map_0 = minus_0 > cons_1 = filter2_1 = if_gcd_1 = le_1 = map_1 = minus_1 = gcd_2 > le_2# = s_1 = filter2_2 = if_gcd_0 > filter2_3 = if_gcd_2 = s_0 = |0|_0 > true_0 = false_0
      
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = [1]
      pi(app) = [2]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(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(s_0) = []
      pi(le_2) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(gcd_2) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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, r3, r9, r12, r14, r16, r18, r22, r23, r24, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2 + 8
          s_1_A(x1) = x1 + 7
          app_A(x1,x2) = 5
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          gcd_0_A() = 5
          gcd_1_A(x1) = 4
          if_gcd_1_A(x1) = 6
          if_gcd_2_A(x1,x2) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          le_2_A(x1,x2) = x1 + x2 + 18
          |0|_0_A() = 8
          true_0_A() = 1
          false_0_A() = 1
          gcd_2_A(x1,x2) = x1 + x2 + 3
          filter_2_A(x1,x2) = x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        minus_2# = gcd_0 = le_2 > s_1 = cons_0 = gcd_1 > cons_1 = |0|_0 = gcd_2 > app = filter_0 = le_0 = map_0 = minus_0 = true_0 > if_gcd_1 = le_1 = minus_1 > if_gcd_2 = map_1 > filter_2 > filter_1 = false_0 = nil_0
      
  
    partial status:
  
      pi(minus_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      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(gcd_2) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.