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, r5, r6, r9, r11, r16, r21, r22, r23, r24, r26, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{1, x1 - 28, x2 - 33}
            filter_1_A(x1) = max{3, x1 + 2}
            filter_2#_A(x1,x2) = max{1, x1 - 26, x2 - 33}
            cons_2_A(x1,x2) = max{32, x1 + 4, x2}
            map_1_A(x1) = max{27, x1 + 2}
            map_2#_A(x1,x2) = max{1, x1 - 26, x2 - 33}
            filter2_3_A(x1,x2,x3) = max{x1 - 35, x2 + 2, x3 + 26}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 26, x3 - 29, x4 - 33}
            false_0_A = 36
            app_A(x1,x2) = max{34, x1 - 16, x2 + 26}
            true_0_A = 36
            cons_0_A = 30
            cons_1_A(x1) = x1 + 20
            filter2_0_A = 0
            filter2_1_A(x1) = max{15, x1 - 3}
            filter2_2_A(x1,x2) = max{x1 - 19, x2 + 26}
            gcd_0_A = 0
            gcd_1_A(x1) = max{15, x1 - 3}
            if_gcd_0_A = 15
            if_gcd_1_A(x1) = max{27, x1 - 3}
            map_0_A = 15
            s_0_A = 0
            s_1_A(x1) = x1
            le_2_A(x1,x2) = max{x1 - 17, x2 + 69}
            |0|_0_A = 0
            minus_2_A(x1,x2) = max{27, x1 + 10, x2 - 3}
            gcd_2_A(x1,x2) = max{29, x1 + 2, x2 + 1}
            filter_2_A(x1,x2) = max{34, x1 - 31, x2 + 1}
            nil_0_A = 35
            app#!_A(x1,x2) = 21
            filter_1!_A(x1) = x1 + 30
            filter_2#!_A(x1,x2) = 21
            cons_2!_A(x1,x2) = x1 + 22
            map_1!_A(x1) = 20
            map_2#!_A(x1,x2) = 21
            filter2_3!_A(x1,x2,x3) = max{0, x1 - 1, x2 - 1, x3 - 1}
            filter2_4#!_A(x1,x2,x3,x4) = 21
            false_0!_A = 25
            app!_A(x1,x2) = max{x1 + 2, x2 + 30}
            true_0!_A = 20
            cons_0!_A = 30
            cons_1!_A(x1) = max{31, x1 + 29}
            filter2_0!_A = 32
            filter2_1!_A(x1) = 31
            filter2_2!_A(x1,x2) = max{0, x1 - 2}
            gcd_0!_A = 2
            gcd_1!_A(x1) = max{1, x1 - 29}
            if_gcd_0!_A = 17
            if_gcd_1!_A(x1) = max{16, x1 - 1}
            map_0!_A = 1
            s_0!_A = 1
            s_1!_A(x1) = 26
            le_2!_A(x1,x2) = 31
            |0|_0!_A = 19
            minus_2!_A(x1,x2) = max{32, x1 - 19, x2 - 32}
            gcd_2!_A(x1,x2) = max{26, x1 - 28, x2 - 32}
            filter_2!_A(x1,x2) = max{x1 + 3, x2 + 27}
            nil_0!_A = 25
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{1, x1 - 10, x2 - 17}
            filter_1_A(x1) = x1 + 9
            filter_2#_A(x1,x2) = max{1, x1 - 8, x2 - 17}
            cons_2_A(x1,x2) = max{16, x1, x2}
            map_1_A(x1) = max{4, x1 + 2}
            map_2#_A(x1,x2) = max{1, x1 - 8, x2 - 17}
            filter2_3_A(x1,x2,x3) = max{x1 - 21, x2 + 2, x3 - 30}
            filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 8, x3 - 40, x4 - 17}
            false_0_A = 14
            app_A(x1,x2) = max{x1 - 10, x2 + 14}
            true_0_A = 14
            cons_0_A = 9
            cons_1_A(x1) = x1 + 7
            filter2_0_A = 2
            filter2_1_A(x1) = max{2, x1 - 1}
            filter2_2_A(x1,x2) = max{1, x1 - 11, x2 - 1}
            gcd_0_A = 2
            gcd_1_A(x1) = x1 + 14
            if_gcd_0_A = 2
            if_gcd_1_A(x1) = max{2, x1 - 1}
            map_0_A = 2
            s_0_A = 2
            s_1_A(x1) = max{9, x1 + 4}
            le_2_A(x1,x2) = max{14, x1 - 19, x2 - 8}
            |0|_0_A = 7
            minus_2_A(x1,x2) = max{3, x1 + 1, x2 - 11}
            gcd_2_A(x1,x2) = max{14, x1 + 4, x2 + 1}
            filter_2_A(x1,x2) = max{31, x1 - 18, x2 + 14}
            nil_0_A = 32
            app#!_A(x1,x2) = x1 + 29
            filter_1!_A(x1) = x1 + 31
            filter_2#!_A(x1,x2) = x1 + 30
            cons_2!_A(x1,x2) = 5
            map_1!_A(x1) = 1
            map_2#!_A(x1,x2) = max{32, x1 + 30}
            filter2_3!_A(x1,x2,x3) = max{0, x1 - 1, x2 - 4, x3 - 1}
            filter2_4#!_A(x1,x2,x3,x4) = x2 + 30
            false_0!_A = 29
            app!_A(x1,x2) = max{4, x1 - 2}
            true_0!_A = 31
            cons_0!_A = 7
            cons_1!_A(x1) = 6
            filter2_0!_A = 6
            filter2_1!_A(x1) = max{5, x1 - 4}
            filter2_2!_A(x1,x2) = max{0, x1 - 4}
            gcd_0!_A = 1
            gcd_1!_A(x1) = max{5, x1 - 16}
            if_gcd_0!_A = 3
            if_gcd_1!_A(x1) = max{2, x1 - 1}
            map_0!_A = 2
            s_0!_A = 1
            s_1!_A(x1) = max{0, x1 - 47}
            le_2!_A(x1,x2) = max{28, x1 - 10, x2 - 10}
            |0|_0!_A = 30
            minus_2!_A(x1,x2) = max{10, x1 - 26, x2 - 46}
            gcd_2!_A(x1,x2) = max{6, x1 - 15}
            filter_2!_A(x1,x2) = x1 + 6
            nil_0!_A = 1
        
  
    partial status:
  
      pi(app#) = []
      pi(filter_1) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(map_1) = []
      pi(map_2#) = []
      pi(filter2_3) = [3]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = [1, 2]
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(gcd_0) = []
      pi(gcd_1) = [1]
      pi(if_gcd_0) = []
      pi(if_gcd_1) = [1]
      pi(map_0) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(le_2) = []
      pi(|0|_0) = []
      pi(minus_2) = [1]
      pi(gcd_2) = [1, 2]
      pi(filter_2) = []
      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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            map_2#_A(x1,x2) = max{0, x2 - 7}
            cons_2_A(x1,x2) = max{7, x1 - 33, x2 + 6}
            map_2#!_A(x1,x2) = max{4, x1 - 1}
            cons_2!_A(x1,x2) = max{3, x1 - 20, x2 - 25}
        
        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 - 16, x2 + 11}
            map_2#!_A(x1,x2) = max{2, x1 - 1, x2 - 25}
            cons_2!_A(x1,x2) = max{1, x1 - 26, x2 - 24}
        
  
    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, r3, r5, r7, r11, r16, r19, r20, r22, r23, r24, r25, r26, r27, r28, r29, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. 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 - 1, x4 - 4}
            false_0_A = 4
            filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 6}
            cons_2_A(x1,x2) = max{x1 + 6, x2 + 2}
            app_A(x1,x2) = max{58, x1 - 19, x2 - 7}
            true_0_A = 0
            cons_0_A = 3
            cons_1_A(x1) = max{0, x1 - 7}
            filter_0_A = 0
            filter_1_A(x1) = max{0, x1 - 7}
            filter2_0_A = 0
            filter2_1_A(x1) = max{0, x1 - 7}
            filter2_2_A(x1,x2) = max{x1 + 18, x2 - 7}
            filter2_3_A(x1,x2,x3) = max{0, x1 - 1, x2 - 26, x3 - 7}
            if_gcd_0_A = 0
            if_gcd_1_A(x1) = max{0, x1 - 7}
            map_0_A = 0
            map_1_A(x1) = max{2, x1 - 7}
            minus_1_A(x1) = x1 + 37
            minus_2_A(x1,x2) = max{19, x1 + 18, x2 - 11}
            pred_0_A = 20
            pred_1_A(x1) = max{19, x1 - 7}
            le_2_A(x1,x2) = max{58, x1 - 26, x2 - 8}
            |0|_0_A = 7
            s_1_A(x1) = max{10, x1 + 7}
            gcd_2_A(x1,x2) = max{13, x1 + 2, x2 + 1}
            filter_2_A(x1,x2) = max{7, x1 - 47, x2 + 4}
            nil_0_A = 8
            filter2_4#!_A(x1,x2,x3,x4) = max{x2 + 2, x3 + 3, x4 - 2}
            false_0!_A = 15
            filter_2#!_A(x1,x2) = max{3, x1 + 2, x2 - 2}
            cons_2!_A(x1,x2) = 0
            app!_A(x1,x2) = max{15, x1 - 1, x2 + 11}
            true_0!_A = 1
            cons_0!_A = 2
            cons_1!_A(x1) = max{1, x1 - 1}
            filter_0!_A = 15
            filter_1!_A(x1) = max{14, x1 - 1}
            filter2_0!_A = 13
            filter2_1!_A(x1) = max{12, x1 - 1}
            filter2_2!_A(x1,x2) = max{15, x2 - 1}
            filter2_3!_A(x1,x2,x3) = max{16, x1 - 1, x2 - 9, x3 - 1}
            if_gcd_0!_A = 3
            if_gcd_1!_A(x1) = max{2, x1}
            map_0!_A = 0
            map_1!_A(x1) = 0
            minus_1!_A(x1) = max{16, x1 + 12}
            minus_2!_A(x1,x2) = max{0, x2 - 1}
            pred_0!_A = 7
            pred_1!_A(x1) = 8
            le_2!_A(x1,x2) = max{16, x1 - 62, x2 - 69}
            |0|_0!_A = 8
            s_1!_A(x1) = max{7, x1 + 1}
            gcd_2!_A(x1,x2) = max{21, x1 - 75, x2 - 68}
            filter_2!_A(x1,x2) = 14
            nil_0!_A = 13
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            filter2_4#_A(x1,x2,x3,x4) = max{90, x2 - 1, x3 - 1, x4 + 54}
            false_0_A = 11
            filter_2#_A(x1,x2) = max{90, x1 - 1, x2 + 54}
            cons_2_A(x1,x2) = max{35, x1 - 54, x2 + 18}
            app_A(x1,x2) = max{53, x1 - 5, x2 + 16}
            true_0_A = 0
            cons_0_A = 0
            cons_1_A(x1) = max{0, x1 - 1}
            filter_0_A = 0
            filter_1_A(x1) = max{0, x1 - 1}
            filter2_0_A = 0
            filter2_1_A(x1) = max{4, x1 - 1}
            filter2_2_A(x1,x2) = max{2, x1 - 6, x2 - 1}
            filter2_3_A(x1,x2,x3) = max{0, x1 - 11, x2 - 6, x3 - 1}
            if_gcd_0_A = 3
            if_gcd_1_A(x1) = max{0, x1 - 1}
            map_0_A = 0
            map_1_A(x1) = max{0, x1 - 1}
            minus_1_A(x1) = max{17, x1 + 16}
            minus_2_A(x1,x2) = max{18, x1 + 11, x2 - 18}
            pred_0_A = 0
            pred_1_A(x1) = max{17, x1 - 14}
            le_2_A(x1,x2) = max{94, x1 - 22, x2 - 27}
            |0|_0_A = 0
            s_1_A(x1) = max{17, x1 + 14}
            gcd_2_A(x1,x2) = max{20, x1 + 2, x2 + 1}
            filter_2_A(x1,x2) = max{35, x1 - 20, x2 - 19}
            nil_0_A = 35
            filter2_4#!_A(x1,x2,x3,x4) = x4 + 60
            false_0!_A = 101
            filter_2#!_A(x1,x2) = x2 + 59
            cons_2!_A(x1,x2) = max{57, x1 + 3}
            app!_A(x1,x2) = max{95, x1 - 1, x2 + 58}
            true_0!_A = 99
            cons_0!_A = 1
            cons_1!_A(x1) = max{0, x1 - 1}
            filter_0!_A = 1
            filter_1!_A(x1) = max{96, x1 - 19}
            filter2_0!_A = 2
            filter2_1!_A(x1) = max{1, x1 - 3}
            filter2_2!_A(x1,x2) = max{2, x1 - 2, x2 + 1}
            filter2_3!_A(x1,x2,x3) = max{96, x1 - 1, x2 + 2}
            if_gcd_0!_A = 2
            if_gcd_1!_A(x1) = max{1, x1 - 96}
            map_0!_A = 1
            map_1!_A(x1) = max{60, x1 + 58}
            minus_1!_A(x1) = max{0, x1 - 1}
            minus_2!_A(x1,x2) = x1 + 4
            pred_0!_A = 1
            pred_1!_A(x1) = 2
            le_2!_A(x1,x2) = max{96, x1 - 18, x2 - 122}
            |0|_0!_A = 100
            s_1!_A(x1) = max{0, x1 - 122}
            gcd_2!_A(x1,x2) = max{98, x1 - 103, x2 - 19}
            filter_2!_A(x1,x2) = 0
            nil_0!_A = 1
        
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [1, 2]
      pi(true_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) = [1, 2, 3]
      pi(if_gcd_0) = []
      pi(if_gcd_1) = [1]
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_1) = [1]
      pi(minus_2) = [1]
      pi(pred_0) = []
      pi(pred_1) = [1]
      pi(le_2) = []
      pi(|0|_0) = []
      pi(s_1) = [1]
      pi(gcd_2) = [1, 2]
      pi(filter_2) = []
      pi(nil_0) = []

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:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gcd_2#_A(x1,x2) = max{0, x1 - 32, x2 - 61}
            s_1_A(x1) = x1 + 30
            if_gcd_3#_A(x1,x2,x3) = max{0, x1 - 48, x2 - 48, x3 - 61}
            le_2_A(x1,x2) = max{47, x1 - 31, x2 - 95}
            false_0_A = 0
            minus_2_A(x1,x2) = max{15, x1 + 1, x2 - 32}
            true_0_A = 4
            |0|_0_A = 30
            pred_1_A(x1) = max{2, x1 - 16}
            gcd_2#!_A(x1,x2) = max{8, x1 - 19, x2 - 21}
            s_1!_A(x1) = x1 + 12
            if_gcd_3#!_A(x1,x2,x3) = max{6, x2 - 20, x3 - 21}
            le_2!_A(x1,x2) = max{130, x1 - 188, x2 - 128}
            false_0!_A = 6
            minus_2!_A(x1,x2) = max{14, x1 - 157, x2 - 34}
            true_0!_A = 132
            |0|_0!_A = 133
            pred_1!_A(x1) = max{13, x1 - 159}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gcd_2#_A(x1,x2) = max{0, x1 - 16, x2 - 24}
            s_1_A(x1) = max{14, x1 + 9}
            if_gcd_3#_A(x1,x2,x3) = max{0, x1 - 25, x2 - 23, x3 - 24}
            le_2_A(x1,x2) = max{24, x1 - 15, x2 - 12}
            false_0_A = 0
            minus_2_A(x1,x2) = max{8, x1 + 1, x2 - 16}
            true_0_A = 0
            |0|_0_A = 0
            pred_1_A(x1) = max{5, x1 - 9}
            gcd_2#!_A(x1,x2) = max{25, x1 - 68, x2 - 72}
            s_1!_A(x1) = max{29, x1 - 57}
            if_gcd_3#!_A(x1,x2,x3) = max{25, x1 - 48, x2 - 70, x3 - 72}
            le_2!_A(x1,x2) = max{30, x1 - 55, x2 - 33}
            false_0!_A = 31
            minus_2!_A(x1,x2) = max{27, x1 - 8, x2 - 58}
            true_0!_A = 26
            |0|_0!_A = 25
            pred_1!_A(x1) = max{26, x1 - 39}
        
  
    partial status:
  
      pi(gcd_2#) = []
      pi(s_1) = [1]
      pi(if_gcd_3#) = []
      pi(le_2) = [1]
      pi(false_0) = []
      pi(minus_2) = [1, 2]
      pi(true_0) = []
      pi(|0|_0) = []
      pi(pred_1) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p2: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: gcd_2#(s_1(x),s_1(y)) -> if_gcd_3#(le_2(y,x),s_1(x),s_1(y))
p2: if_gcd_3#(true_0(),s_1(x),s_1(y)) -> gcd_2#(minus_2(x,y),s_1(y))

and R consists of:

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

The set of usable rules consists of

  r22, r23, r24, r25, r26, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gcd_2#_A(x1,x2) = max{5, x1 - 5, x2 - 10}
            s_1_A(x1) = max{9, x1 + 7}
            if_gcd_3#_A(x1,x2,x3) = max{5, x1 - 3, x2 - 11, x3 - 10}
            le_2_A(x1,x2) = 8
            true_0_A = 1
            minus_2_A(x1,x2) = max{6, x1 + 1, x2 + 1}
            |0|_0_A = 23
            false_0_A = 3
            pred_1_A(x1) = max{10, x1 - 7}
            gcd_2#!_A(x1,x2) = 11
            s_1!_A(x1) = 14
            if_gcd_3#!_A(x1,x2,x3) = 10
            le_2!_A(x1,x2) = max{16, x1 - 10, x2 - 25}
            true_0!_A = 12
            minus_2!_A(x1,x2) = 9
            |0|_0!_A = 13
            false_0!_A = 8
            pred_1!_A(x1) = 1
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gcd_2#_A(x1,x2) = max{0, x1 - 28, x2 - 41}
            s_1_A(x1) = max{26, x1 + 16}
            if_gcd_3#_A(x1,x2,x3) = max{0, x1 - 41, x2 - 40, x3 - 41}
            le_2_A(x1,x2) = max{40, x1 - 27, x2 - 29}
            true_0_A = 0
            minus_2_A(x1,x2) = max{15, x1 + 4, x2 - 28}
            |0|_0_A = 17
            false_0_A = 2
            pred_1_A(x1) = max{5, x1 - 16}
            gcd_2#!_A(x1,x2) = max{11, x1 - 6, x2 - 5}
            s_1!_A(x1) = 5
            if_gcd_3#!_A(x1,x2,x3) = max{x1 + 39, x2 + 6, x3 + 6}
            le_2!_A(x1,x2) = max{x1 + 12, x2 + 11}
            true_0!_A = 10
            minus_2!_A(x1,x2) = max{11, x2 - 11}
            |0|_0!_A = 11
            false_0!_A = 2
            pred_1!_A(x1) = 4
        
  
    partial status:
  
      pi(gcd_2#) = [1]
      pi(s_1) = [1]
      pi(if_gcd_3#) = [2]
      pi(le_2) = [1]
      pi(true_0) = []
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(false_0) = []
      pi(pred_1) = [1]

The following rules are removed:

  p1, p2

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

  r22, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = max{0, x1 - 5, x2 - 5}
            s_1_A(x1) = max{25, x1 + 4}
            le_2_A(x1,x2) = max{33, x1 - 26, x2 - 5}
            |0|_0_A = 4
            true_0_A = 0
            false_0_A = 2
            le_2#!_A(x1,x2) = max{10, x2 - 79}
            s_1!_A(x1) = max{9, x1 - 80}
            le_2!_A(x1,x2) = max{10, x1 - 80, x2 - 83}
            |0|_0!_A = 10
            true_0!_A = 1
            false_0!_A = 4
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            le_2#_A(x1,x2) = max{0, x1 - 4, x2 - 4}
            s_1_A(x1) = max{24, x1 + 3}
            le_2_A(x1,x2) = max{17, x1 - 38, x2 - 4}
            |0|_0_A = 0
            true_0_A = 0
            false_0_A = 2
            le_2#!_A(x1,x2) = max{7, x1 - 25, x2 - 55}
            s_1!_A(x1) = max{6, x1 - 56}
            le_2!_A(x1,x2) = max{8, x1 - 25, x2 - 25}
            |0|_0!_A = 5
            true_0!_A = 4
            false_0!_A = 4
        
  
    partial status:
  
      pi(le_2#) = [1]
      pi(s_1) = [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.

-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_2#_A(x1,x2) = max{0, x2 - 4}
            s_1_A(x1) = max{19, x1 + 3}
            minus_2#!_A(x1,x2) = max{0, x2 - 20}
            s_1!_A(x1) = max{10, x1 - 125}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus_2#_A(x1,x2) = max{0, x2 - 2}
            s_1_A(x1) = max{12, x1 + 1}
            minus_2#!_A(x1,x2) = max{8, x1 - 1, x2 - 32}
            s_1!_A(x1) = max{7, x1 - 31}
        
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.