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

  r29

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{20, x2 + 1}
        filter_1_A(x1) = 30
        filter_2#_A(x1,x2) = max{18, x2 - 20}
        cons_2_A(x1,x2) = max{61, x1 + 22, x2 + 38}
        map_1_A(x1) = x1 + 66
        map_2#_A(x1,x2) = max{19, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{2, x1 - 67, x2 - 38}
        filter2_4#_A(x1,x2,x3,x4) = max{19, x4 - 1}
        false_0_A = 96
        app_A(x1,x2) = max{x1 - 38, x2 + 65}
        true_0_A = 63
        gcd_2_A(x1,x2) = max{36, x1}
        s_1_A(x1) = 35
        |0|_0_A = 97
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app#_A(x1,x2) = x1
            filter_1_A(x1) = x1 + 5
            filter_2#_A(x1,x2) = x1
            cons_2_A(x1,x2) = x1 + 11
            map_1_A(x1) = x1 + 9
            map_2#_A(x1,x2) = x1
            filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 6
            filter2_4#_A(x1,x2,x3,x4) = x2
            false_0_A() = 16
            app_A(x1,x2) = x1 + x2 + 3
            true_0_A() = 11
            gcd_2_A(x1,x2) = x1 + 9
            s_1_A(x1) = x1 + 5
            |0|_0_A() = 1
            app#!_A(x1,x2) = x1 + 27
            filter_1!_A(x1) = 1
            filter_2#!_A(x1,x2) = x2 + 33
            cons_2!_A(x1,x2) = x1 + 14
            map_1!_A(x1) = x1 + 16
            map_2#!_A(x1,x2) = x2 + 15
            filter2_3!_A(x1,x2,x3) = 35
            filter2_4#!_A(x1,x2,x3,x4) = x4 + 34
            false_0!_A() = 1
            app!_A(x1,x2) = x1 + 7
            true_0!_A() = 13
            gcd_2!_A(x1,x2) = x1 + 2
            s_1!_A(x1) = 7
            |0|_0!_A() = 14
  
    partial status:
  
      pi(app#) = [2]
      pi(filter_1) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(gcd_2) = [1]
      pi(s_1) = []
      pi(|0|_0) = []

The following rules are removed:

  p1, p2, p3, p4, p5, p6, p7, p8, p9

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

  r7, r9, r14, r16, r20, r23, r25, r26, r27, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gcd_2#_A(x1,x2) = max{x1 + 1, x2 - 14}
        s_1_A(x1) = x1 + 34
        if_gcd_3#_A(x1,x2,x3) = max{35, x2 - 1, x3 - 14}
        le_2_A(x1,x2) = max{75, x1 + 36, x2 + 33}
        false_0_A = 71
        minus_2_A(x1,x2) = max{17, x1 + 1}
        true_0_A = 76
        app_A(x1,x2) = max{x1 + 11, x2 + 78}
        filter2_2_A(x1,x2) = max{76, x1 + 1}
        filter2_3_A(x1,x2,x3) = max{2, x1 - 5}
        gcd_0_A = 80
        gcd_1_A(x1) = max{79, x1 + 67}
        le_0_A = 1
        le_1_A(x1) = x1 + 37
        map_0_A = 80
        map_1_A(x1) = 79
        pred_0_A = 1
        pred_1_A(x1) = max{2, x1 - 18}
        |0|_0_A = 13
        filter_2_A(x1,x2) = max{81, x1 + 4}
        nil_0_A = 80
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            gcd_2#_A(x1,x2) = 0
            s_1_A(x1) = x1 + 12
            if_gcd_3#_A(x1,x2,x3) = 0
            le_2_A(x1,x2) = x1 + 3
            false_0_A() = 1
            minus_2_A(x1,x2) = x1 + 7
            true_0_A() = 5
            app_A(x1,x2) = x1 + x2 + 2
            filter2_2_A(x1,x2) = x1 + x2 + 7
            filter2_3_A(x1,x2,x3) = x1 + x2 + x3
            gcd_0_A() = 1
            gcd_1_A(x1) = x1 + 1
            le_0_A() = 1
            le_1_A(x1) = x1
            map_0_A() = 0
            map_1_A(x1) = x1
            pred_0_A() = 1
            pred_1_A(x1) = x1
            |0|_0_A() = 1
            filter_2_A(x1,x2) = x2 + 19
            nil_0_A() = 9
            gcd_2#!_A(x1,x2) = x1 + x2 + 1
            s_1!_A(x1) = x1 + 3
            if_gcd_3#!_A(x1,x2,x3) = x2 + x3
            le_2!_A(x1,x2) = x1 + 2
            false_0!_A() = 4
            minus_2!_A(x1,x2) = 18
            true_0!_A() = 19
            app!_A(x1,x2) = 17
            filter2_2!_A(x1,x2) = 15
            filter2_3!_A(x1,x2,x3) = x1 + x3 + 14
            gcd_0!_A() = 0
            gcd_1!_A(x1) = 16
            le_0!_A() = 0
            le_1!_A(x1) = 16
            map_0!_A() = 19
            map_1!_A(x1) = x1 + 18
            pred_0!_A() = 0
            pred_1!_A(x1) = x1 + 18
            |0|_0!_A() = 0
            filter_2!_A(x1,x2) = x2 + 5
            nil_0!_A() = 13
  
    partial status:
  
      pi(gcd_2#) = []
      pi(s_1) = [1]
      pi(if_gcd_3#) = []
      pi(le_2) = []
      pi(false_0) = []
      pi(minus_2) = []
      pi(true_0) = []
      pi(app) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(gcd_0) = []
      pi(gcd_1) = [1]
      pi(le_0) = []
      pi(le_1) = [1]
      pi(map_0) = []
      pi(map_1) = []
      pi(pred_0) = []
      pi(pred_1) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      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: 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

  r18, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le_2#_A(x1,x2) = max{x1 + 6, x2 + 6}
        s_1_A(x1) = x1 + 4
        app_A(x1,x2) = max{x1 - 10, x2 + 20}
        minus_0_A = 8
        minus_1_A(x1) = x1 + 17
        filter_2_A(x1,x2) = max{6, x2 - 32}
        nil_0_A = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            le_2#_A(x1,x2) = 0
            s_1_A(x1) = 3
            app_A(x1,x2) = x1 + x2 + 1
            minus_0_A() = 12
            minus_1_A(x1) = x1 + 11
            filter_2_A(x1,x2) = x1 + x2 + 10
            nil_0_A() = 6
            le_2#!_A(x1,x2) = x2 + 6
            s_1!_A(x1) = 5
            app!_A(x1,x2) = x1 + 13
            minus_0!_A() = 0
            minus_1!_A(x1) = 24
            filter_2!_A(x1,x2) = x2 + 7
            nil_0!_A() = 0
  
    partial status:
  
      pi(le_2#) = []
      pi(s_1) = []
      pi(app) = [2]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus_2#_A(x1,x2) = x2 + 3
        s_1_A(x1) = max{24, x1 + 3}
        filter_2_A(x1,x2) = 25
        nil_0_A = 24
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            minus_2#_A(x1,x2) = 0
            s_1_A(x1) = 2
            filter_2_A(x1,x2) = x2 + 3
            nil_0_A() = 3
            minus_2#!_A(x1,x2) = 0
            s_1!_A(x1) = x1 + 18
            filter_2!_A(x1,x2) = 9
            nil_0!_A() = 1
  
    partial status:
  
      pi(minus_2#) = [2]
      pi(s_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.