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, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + 6
          filter_1_A(x1) = x1 + 6
          filter_2#_A(x1,x2) = x1 + 7
          cons_2_A(x1,x2) = 5
          map_1_A(x1) = x1 + 8
          map_2#_A(x1,x2) = x1 + 7
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 10
          filter2_4#_A(x1,x2,x3,x4) = x2 + 7
          false_0_A() = 8
          app_A(x1,x2) = x1 + x2 + 4
          true_0_A() = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 7
          filter_2_A(x1,x2) = x1 + 9
          filter2_0_A() = 5
          filter2_1_A(x1) = x1 + 8
          filter2_2_A(x1,x2) = x1 + x2 + 11
          filter2_4_A(x1,x2,x3,x4) = x2 + 9
          gcd_0_A() = 1
          gcd_1_A(x1) = x1 + 2
          gcd_2_A(x1,x2) = x1 + x2 + 4
          if_gcd_0_A() = 2
          if_gcd_1_A(x1) = 1
          if_gcd_2_A(x1,x2) = x2 + 2
          if_gcd_3_A(x1,x2,x3) = x2 + x3 + 3
          le_0_A() = 7
          le_1_A(x1) = 6
          le_2_A(x1,x2) = 9
          map_0_A() = 5
          map_2_A(x1,x2) = 6
          minus_0_A() = 7
          minus_1_A(x1) = x1 + 10
          minus_2_A(x1,x2) = x1 + 9
          pred_0_A() = 1
          pred_1_A(x1) = x1
          s_0_A() = 8
          s_1_A(x1) = x1 + 11
          |0|_0_A() = 9
          nil_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{0, x1 - 125, x2 - 37}
          filter_1_A(x1) = x1 + 5
          filter_2#_A(x1,x2) = max{0, x1 - 120, x2 - 37}
          cons_2_A(x1,x2) = max{2, x1, x2}
          map_1_A(x1) = max{9, x1 + 5}
          map_2#_A(x1,x2) = max{0, x1 - 120, x2 - 37}
          filter2_3_A(x1,x2,x3) = max{122, x1 - 33, x2 + 7, x3 + 17}
          filter2_4#_A(x1,x2,x3,x4) = max{0, x1 - 158, x2 - 120, x3 - 108, x4 - 37}
          false_0_A = 157
          app_A(x1,x2) = max{124, x1 - 16, x2 + 121}
          true_0_A = 121
          cons_0_A = 19
          cons_1_A(x1) = x1 + 17
          filter_0_A = 5
          filter_2_A(x1,x2) = max{122, x1 - 35, x2 + 1}
          filter2_0_A = 15
          filter2_1_A(x1) = max{5, x1 - 1}
          filter2_2_A(x1,x2) = max{x1 - 17, x2 + 121}
          filter2_4_A(x1,x2,x3,x4) = max{122, x1 - 158, x2 - 35, x3 + 1, x4 + 1}
          gcd_0_A = 5
          gcd_1_A(x1) = x1 + 18
          gcd_2_A(x1,x2) = max{124, x1 + 2, x2 + 1}
          if_gcd_0_A = 7
          if_gcd_1_A(x1) = max{5, x1 - 1}
          if_gcd_2_A(x1,x2) = max{x1 - 106, x2 + 18}
          if_gcd_3_A(x1,x2,x3) = max{124, x1 - 122, x2 + 2, x3 + 1}
          le_0_A = 15
          le_1_A(x1) = max{15, x1 - 116}
          le_2_A(x1,x2) = x2 + 121
          map_0_A = 8
          map_2_A(x1,x2) = max{124, x1 - 16, x2 + 121}
          minus_0_A = 5
          minus_1_A(x1) = x1 + 71
          minus_2_A(x1,x2) = x1 + 55
          pred_0_A = 5
          pred_1_A(x1) = max{55, x1 - 55}
          s_0_A = 5
          s_1_A(x1) = x1 + 121
          |0|_0_A = 36
          nil_0_A = 0
          app#!_A(x1,x2) = max{14, x1 - 123}
          filter_1!_A(x1) = max{8, x1 - 174}
          filter_2#!_A(x1,x2) = max{1, x1 - 173, x2 - 3}
          cons_2!_A(x1,x2) = max{12, x1 - 235, x2 - 110}
          map_1!_A(x1) = max{1, x1 - 1}
          map_2#!_A(x1,x2) = max{13, x2 - 110}
          filter2_3!_A(x1,x2,x3) = max{0, x1 - 129, x2 - 5}
          filter2_4#!_A(x1,x2,x3,x4) = max{1, x2 - 173, x3 - 4, x4 - 3}
          false_0!_A = 2
          app!_A(x1,x2) = max{x1 - 6, x2 + 7}
          true_0!_A = 14
          cons_0!_A = 1
          cons_1!_A(x1) = max{13, x1 - 236}
          filter_0!_A = 1
          filter_2!_A(x1,x2) = max{13, x1 - 5, x2 - 3}
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{8, x1 - 2}
          filter2_2!_A(x1,x2) = max{9, x1 - 1, x2 - 1}
          filter2_4!_A(x1,x2,x3,x4) = max{13, x1 - 128, x2 - 5, x3 - 235, x4 - 3}
          gcd_0!_A = 1
          gcd_1!_A(x1) = max{7, x1 - 1}
          gcd_2!_A(x1,x2) = max{121, x2 - 37}
          if_gcd_0!_A = 1
          if_gcd_1!_A(x1) = 0
          if_gcd_2!_A(x1,x2) = max{5, x2 - 55}
          if_gcd_3!_A(x1,x2,x3) = max{6, x1 - 117, x2 - 54, x3}
          le_0!_A = 4
          le_1!_A(x1) = 3
          le_2!_A(x1,x2) = 15
          map_0!_A = 2
          map_2!_A(x1,x2) = max{2, x1 - 125, x2 - 1}
          minus_0!_A = 9
          minus_1!_A(x1) = x1 + 67
          minus_2!_A(x1,x2) = x1 + 66
          pred_0!_A = 3
          pred_1!_A(x1) = max{2, x1 - 55}
          s_0!_A = 1
          s_1!_A(x1) = max{41, x1 - 122}
          |0|_0!_A = 15
          nil_0!_A = 14
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = [1]
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(map_1) = []
      pi(map_2#) = [1]
      pi(filter2_3) = [1]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_4) = []
      pi(gcd_0) = []
      pi(gcd_1) = []
      pi(gcd_2) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(if_gcd_2) = []
      pi(if_gcd_3) = [2]
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_2) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(minus_2) = [1]
      pi(pred_0) = []
      pi(pred_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p3, p5, p6



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: filter2_4#(false_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#(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}


-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    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) = x1 + x2 + 9
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x2 - 12}
          cons_2_A(x1,x2) = max{x1 - 84, x2 + 11}
          map_2#!_A(x1,x2) = max{29, x1 - 1, x2 - 12}
          cons_2!_A(x1,x2) = max{28, x1 - 9, x2 - 13}
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filter2_0(),y) -> filter2_1(y)
r6: app(filter2_1(x1),y) -> filter2_2(x1,y)
r7: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r8: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r9: app(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, r11, r14, r15, r16, r19, r20, r22, r23, r24, r25, r26, r27, r28, r29, r33, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    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 + 4
          false_0_A() = 2
          filter_2#_A(x1,x2) = x2 + 3
          cons_2_A(x1,x2) = x2 + 5
          app_A(x1,x2) = x1 + x2 + 12
          true_0_A() = 2
          cons_0_A() = 2
          cons_1_A(x1) = 1
          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 + 1
          if_gcd_0_A() = 2
          if_gcd_1_A(x1) = 1
          le_0_A() = 2
          le_1_A(x1) = 1
          le_2_A(x1,x2) = x2 + 3
          map_0_A() = 2
          map_1_A(x1) = x1 + 1
          minus_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + x2 + 3
          pred_0_A() = 4
          pred_1_A(x1) = x1 + 3
          |0|_0_A() = 1
          s_1_A(x1) = x1 + 5
          gcd_2_A(x1,x2) = x1 + x2 + 1
          map_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 1, x3 - 77, x4 - 5}
          false_0_A = 53
          filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 5}
          cons_2_A(x1,x2) = max{4, x1 - 72, x2}
          app_A(x1,x2) = max{78, x1 - 4, x2 + 2}
          true_0_A = 53
          cons_0_A = 3
          cons_1_A(x1) = max{0, x1 - 45}
          filter_0_A = 3
          filter_1_A(x1) = max{2, x1 - 8}
          filter2_0_A = 3
          filter2_1_A(x1) = max{3, x1 - 1}
          filter2_2_A(x1,x2) = max{2, x1 - 5, x2 - 1}
          filter2_3_A(x1,x2,x3) = max{2, x1 - 9, x2 - 5, x3 - 39}
          if_gcd_0_A = 3
          if_gcd_1_A(x1) = max{2, x1 - 45}
          le_0_A = 3
          le_1_A(x1) = max{2, x1 - 1}
          le_2_A(x1,x2) = max{78, x1 - 39, x2 - 39}
          map_0_A = 3
          map_1_A(x1) = max{3, x1 - 52}
          minus_1_A(x1) = x1 + 31
          minus_2_A(x1,x2) = max{28, x1 + 1, x2 - 40}
          pred_0_A = 3
          pred_1_A(x1) = max{3, x1 - 29}
          |0|_0_A = 8
          s_1_A(x1) = x1 + 38
          gcd_2_A(x1,x2) = max{66, x1 + 38, x2 + 1}
          map_2_A(x1,x2) = max{3, x1 - 56, x2 + 2}
          nil_0_A = 7
          filter_2_A(x1,x2) = max{6, x1 - 40, x2 + 1}
          filter2_4#!_A(x1,x2,x3,x4) = max{11, x2 - 44, x3 - 82, x4 - 10}
          false_0!_A = 2
          filter_2#!_A(x1,x2) = max{11, x1 - 44, x2 - 10}
          cons_2!_A(x1,x2) = max{8, x1 - 89, x2 - 10}
          app!_A(x1,x2) = max{x1 - 43, x2 + 15}
          true_0!_A = 7
          cons_0!_A = 16
          cons_1!_A(x1) = 15
          filter_0!_A = 8
          filter_1!_A(x1) = max{9, x1 - 51}
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{2, x1 - 46}
          filter2_2!_A(x1,x2) = max{14, x1 - 45, x2 - 1}
          filter2_3!_A(x1,x2,x3) = max{13, x1 - 49, x2 - 50}
          if_gcd_0!_A = 12
          if_gcd_1!_A(x1) = 11
          le_0!_A = 3
          le_1!_A(x1) = 2
          le_2!_A(x1,x2) = max{3, x2 - 9}
          map_0!_A = 1
          map_1!_A(x1) = max{2, x1 - 1}
          minus_1!_A(x1) = max{2, x1 - 1}
          minus_2!_A(x1,x2) = max{x1 + 71, x2 - 14}
          pred_0!_A = 1
          pred_1!_A(x1) = x1 + 42
          |0|_0!_A = 3
          s_1!_A(x1) = max{8, x1 + 1}
          gcd_2!_A(x1,x2) = max{4, x1 - 29, x2 - 52}
          map_2!_A(x1,x2) = max{9, x1 - 94, x2 - 9}
          nil_0!_A = 9
          filter_2!_A(x1,x2) = max{11, x1 - 50, x2 - 8}
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(app) = [1]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = []
      pi(if_gcd_0) = []
      pi(if_gcd_1) = []
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(pred_0) = []
      pi(pred_1) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(gcd_2) = []
      pi(map_2) = [2]
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- 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

  r22, r23, r24, r25, r26, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    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) = 10
          false_0_A() = 9
          minus_2_A(x1,x2) = x1 + 2
          true_0_A() = 0
          |0|_0_A() = 0
          pred_1_A(x1) = x1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gcd_2#_A(x1,x2) = max{0, x1 - 33, x2 - 39}
          s_1_A(x1) = max{31, x1 + 13}
          if_gcd_3#_A(x1,x2,x3) = max{0, x1 - 33, x2 - 38, x3 - 39}
          le_2_A(x1,x2) = max{32, x1 - 32, x2 - 24}
          false_0_A = 0
          minus_2_A(x1,x2) = max{9, x1 + 1, x2 - 33}
          true_0_A = 0
          |0|_0_A = 23
          pred_1_A(x1) = max{3, x1 - 10}
          gcd_2#!_A(x1,x2) = max{27, x1 - 2}
          s_1!_A(x1) = 27
          if_gcd_3#!_A(x1,x2,x3) = max{6, x1 - 33, x3 - 49}
          le_2!_A(x1,x2) = max{28, x2 - 32}
          false_0!_A = 31
          minus_2!_A(x1,x2) = max{5, x1 - 36, x2 - 37}
          true_0!_A = 29
          |0|_0!_A = 30
          pred_1!_A(x1) = max{4, x1 - 46}
  
    partial status:
  
      pi(gcd_2#) = [1]
      pi(s_1) = [1]
      pi(if_gcd_3#) = [2]
      pi(le_2) = []
      pi(false_0) = []
      pi(minus_2) = []
      pi(true_0) = []
      pi(|0|_0) = []
      pi(pred_1) = []

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.

-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          le_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 10
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          le_2#_A(x1,x2) = 0
          s_1_A(x1) = max{4, x1 - 29}
          le_2#!_A(x1,x2) = max{22, x1 - 5, x2 - 5}
          s_1!_A(x1) = max{21, x1 + 18}
  
    partial status:
  
      pi(le_2#) = [2]
      pi(s_1) = [1]

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

  r14, r22, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x2
          s_1_A(x1) = x1 + 4
          app_A(x1,x2) = x2 + 6
          le_0_A() = 2
          le_1_A(x1) = 1
          le_2_A(x1,x2) = x1 + x2 + 13
          |0|_0_A() = 2
          true_0_A() = 1
          false_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_2#_A(x1,x2) = max{0, x2 - 11}
          s_1_A(x1) = max{37, x1 + 10}
          app_A(x1,x2) = max{1, x1 - 3, x2 - 90}
          le_0_A = 2
          le_1_A(x1) = max{1, x1 - 90}
          le_2_A(x1,x2) = max{26, x1 - 38, x2 - 1}
          |0|_0_A = 0
          true_0_A = 0
          false_0_A = 0
          minus_2#!_A(x1,x2) = max{4, x1 - 1}
          s_1!_A(x1) = max{3, x1 - 34}
          app!_A(x1,x2) = max{x1 - 108, x2}
          le_0!_A = 1
          le_1!_A(x1) = max{2, x1 - 39}
          le_2!_A(x1,x2) = max{5, x1 - 38, x2 - 38}
          |0|_0!_A = 2
          true_0!_A = 1
          false_0!_A = 1
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]
      pi(app) = [2]
      pi(le_0) = []
      pi(le_1) = []
      pi(le_2) = [1]
      pi(|0|_0) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.