YES
We show the termination of the TRS R:

  |app'|(|app'|(minus(),x),|0|()) -> x
  |app'|(|app'|(minus(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(|app'|(minus(),x),y)
  |app'|(|app'|(minus(),|app'|(|app'|(minus(),x),y)),z) -> |app'|(|app'|(minus(),x),|app'|(|app'|(plus(),y),z))
  |app'|(|app'|(quot(),|0|()),|app'|(s(),y)) -> |0|()
  |app'|(|app'|(quot(),|app'|(s(),x)),|app'|(s(),y)) -> |app'|(s(),|app'|(|app'|(quot(),|app'|(|app'|(minus(),x),y)),|app'|(s(),y)))
  |app'|(|app'|(plus(),|0|()),y) -> y
  |app'|(|app'|(plus(),|app'|(s(),x)),y) -> |app'|(s(),|app'|(|app'|(plus(),x),y))
  |app'|(|app'|(app(),nil()),k) -> k
  |app'|(|app'|(app(),l),nil()) -> l
  |app'|(|app'|(app(),|app'|(|app'|(cons(),x),l)),k) -> |app'|(|app'|(cons(),x),|app'|(|app'|(app(),l),k))
  |app'|(sum(),|app'|(|app'|(cons(),x),nil())) -> |app'|(|app'|(cons(),x),nil())
  |app'|(sum(),|app'|(|app'|(cons(),x),|app'|(|app'|(cons(),y),l))) -> |app'|(sum(),|app'|(|app'|(cons(),|app'|(|app'|(plus(),x),y)),l))
  |app'|(sum(),|app'|(|app'|(app(),l),|app'|(|app'|(cons(),x),|app'|(|app'|(cons(),y),k)))) -> |app'|(sum(),|app'|(|app'|(app(),l),|app'|(sum(),|app'|(|app'|(cons(),x),|app'|(|app'|(cons(),y),k)))))
  |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'|(app_0(),y) -> app_1(y)
  |app'|(app_1(x1),y) -> app_2(x1,y)
  |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'|(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'|(plus_0(),y) -> plus_1(y)
  |app'|(plus_1(x1),y) -> plus_2(x1,y)
  |app'|(quot_0(),y) -> quot_1(y)
  |app'|(quot_1(x1),y) -> quot_2(x1,y)
  |app'|(s_0(),y) -> s_1(y)
  |app'|(sum_0(),y) -> sum_1(y)
  minus_2(x,|0|_0()) -> x
  minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
  minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
  quot_2(|0|_0(),s_1(y)) -> |0|_0()
  quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
  plus_2(|0|_0(),y) -> y
  plus_2(s_1(x),y) -> s_1(plus_2(x,y))
  app_2(nil_0(),k) -> k
  app_2(l,nil_0()) -> l
  app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
  sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
  sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
  sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
  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'|#(app_1(x1),y) -> app_2#(x1,y)
p2: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p3: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p5: |app'|#(minus_1(x1),y) -> minus_2#(x1,y)
p6: |app'|#(plus_1(x1),y) -> plus_2#(x1,y)
p7: |app'|#(quot_1(x1),y) -> quot_2#(x1,y)
p8: |app'|#(sum_0(),y) -> sum_1#(y)
p9: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p10: minus_2#(minus_2(x,y),z) -> minus_2#(x,plus_2(y,z))
p11: minus_2#(minus_2(x,y),z) -> plus_2#(y,z)
p12: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))
p13: quot_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p14: plus_2#(s_1(x),y) -> plus_2#(x,y)
p15: app_2#(cons_2(x,l),k) -> app_2#(l,k)
p16: sum_1#(cons_2(x,cons_2(y,l))) -> sum_1#(cons_2(plus_2(x,y),l))
p17: sum_1#(cons_2(x,cons_2(y,l))) -> plus_2#(x,y)
p18: sum_1#(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1#(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
p19: sum_1#(app_2(l,cons_2(x,cons_2(y,k)))) -> app_2#(l,sum_1(cons_2(x,cons_2(y,k))))
p20: sum_1#(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1#(cons_2(x,cons_2(y,k)))
p21: map_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p22: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p23: filter_2#(f,cons_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p24: filter_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p25: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p26: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p21, p22, p23, p24, p25, p26}
  {p18}
  {p15}
  {p12}
  {p9, p10}
  {p16}
  {p14}


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p2: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p4: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: filter_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p7: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
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'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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, r39

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) = x1 + 1
          cons_2_A(x1,x2) = 2
          |app'|#_A(x1,x2) = x1
          map_1_A(x1) = x1 + 4
          filter2_3_A(x1,x2,x3) = x2 + x3 + 6
          filter2_4#_A(x1,x2,x3,x4) = x2 + 5
          false_0_A() = 4
          filter_2#_A(x1,x2) = x1 + 5
          filter_1_A(x1) = x1 + 6
          |app'|_A(x1,x2) = x1 + x2 + 6
          true_0_A() = 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 3
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 3
          filter_0_A() = 1
          filter_2_A(x1,x2) = 3
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x2 + 2
          filter2_4_A(x1,x2,x3,x4) = 3
          map_0_A() = 1
          map_2_A(x1,x2) = 3
          minus_0_A() = 1
          minus_1_A(x1) = x1 + 5
          minus_2_A(x1,x2) = x1 + 4
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 2
          plus_2_A(x1,x2) = x2 + 7
          quot_0_A() = 4
          quot_1_A(x1) = x1 + 3
          quot_2_A(x1,x2) = 2
          s_0_A() = 1
          s_1_A(x1) = x1
          sum_0_A() = 7
          sum_1_A(x1) = 6
          |0|_0_A() = 1
          nil_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x1 - 48, x2 - 58}
          cons_2_A(x1,x2) = max{x1 + 13, x2}
          |app'|#_A(x1,x2) = max{0, x1 - 61, x2 - 58}
          map_1_A(x1) = x1 + 13
          filter2_3_A(x1,x2,x3) = max{x1 - 57, x2 + 13, x3 + 60}
          filter2_4#_A(x1,x2,x3,x4) = max{0, x1 - 118, x2 - 61, x3 - 58, x4 - 71}
          false_0_A = 61
          filter_2#_A(x1,x2) = max{0, x1 - 61, x2 - 71}
          filter_1_A(x1) = x1 + 13
          |app'|_A(x1,x2) = max{x1 - 19, x2 + 60}
          true_0_A = 61
          app_0_A = 20
          app_1_A(x1) = x1 + 60
          app_2_A(x1,x2) = max{x1 + 41, x2 + 9}
          cons_0_A = 13
          cons_1_A(x1) = x1 + 32
          filter_0_A = 13
          filter_2_A(x1,x2) = max{x1 - 19, x2 + 60}
          filter2_0_A = 13
          filter2_1_A(x1) = max{13, x1 - 1}
          filter2_2_A(x1,x2) = max{x1 - 20, x2 + 36}
          filter2_4_A(x1,x2,x3,x4) = max{x1 - 76, x2 - 19, x3 + 13, x4 + 60}
          map_0_A = 13
          map_2_A(x1,x2) = max{x1 - 6, x2 + 60}
          minus_0_A = 13
          minus_1_A(x1) = x1 + 32
          minus_2_A(x1,x2) = max{59, x1}
          plus_0_A = 13
          plus_1_A(x1) = max{13, x1 - 41}
          plus_2_A(x1,x2) = max{18, x1 - 60, x2}
          quot_0_A = 13
          quot_1_A(x1) = max{13, x1 - 1}
          quot_2_A(x1,x2) = max{60, x1 - 60, x2 - 2}
          s_0_A = 13
          s_1_A(x1) = x1
          sum_0_A = 13
          sum_1_A(x1) = max{31, x1}
          |0|_0_A = 1
          nil_0_A = 76
          map_2#!_A(x1,x2) = x1 + 70
          cons_2!_A(x1,x2) = max{0, x1 - 79}
          |app'|#!_A(x1,x2) = max{129, x1 + 56, x2 - 76}
          map_1!_A(x1) = x1 + 70
          filter2_3!_A(x1,x2,x3) = max{4, x1 - 35, x2 - 16, x3 - 79}
          filter2_4#!_A(x1,x2,x3,x4) = max{128, x3 - 78, x4 - 90}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{128, x2 - 90}
          filter_1!_A(x1) = max{127, x1 - 76}
          |app'|!_A(x1,x2) = max{0, x1 - 14, x2 - 76}
          true_0!_A = 1
          app_0!_A = 1
          app_1!_A(x1) = max{2, x1 - 78}
          app_2!_A(x1,x2) = max{3, x1 - 79, x2 - 88}
          cons_0!_A = 1
          cons_1!_A(x1) = max{0, x1 - 76}
          filter_0!_A = 128
          filter_2!_A(x1,x2) = max{126, x1 - 15, x2 + 48}
          filter2_0!_A = 1
          filter2_1!_A(x1) = x1 + 4
          filter2_2!_A(x1,x2) = max{3, x1 - 36}
          filter2_4!_A(x1,x2,x3,x4) = max{126, x1 + 1, x2 - 15, x3 - 1, x4 + 48}
          map_0!_A = 1
          map_2!_A(x1,x2) = max{0, x1 - 99}
          minus_0!_A = 1
          minus_1!_A(x1) = max{2, x1 - 60}
          minus_2!_A(x1,x2) = max{18, x1 - 60, x2}
          plus_0!_A = 128
          plus_1!_A(x1) = max{127, x1 - 61}
          plus_2!_A(x1,x2) = max{127, x1 - 61}
          quot_0!_A = 2
          quot_1!_A(x1) = max{1, x1 - 1}
          quot_2!_A(x1,x2) = x2 + 17
          s_0!_A = 0
          s_1!_A(x1) = 0
          sum_0!_A = 1
          sum_1!_A(x1) = max{126, x1 - 75}
          |0|_0!_A = 1
          nil_0!_A = 125
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(|app'|#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = []
      pi(|app'|) = [1, 2]
      pi(true_0) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_4) = []
      pi(map_0) = []
      pi(map_2) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = []
      pi(plus_0) = []
      pi(plus_1) = [1]
      pi(plus_2) = []
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(quot_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(sum_0) = []
      pi(sum_1) = []
      pi(|0|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p4, p6, p7



-- 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'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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) = x2 + 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = 0
          cons_2_A(x1,x2) = max{7, x1 - 29}
          map_2#!_A(x1,x2) = max{17, x1 - 1, x2 - 18}
          cons_2!_A(x1,x2) = max{33, x1 + 16, x2 - 6}
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_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'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r8, r11, r13, r15, r17, r19, r20, r24, r26, r27, r28, r29, r30, r31, r32, r33, r34, r36

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) = x2 + x4 + 4
          false_0_A() = 3
          filter_2#_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x2 + 3
          |app'|_A(x1,x2) = x1 + x2 + 6
          true_0_A() = 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 3
          app_2_A(x1,x2) = x1 + x2 + 2
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 5
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 4
          map_0_A() = 2
          map_1_A(x1) = x1 + 1
          minus_0_A() = 6
          minus_1_A(x1) = 5
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 4
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 4
          s_0_A() = 1
          s_1_A(x1) = 2
          sum_0_A() = 6
          sum_1_A(x1) = 5
          quot_2_A(x1,x2) = 3
          |0|_0_A() = 1
          plus_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
          map_2_A(x1,x2) = x1 + 4
          filter_2_A(x1,x2) = x1 + 4
  
    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 - 80, x4 - 18}
          false_0_A = 37
          filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 18}
          cons_2_A(x1,x2) = max{17, x1 - 62, x2}
          |app'|_A(x1,x2) = max{37, x1 - 4, x2 + 15}
          true_0_A = 37
          app_0_A = 0
          app_1_A(x1) = x1 + 15
          app_2_A(x1,x2) = max{33, x1 + 1, x2 + 15}
          cons_0_A = 3
          cons_1_A(x1) = max{3, x1 - 4}
          filter_0_A = 0
          filter_1_A(x1) = max{3, x1 - 6}
          filter2_0_A = 0
          filter2_1_A(x1) = max{0, x1 - 6}
          filter2_2_A(x1,x2) = max{0, x1 - 42, x2 - 6}
          map_0_A = 0
          map_1_A(x1) = max{3, x1 - 4}
          minus_0_A = 0
          minus_1_A(x1) = x1 + 5
          plus_0_A = 0
          plus_1_A(x1) = max{0, x1 - 19}
          quot_0_A = 0
          quot_1_A(x1) = max{3, x1 - 6}
          s_0_A = 0
          s_1_A(x1) = max{12, x1}
          sum_0_A = 0
          sum_1_A(x1) = max{18, x1}
          quot_2_A(x1,x2) = max{24, x1 + 11, x2 - 13}
          |0|_0_A = 0
          plus_2_A(x1,x2) = max{15, x1 - 13, x2}
          nil_0_A = 34
          map_2_A(x1,x2) = max{16, x1 - 26, x2 + 15}
          filter_2_A(x1,x2) = max{33, x1 - 47, x2 + 15}
          filter2_4#!_A(x1,x2,x3,x4) = max{123, x3 + 3, x4 + 120}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{x1 + 119, x2 + 21}
          cons_2!_A(x1,x2) = max{122, x1 + 2}
          |app'|!_A(x1,x2) = max{120, x1 + 5, x2 - 40}
          true_0!_A = 1
          app_0!_A = 5
          app_1!_A(x1) = max{4, x1 - 41}
          app_2!_A(x1,x2) = max{123, x1 + 65, x2 - 19}
          cons_0!_A = 6
          cons_1!_A(x1) = max{7, x1 - 1}
          filter_0!_A = 1
          filter_1!_A(x1) = max{6, x1 - 41}
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{121, x1 - 39}
          filter2_2!_A(x1,x2) = max{119, x1 - 38}
          map_0!_A = 5
          map_1!_A(x1) = max{6, x1 - 1}
          minus_0!_A = 122
          minus_1!_A(x1) = max{121, x1 + 1}
          plus_0!_A = 1
          plus_1!_A(x1) = max{2, x1 - 41}
          quot_0!_A = 1
          quot_1!_A(x1) = x1 + 11
          s_0!_A = 1
          s_1!_A(x1) = max{0, x1 - 40}
          sum_0!_A = 122
          sum_1!_A(x1) = 121
          quot_2!_A(x1,x2) = max{x1 + 11, x2 - 15}
          |0|_0!_A = 0
          plus_2!_A(x1,x2) = max{123, x1 - 13, x2 - 14}
          nil_0!_A = 137
          map_2!_A(x1,x2) = max{123, x1 + 1, x2 + 102}
          filter_2!_A(x1,x2) = max{138, x1 + 4, x2 - 35}
  
    partial status:
  
      pi(filter2_4#) = [4]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(|app'|) = [1, 2]
      pi(true_0) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(app_2) = [2]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = [1]
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(sum_0) = []
      pi(sum_1) = []
      pi(quot_2) = []
      pi(|0|_0) = []
      pi(plus_2) = []
      pi(nil_0) = []
      pi(map_2) = []
      pi(filter_2) = [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: sum_1#(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1#(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r26, r27, r28, r29, r30, r31, r32

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:
          sum_1#_A(x1) = x1 + 1
          app_2_A(x1,x2) = x1 + x2 + 7
          cons_2_A(x1,x2) = 6
          sum_1_A(x1) = 6
          plus_2_A(x1,x2) = x2 + 3
          |0|_0_A() = 0
          s_1_A(x1) = 2
          nil_0_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum_1#_A(x1) = max{0, x1 - 37}
          app_2_A(x1,x2) = max{x1 + 8, x2 + 17}
          cons_2_A(x1,x2) = max{18, x1 - 9, x2}
          sum_1_A(x1) = max{19, x1}
          plus_2_A(x1,x2) = max{x1 - 35, x2}
          |0|_0_A = 32
          s_1_A(x1) = max{0, x1 - 15}
          nil_0_A = 5
          sum_1#!_A(x1) = 0
          app_2!_A(x1,x2) = 0
          cons_2!_A(x1,x2) = max{43, x2 + 38}
          sum_1!_A(x1) = 47
          plus_2!_A(x1,x2) = max{30, x1 - 20, x2 - 1}
          |0|_0!_A = 32
          s_1!_A(x1) = max{49, x1}
          nil_0!_A = 42
  
    partial status:
  
      pi(sum_1#) = [1]
      pi(app_2) = [2]
      pi(cons_2) = []
      pi(sum_1) = []
      pi(plus_2) = []
      pi(|0|_0) = []
      pi(s_1) = []
      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: app_2#(cons_2(x,l),k) -> app_2#(l,k)

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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:
          app_2#_A(x1,x2) = x1
          cons_2_A(x1,x2) = x1 + x2 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app_2#_A(x1,x2) = max{0, x1 - 3}
          cons_2_A(x1,x2) = max{4, x1 - 24, x2 + 2}
          app_2#!_A(x1,x2) = max{0, x1 - 16}
          cons_2!_A(x1,x2) = max{3, x1 - 143, x2 - 15}
  
    partial status:
  
      pi(app_2#) = [1]
      pi(cons_2) = [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: quot_2#(s_1(x),s_1(y)) -> quot_2#(minus_2(x,y),s_1(y))

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r21, r22, r23, 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:
          quot_2#_A(x1,x2) = x1 + 1
          s_1_A(x1) = x1 + 10
          minus_2_A(x1,x2) = x1 + 8
          |0|_0_A() = 0
          plus_2_A(x1,x2) = x1 + x2 + 11
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quot_2#_A(x1,x2) = max{0, x1 - 16, x2 - 15}
          s_1_A(x1) = max{13, x1}
          minus_2_A(x1,x2) = max{15, x1, x2 - 139}
          |0|_0_A = 15
          plus_2_A(x1,x2) = max{140, x1 - 137, x2}
          quot_2#!_A(x1,x2) = 1
          s_1!_A(x1) = 19
          minus_2!_A(x1,x2) = max{18, x1 + 3, x2 - 595}
          |0|_0!_A = 1
          plus_2!_A(x1,x2) = max{195, x1 - 135, x2 - 1}
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = [1]
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(plus_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: minus_2#(s_1(x),s_1(y)) -> minus_2#(x,y)
p2: minus_2#(minus_2(x,y),z) -> minus_2#(x,plus_2(y,z))

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  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:
          minus_2#_A(x1,x2) = 1
          s_1_A(x1) = 3
          minus_2_A(x1,x2) = 2
          plus_2_A(x1,x2) = x2 + 4
          |0|_0_A() = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_2#_A(x1,x2) = max{0, x1 - 16, x2 - 60}
          s_1_A(x1) = max{6, x1}
          minus_2_A(x1,x2) = max{x1 + 14, x2 - 60}
          plus_2_A(x1,x2) = max{59, x1 - 16, x2}
          |0|_0_A = 6
          minus_2#!_A(x1,x2) = x1 + 36
          s_1!_A(x1) = max{19, x1 - 85}
          minus_2!_A(x1,x2) = max{14, x1 + 1}
          plus_2!_A(x1,x2) = max{21, x1 - 25, x2 - 1}
          |0|_0!_A = 21
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(plus_2) = []
      pi(|0|_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

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

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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:
          minus_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus_2#_A(x1,x2) = max{0, x1 - 24, x2 - 24}
          s_1_A(x1) = max{33, x1 + 23}
          minus_2#!_A(x1,x2) = max{21, x1 - 34, x2 - 1}
          s_1!_A(x1) = max{44, x1 - 133}
  
    partial status:
  
      pi(minus_2#) = [1]
      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: sum_1#(cons_2(x,cons_2(y,l))) -> sum_1#(cons_2(plus_2(x,y),l))

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r26, r27, r36

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:
          sum_1#_A(x1) = 1
          cons_2_A(x1,x2) = x1 + 4
          plus_2_A(x1,x2) = x1 + x2 + 8
          |0|_0_A() = 1
          s_1_A(x1) = 2
          filter_2_A(x1,x2) = x2 + 2
          nil_0_A() = 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          sum_1#_A(x1) = max{0, x1 - 55}
          cons_2_A(x1,x2) = max{40, x1 - 74, x2 + 14}
          plus_2_A(x1,x2) = max{113, x1 - 77, x2 + 1}
          |0|_0_A = 4
          s_1_A(x1) = max{4, x1 - 114}
          filter_2_A(x1,x2) = max{354, x1 - 187, x2 - 149}
          nil_0_A = 77
          sum_1#!_A(x1) = x1 + 76
          cons_2!_A(x1,x2) = max{105, x2 - 247}
          plus_2!_A(x1,x2) = max{106, x1 - 119}
          |0|_0!_A = 1
          s_1!_A(x1) = max{102, x1 - 6}
          filter_2!_A(x1,x2) = max{98, x2 - 78}
          nil_0!_A = 76
  
    partial status:
  
      pi(sum_1#) = []
      pi(cons_2) = []
      pi(plus_2) = [1, 2]
      pi(|0|_0) = []
      pi(s_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: plus_2#(s_1(x),y) -> plus_2#(x,y)

and R consists of:

r1: |app'|(app_0(),y) -> app_1(y)
r2: |app'|(app_1(x1),y) -> app_2(x1,y)
r3: |app'|(cons_0(),y) -> cons_1(y)
r4: |app'|(cons_1(x1),y) -> cons_2(x1,y)
r5: |app'|(filter_0(),y) -> filter_1(y)
r6: |app'|(filter_1(x1),y) -> filter_2(x1,y)
r7: |app'|(filter2_0(),y) -> filter2_1(y)
r8: |app'|(filter2_1(x1),y) -> filter2_2(x1,y)
r9: |app'|(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r10: |app'|(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r11: |app'|(map_0(),y) -> map_1(y)
r12: |app'|(map_1(x1),y) -> map_2(x1,y)
r13: |app'|(minus_0(),y) -> minus_1(y)
r14: |app'|(minus_1(x1),y) -> minus_2(x1,y)
r15: |app'|(plus_0(),y) -> plus_1(y)
r16: |app'|(plus_1(x1),y) -> plus_2(x1,y)
r17: |app'|(quot_0(),y) -> quot_1(y)
r18: |app'|(quot_1(x1),y) -> quot_2(x1,y)
r19: |app'|(s_0(),y) -> s_1(y)
r20: |app'|(sum_0(),y) -> sum_1(y)
r21: minus_2(x,|0|_0()) -> x
r22: minus_2(s_1(x),s_1(y)) -> minus_2(x,y)
r23: minus_2(minus_2(x,y),z) -> minus_2(x,plus_2(y,z))
r24: quot_2(|0|_0(),s_1(y)) -> |0|_0()
r25: quot_2(s_1(x),s_1(y)) -> s_1(quot_2(minus_2(x,y),s_1(y)))
r26: plus_2(|0|_0(),y) -> y
r27: plus_2(s_1(x),y) -> s_1(plus_2(x,y))
r28: app_2(nil_0(),k) -> k
r29: app_2(l,nil_0()) -> l
r30: app_2(cons_2(x,l),k) -> cons_2(x,app_2(l,k))
r31: sum_1(cons_2(x,nil_0())) -> cons_2(x,nil_0())
r32: sum_1(cons_2(x,cons_2(y,l))) -> sum_1(cons_2(plus_2(x,y),l))
r33: sum_1(app_2(l,cons_2(x,cons_2(y,k)))) -> sum_1(app_2(l,sum_1(cons_2(x,cons_2(y,k)))))
r34: map_2(f,nil_0()) -> nil_0()
r35: map_2(f,cons_2(x,xs)) -> cons_2(|app'|(f,x),map_2(f,xs))
r36: filter_2(f,nil_0()) -> nil_0()
r37: filter_2(f,cons_2(x,xs)) -> filter2_4(|app'|(f,x),f,x,xs)
r38: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r39: 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:
          plus_2#_A(x1,x2) = x1
          s_1_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus_2#_A(x1,x2) = max{0, x1 - 7}
          s_1_A(x1) = max{195, x1 + 6}
          plus_2#!_A(x1,x2) = max{4, x1 - 409, x2 - 1}
          s_1!_A(x1) = max{3, x1 - 410}
  
    partial status:
  
      pi(plus_2#) = [1]
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.