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, r3, r4, r5, r7, r8, r9, r11, r13, r14, r15, r16, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x1 + x2 + 5
          cons_2_A(x1,x2) = x1 + x2 + 4
          |app'|#_A(x1,x2) = x1 + x2 + 5
          map_1_A(x1) = x1 + 6
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 1
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5
          false_0_A() = 1
          filter_2#_A(x1,x2) = x1 + x2 + 2
          filter_1_A(x1) = x1 + 4
          |app'|_A(x1,x2) = x1 + x2 + 5
          true_0_A() = 1
          app_0_A() = 1
          app_1_A(x1) = x1 + 4
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          map_0_A() = 2
          minus_0_A() = 2
          minus_1_A(x1) = x1 + 6
          minus_2_A(x1,x2) = x1 + x2 + 10
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 5
          plus_2_A(x1,x2) = x1 + x2 + 9
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 1
          app_2_A(x1,x2) = x1 + 3
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + 3
  
    base order 2:
  
      precedence:
      
        |app'|# > filter2_4# > map_2# = filter_2# = |app'| > map_1 = filter_1 = app_1 = filter2_1 = plus_2 = quot_1 > app_0 = filter_0 = filter2_0 = map_0 = minus_2 = plus_1 = quot_0 = app_2 = filter_2 > cons_2 = false_0 = minus_1 = plus_0 = s_1 = quot_2 > true_0 = cons_1 = minus_0 = s_0 = |0|_0 > filter2_3 = cons_0 = filter2_2 = nil_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = []
      pi(|app'|#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [1]
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(|app'|) = [2]
      pi(true_0) = []
      pi(app_0) = []
      pi(app_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = [2]
      pi(plus_0) = []
      pi(plus_1) = [1]
      pi(plus_2) = []
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: |app'|#(map_1(x1),y) -> map_2#(x1,y)
p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p3: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p6: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: |app'|(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:

  {p3, p4, p5, p6, p7, p8}
  {p2}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> |app'|#(f,x)
p2: |app'|#(filter_1(x1),y) -> filter_2#(x1,y)
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)
p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p6: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,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

  r1, r3, r4, r5, r7, r8, r9, r11, r13, r14, r15, r16, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x1 + x2 + 8
          cons_2_A(x1,x2) = x1 + x2 + 8
          |app'|#_A(x1,x2) = x1 + x2 + 9
          filter_1_A(x1) = x1 + 7
          filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 9
          |app'|_A(x1,x2) = x1 + x2 + 2
          true_0_A() = 1
          false_0_A() = 1
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          app_0_A() = 17
          app_1_A(x1) = x1 + 18
          cons_0_A() = 6
          cons_1_A(x1) = x1 + 7
          filter_0_A() = 6
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          map_0_A() = 7
          map_1_A(x1) = x1 + 8
          minus_0_A() = 2
          minus_1_A(x1) = x1 + 3
          minus_2_A(x1,x2) = x1 + x2 + 4
          plus_0_A() = 1
          plus_1_A(x1) = x1 + 2
          plus_2_A(x1,x2) = x1 + x2 + 3
          quot_0_A() = 1
          quot_1_A(x1) = x1 + 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 1
          app_2_A(x1,x2) = x1 + 17
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 8
  
    base order 2:
  
      precedence:
      
        |app'|# = plus_2 > filter2_4# > cons_2 = app_2 > nil_0 > filter_2# = filter2_3 = map_1 > |app'| = map_0 > app_1 = cons_1 = filter2_2 = minus_2 = plus_1 = quot_1 = filter_2 > filter_1 = true_0 = false_0 = app_0 = cons_0 = filter2_1 = minus_1 = plus_0 = quot_0 = quot_2 > filter_0 = filter2_0 = minus_0 = s_1 > s_0 = |0|_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(|app'|#) = []
      pi(filter_1) = [1]
      pi(filter2_4#) = []
      pi(|app'|) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [3]
      pi(app_0) = []
      pi(app_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [1, 2]
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = [1]
      pi(minus_2) = [2]
      pi(plus_0) = []
      pi(plus_1) = [1]
      pi(plus_2) = [1]
      pi(quot_0) = []
      pi(quot_1) = [1]
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(app_2) = [1]
      pi(nil_0) = []
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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)) -> filter2_4#(|app'|(f,x),f,x,xs)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: |app'|#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,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:

  {p2, p3, p4}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(|app'|(f,x),f,x,xs)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
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, r3, r5, r7, r8, r9, r11, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          filter_2#_A(x1,x2) = x2 + 2
          cons_2_A(x1,x2) = x1 + x2 + 4
          filter2_4#_A(x1,x2,x3,x4) = x4 + 3
          |app'|_A(x1,x2) = x2 + 10
          false_0_A() = 1
          true_0_A() = 3
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 7
          map_1_A(x1) = 6
          minus_0_A() = 2
          minus_1_A(x1) = 1
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 14
          |0|_0_A() = 1
          plus_2_A(x1,x2) = x1 + x2 + 3
          quot_2_A(x1,x2) = x1 + 11
          app_2_A(x1,x2) = x1 + 2
          nil_0_A() = 2
          filter_2_A(x1,x2) = x1 + x2 + 5
  
    base order 2:
  
      precedence:
      
        map_0 > map_1 = minus_2 > |app'| = filter2_0 = minus_0 = nil_0 > cons_2 = app_0 = cons_0 = filter_0 = filter2_1 = minus_1 = plus_0 = quot_0 = app_2 = filter_2 > true_0 = app_1 = cons_1 = filter_1 = filter2_2 = plus_2 > filter_2# = filter2_3 = plus_1 = s_1 > filter2_4# = false_0 = s_0 = quot_2 > quot_1 = |0|_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = []
      pi(filter2_4#) = [4]
      pi(|app'|) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = [2]
      pi(|0|_0) = []
      pi(plus_2) = [2]
      pi(quot_2) = []
      pi(app_2) = [1]
      pi(nil_0) = []
      pi(filter_2) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

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

  (no SCCs)

-- 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

  r1, r3, r5, r7, r8, r11, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 7
          |app'|_A(x1,x2) = x2 + 4
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 4
          filter_1_A(x1) = 3
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          map_0_A() = 3
          map_1_A(x1) = 2
          minus_0_A() = 1
          minus_1_A(x1) = 2
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          minus_2_A(x1,x2) = x1 + x2 + 8
          |0|_0_A() = 1
          plus_2_A(x1,x2) = x1 + x2 + 5
          quot_2_A(x1,x2) = x1 + 5
          app_2_A(x1,x2) = x1 + 15
          nil_0_A() = 7
          filter_2_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        minus_1 > app_0 = cons_0 = filter_0 = filter2_0 = map_0 = minus_0 = plus_0 = quot_0 = minus_2 = |0|_0 > s_1 = plus_2 = quot_2 = filter_2 > cons_2 = s_0 = app_2 = nil_0 > map_2# > |app'| > app_1 = cons_1 = filter_1 = filter2_1 = map_1 = plus_1 = quot_1 > filter2_2
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(|app'|) = [2]
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(plus_2) = []
      pi(quot_2) = [1]
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_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: 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

  r9, r21, r24, r27, r28, r29, r30, r31, r32, r33, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sum_1#_A(x1) = x1 + 2
          app_2_A(x1,x2) = x1 + x2 + 2
          cons_2_A(x1,x2) = x2 + 5
          sum_1_A(x1) = 7
          |app'|_A(x1,x2) = 0
          filter2_2_A(x1,x2) = x1 + 1
          filter2_3_A(x1,x2,x3) = 0
          minus_2_A(x1,x2) = x1 + x2 + 3
          |0|_0_A() = 0
          quot_2_A(x1,x2) = 1
          s_1_A(x1) = 2
          plus_2_A(x1,x2) = x2 + 2
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        plus_2 > app_2 = minus_2 > sum_1# = quot_2 > cons_2 = sum_1 = s_1 > |0|_0 > filter2_2 > |app'| > filter2_3 = filter_2 > nil_0
      
  
    partial status:
  
      pi(sum_1#) = [1]
      pi(app_2) = []
      pi(cons_2) = []
      pi(sum_1) = []
      pi(|app'|) = []
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(minus_2) = [1, 2]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(s_1) = []
      pi(plus_2) = []
      pi(nil_0) = []
      pi(filter_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: 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

  r1, r3, r7, r8, r9, r11, r13, r15, r17, r19, r21, r24, r25, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app_2#_A(x1,x2) = x1 + x2
          cons_2_A(x1,x2) = x1 + x2 + 16
          |app'|_A(x1,x2) = x2 + 13
          app_0_A() = 13
          app_1_A(x1) = 12
          cons_0_A() = 13
          cons_1_A(x1) = 12
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 3
          map_1_A(x1) = 2
          minus_0_A() = 11
          minus_1_A(x1) = 10
          plus_0_A() = 13
          plus_1_A(x1) = 12
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 13
          s_1_A(x1) = 12
          minus_2_A(x1,x2) = x1 + x2 + 9
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 2
          plus_2_A(x1,x2) = 17
          app_2_A(x1,x2) = x1 + 33
          nil_0_A() = 19
          filter_2_A(x1,x2) = x2 + 14
  
    base order 2:
  
      precedence:
      
        filter_2 > app_2# = filter2_0 = app_2 = nil_0 > cons_2 = app_0 = filter2_1 = minus_0 = quot_0 = s_0 > filter2_2 > filter2_3 > |app'| = cons_0 = map_0 = plus_0 > app_1 = cons_1 = map_1 = minus_1 = plus_1 > minus_2 = plus_2 > quot_2 > quot_1 = s_1 > |0|_0
      
  
    partial status:
  
      pi(app_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(|app'|) = [2]
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = [1]
      pi(|0|_0) = []
      pi(quot_2) = [1]
      pi(plus_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_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: 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

  r1, r3, r5, r7, r8, r9, r11, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quot_2#_A(x1,x2) = x1 + x2
          s_1_A(x1) = x1 + 3
          minus_2_A(x1,x2) = x1 + 2
          |app'|_A(x1,x2) = x2 + 7
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 6
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 7
          filter2_1_A(x1) = 6
          filter2_2_A(x1,x2) = 5
          filter2_3_A(x1,x2,x3) = 4
          map_0_A() = 7
          map_1_A(x1) = 6
          minus_0_A() = 2
          minus_1_A(x1) = 1
          plus_0_A() = 7
          plus_1_A(x1) = 6
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          |0|_0_A() = 1
          plus_2_A(x1,x2) = x1 + x2 + 5
          quot_2_A(x1,x2) = x1 + 8
          app_2_A(x1,x2) = x1 + 8
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 2
  
    base order 2:
  
      precedence:
      
        quot_2# = filter2_0 = s_0 = plus_2 > s_1 = filter2_1 = quot_2 > minus_2 = filter2_2 = minus_0 = app_2 > filter2_3 = minus_1 > |app'| = cons_0 > app_0 = cons_1 = plus_0 = quot_0 > app_1 = filter_0 = map_1 = plus_1 = quot_1 = |0|_0 = nil_0 = filter_2 > filter_1 = map_0
      
  
    partial status:
  
      pi(quot_2#) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(|0|_0) = []
      pi(plus_2) = [2]
      pi(quot_2) = [1]
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_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

  r1, r3, r5, r7, r8, r11, r13, r15, r17, r21, r22, r23, r24, r26, r27, r29, r30, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2 + 5
          s_1_A(x1) = x1 + 4
          minus_2_A(x1,x2) = x1 + x2 + 9
          plus_2_A(x1,x2) = x1 + x2 + 7
          |app'|_A(x1,x2) = 3
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          map_0_A() = 1
          map_1_A(x1) = 2
          minus_0_A() = 2
          minus_1_A(x1) = 1
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 4
          app_2_A(x1,x2) = x1 + 5
          nil_0_A() = 1
          cons_2_A(x1,x2) = x1 + 6
          filter_2_A(x1,x2) = x2 + 4
  
    base order 2:
  
      precedence:
      
        minus_2# > minus_2 = app_0 > plus_2 = app_1 > app_2 > nil_0 > cons_2 > s_1 = quot_2 = filter_2 > |app'| = filter_0 = filter2_0 = minus_0 = plus_0 = quot_0 = |0|_0 > cons_1 = filter_1 = filter2_1 = map_1 = minus_1 = plus_1 = quot_1 > cons_0 = filter2_2 = map_0
      
  
    partial status:
  
      pi(minus_2#) = []
      pi(s_1) = []
      pi(minus_2) = []
      pi(plus_2) = []
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(|0|_0) = []
      pi(quot_2) = [1]
      pi(app_2) = [1]
      pi(nil_0) = []
      pi(cons_2) = [1]
      pi(filter_2) = [2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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

  r1, r3, r5, r7, r8, r9, r11, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus_2#_A(x1,x2) = x1 + x2 + 10
          minus_2_A(x1,x2) = x1 + x2 + 12
          plus_2_A(x1,x2) = x1 + x2 + 3
          |app'|_A(x1,x2) = x2 + 8
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 8
          map_1_A(x1) = 7
          minus_0_A() = 1
          minus_1_A(x1) = 2
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 1
          quot_1_A(x1) = 2
          s_0_A() = 1
          s_1_A(x1) = x1 + 2
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 9
          app_2_A(x1,x2) = x1 + 11
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 9
  
    base order 2:
  
      precedence:
      
        map_0 > map_1 = app_2 > minus_2# > plus_2 = filter_0 = filter2_0 > minus_2 > quot_2 = filter_2 > |app'| = app_0 = cons_0 = plus_0 = nil_0 > app_1 = cons_1 = filter_1 = filter2_1 = minus_1 = plus_1 = quot_1 = s_1 > filter2_2 = minus_0 = quot_0 = s_0 = |0|_0 > filter2_3
      
  
    partial status:
  
      pi(minus_2#) = []
      pi(minus_2) = []
      pi(plus_2) = []
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = [1]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_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: 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

  r1, r5, r7, r8, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36, r37, r38, r39

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          sum_1#_A(x1) = x1
          cons_2_A(x1,x2) = x1 + x2 + 10
          plus_2_A(x1,x2) = x1 + x2 + 3
          |app'|_A(x1,x2) = x2 + 8
          app_0_A() = 8
          app_1_A(x1) = 7
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          s_1_A(x1) = x1 + 7
          minus_2_A(x1,x2) = x1 + x2 + 4
          |0|_0_A() = 1
          quot_2_A(x1,x2) = x1 + 2
          app_2_A(x1,x2) = x1 + 6
          nil_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 9
          filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 19
          true_0_A() = 1
          false_0_A() = 1
  
    base order 2:
  
      precedence:
      
        sum_1# > plus_2 = app_0 = filter_0 = quot_0 > minus_2 = app_2 > |app'| = filter2_0 = minus_0 = plus_0 > app_1 = filter2_1 = minus_1 = plus_1 = s_1 = filter_2 = filter2_4 > cons_2 = filter_1 = s_0 = quot_2 = false_0 > filter2_2 = quot_1 = |0|_0 = true_0 > nil_0
      
  
    partial status:
  
      pi(sum_1#) = []
      pi(cons_2) = [1]
      pi(plus_2) = [1]
      pi(|app'|) = [2]
      pi(app_0) = []
      pi(app_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(minus_2) = [2]
      pi(|0|_0) = []
      pi(quot_2) = []
      pi(app_2) = [1]
      pi(nil_0) = []
      pi(filter_2) = [2]
      pi(filter2_4) = []
      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: 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

  r1, r3, r5, r7, r8, r9, r11, r13, r15, r17, r19, r21, r22, r23, r24, r26, r27, r29, r36

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus_2#_A(x1,x2) = x1
          s_1_A(x1) = x1 + 2
          |app'|_A(x1,x2) = x2 + 4
          app_0_A() = 2
          app_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          filter_0_A() = 4
          filter_1_A(x1) = 3
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = 2
          filter2_3_A(x1,x2,x3) = 1
          map_0_A() = 4
          map_1_A(x1) = 3
          minus_0_A() = 2
          minus_1_A(x1) = 1
          plus_0_A() = 2
          plus_1_A(x1) = 1
          quot_0_A() = 2
          quot_1_A(x1) = 1
          s_0_A() = 1
          minus_2_A(x1,x2) = x1 + x2 + 8
          |0|_0_A() = 1
          plus_2_A(x1,x2) = x1 + x2 + 5
          quot_2_A(x1,x2) = x1 + 5
          app_2_A(x1,x2) = x1 + 15
          nil_0_A() = 1
          filter_2_A(x1,x2) = x2 + 2
  
    base order 2:
  
      precedence:
      
        app_0 = filter_0 = filter2_0 = plus_0 = quot_0 = plus_2 = app_2 > filter2_2 = filter_2 > |app'| = cons_0 = filter2_3 = map_0 = minus_0 = nil_0 > app_1 = cons_1 = filter_1 = filter2_1 = map_1 = minus_1 = plus_1 = minus_2 > quot_2 > s_1 = quot_1 = |0|_0 > plus_2# = s_0
      
  
    partial status:
  
      pi(plus_2#) = []
      pi(s_1) = []
      pi(|app'|) = []
      pi(app_0) = []
      pi(app_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(plus_0) = []
      pi(plus_1) = []
      pi(quot_0) = []
      pi(quot_1) = []
      pi(s_0) = []
      pi(minus_2) = []
      pi(|0|_0) = []
      pi(plus_2) = []
      pi(quot_2) = []
      pi(app_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.