YES
We show the termination of the TRS R:

  app(D(),t()) -> |1|()
  app(D(),constant()) -> |0|()
  app(D(),app(app(+(),x),y)) -> app(app(+(),app(D(),x)),app(D(),y))
  app(D(),app(app(*(),x),y)) -> app(app(+(),app(app(*(),y),app(D(),x))),app(app(*(),x),app(D(),y)))
  app(D(),app(app(-(),x),y)) -> app(app(-(),app(D(),x)),app(D(),y))
  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(*_0(),y) -> *_1(y)
  app(*_1(x1),y) -> *_2(x1,y)
  app(+_0(),y) -> +_1(y)
  app(+_1(x1),y) -> +_2(x1,y)
  app(-_0(),y) -> -_1(y)
  app(-_1(x1),y) -> -_2(x1,y)
  app(D_0(),y) -> D_1(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)
  D_1(t_0()) -> |1|_0()
  D_1(constant_0()) -> |0|_0()
  D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
  D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
  D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
  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#(D_0(),y) -> D_1#(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: D_1#(+_2(x,y)) -> D_1#(x)
p6: D_1#(+_2(x,y)) -> D_1#(y)
p7: D_1#(*_2(x,y)) -> D_1#(x)
p8: D_1#(*_2(x,y)) -> D_1#(y)
p9: D_1#(-_2(x,y)) -> D_1#(x)
p10: D_1#(-_2(x,y)) -> D_1#(y)
p11: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p12: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p13: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p14: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p15: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p16: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p11, p12, p13, p14, p15, p16}
  {p5, p6, p7, p8, p9, p10}


-- 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(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{5, x2 - 7}
        cons_2_A(x1,x2) = max{x1 + 13, x2 + 29}
        app#_A(x1,x2) = x2 + 5
        map_1_A(x1) = max{13, x1 - 31}
        filter2_3_A(x1,x2,x3) = max{18, x1 - 31, x2 - 31, x3 - 31}
        filter2_4#_A(x1,x2,x3,x4) = x4 + 4
        false_0_A = 32
        filter_2#_A(x1,x2) = x2 + 3
        filter_1_A(x1) = max{15, x1 - 31}
        app_A(x1,x2) = 15
        true_0_A = 32
        D_1_A(x1) = max{13, x1 - 65}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{33, x1 - 30, x2 - 76}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_2 > filter_2 > false_0 = app = nil_0 > map_1 = filter2_3 = filter_1 = true_0 > filter_2# = D_1 = t_0 = constant_0 > filter2_4# = |1|_0 = |0|_0 > app# > map_2#
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]
      pi(app#) = [2]
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(filter_1) = []
      pi(app) = []
      pi(true_0) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> app#(f,x)
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(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- 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(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r10, r12, r13, r14, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 21, x2 + 42}
        cons_2_A(x1,x2) = max{19, x1 - 21, x2}
        app_A(x1,x2) = max{20, x1 - 22, x2 - 43}
        *_0_A = 17
        *_1_A(x1) = max{18, x1 - 44}
        *_2_A(x1,x2) = max{19, x1 - 67, x2 - 75}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 44}
        +_2_A(x1,x2) = max{3, x1 - 85, x2 - 79}
        -_0_A = 6
        -_1_A(x1) = max{7, x1 - 44}
        -_2_A(x1,x2) = max{8, x1 - 76, x2 - 68}
        cons_0_A = 41
        cons_1_A(x1) = max{18, x1 - 44}
        filter_0_A = 1
        filter_1_A(x1) = max{19, x1 - 43}
        filter2_0_A = 16
        filter2_1_A(x1) = max{17, x1 - 44}
        filter2_2_A(x1,x2) = max{18, x1 - 67, x2 - 44}
        filter2_3_A(x1,x2,x3) = max{19, x1 - 90, x2 - 67, x3 - 44}
        D_1_A(x1) = max{7, x1 - 76}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{21, x1 - 2, x2 - 21}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter2_3 = filter_2 = nil_0 > map_2# = cons_2 = app = +_0 = cons_0 > *_2 = -_2 = cons_1 = filter_1 = filter2_2 > *_1 = -_1 = filter_0 = filter2_1 = D_1 = t_0 = constant_0 > *_0 = +_2 = -_0 = filter2_0 = |1|_0 = |0|_0 > +_1
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      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(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_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#(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)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r10, r13, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 5
        filter2_3_A(x1,x2,x3) = max{8, x1 - 1, x2 + 7, x3 - 1}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 6
        true_0_A = 1
        filter_2#_A(x1,x2) = x1 + 6
        cons_2_A(x1,x2) = max{4, x1 - 15, x2 - 15}
        app_A(x1,x2) = max{x1 - 12, x2 + 10}
        false_0_A = 15
        filter_1_A(x1) = max{27, x1 + 7}
        *_0_A = 5
        *_1_A(x1) = max{6, x1 - 2}
        *_2_A(x1,x2) = max{10, x1 - 14, x2 - 14}
        +_0_A = 1
        +_1_A(x1) = max{6, x1 - 2}
        +_2_A(x1,x2) = max{10, x1 - 14, x2 - 15}
        -_0_A = 11
        -_1_A(x1) = max{9, x1 - 1}
        -_2_A(x1,x2) = max{10, x1 - 14, x2 - 14}
        cons_0_A = 1
        cons_1_A(x1) = max{3, x1 - 4}
        filter_0_A = 40
        filter2_1_A(x1) = max{11, x1 - 1}
        filter2_2_A(x1,x2) = max{9, x1 - 14, x2 - 1}
        map_0_A = 4
        map_1_A(x1) = max{5, x1 - 15}
        D_1_A(x1) = max{10, x1 - 14}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 13
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{14, x1 - 43, x2 - 5}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = *_0 = filter_2 > filter2_3 = false_0 = filter_1 = *_1 = *_2 = -_2 > filter2_4# = filter_2# = cons_2 = -_1 = filter_0 = filter2_2 = map_1 = D_1 = t_0 = constant_0 = nil_0 > app# = true_0 = +_2 = -_0 = cons_1 = filter2_1 = map_0 = |1|_0 = |0|_0 > +_1 = cons_0 > +_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

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

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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: filter2_4#(true_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#(false_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = x1 + 2
        true_0_A = 12
        filter_2#_A(x1,x2) = 10
        cons_2_A(x1,x2) = 2
        app_A(x1,x2) = 7
        false_0_A = 11
        *_0_A = 2
        *_1_A(x1) = 2
        *_2_A(x1,x2) = 0
        +_0_A = 2
        +_1_A(x1) = 2
        +_2_A(x1,x2) = 7
        -_0_A = 2
        -_1_A(x1) = 2
        -_2_A(x1,x2) = 0
        D_0_A = 2
        D_1_A(x1) = 7
        cons_0_A = 2
        cons_1_A(x1) = 1
        filter_0_A = 2
        filter_1_A(x1) = 2
        filter_2_A(x1,x2) = 6
        filter2_0_A = 2
        filter2_1_A(x1) = 2
        filter2_2_A(x1,x2) = 0
        filter2_3_A(x1,x2,x3) = 2
        filter2_4_A(x1,x2,x3,x4) = 6
        map_0_A = 5
        map_1_A(x1) = 4
        map_2_A(x1,x2) = 3
        t_0_A = 2
        |1|_0_A = 1
        constant_0_A = 2
        |0|_0_A = 1
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        D_0 = filter_2 = filter2_4 = map_0 = nil_0 > filter2_4# = true_0 = false_0 > filter_2# > cons_2 = app = *_2 = -_2 = D_1 = map_2 = t_0 = constant_0 > *_1 = +_1 = +_2 = -_0 = -_1 = cons_1 = filter_1 = filter2_3 = map_1 = |1|_0 = |0|_0 > *_0 = +_0 = cons_0 = filter_0 = filter2_2 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(filter2_4#) = [1]
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(D_0) = []
      pi(D_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(filter2_4) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: 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)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r9, r10, r12, r13, r14, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{2, x2 + 1}
        cons_2_A(x1,x2) = max{15, x1 + 13, x2 + 4}
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 12, x4 + 3}
        app_A(x1,x2) = max{29, x1 - 31, x2 + 9}
        false_0_A = 10
        *_0_A = 1
        *_1_A(x1) = max{8, x1 - 1}
        *_2_A(x1,x2) = max{25, x1 - 36, x2 - 9}
        +_0_A = 1
        +_1_A(x1) = max{0, x1 - 1}
        +_2_A(x1,x2) = max{28, x1 - 33}
        -_0_A = 9
        -_1_A(x1) = max{10, x1 - 1}
        -_2_A(x1,x2) = max{11, x1 - 59, x2 - 23}
        cons_1_A(x1) = x1 + 45
        filter_0_A = 27
        filter_1_A(x1) = max{28, x1 - 12}
        filter2_0_A = 25
        filter2_1_A(x1) = max{26, x1 - 1}
        filter2_2_A(x1,x2) = max{27, x1 - 33, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{28, x1 - 65, x2 - 33, x3 - 1}
        D_1_A(x1) = max{8, x1 - 27}
        t_0_A = 26
        |1|_0_A = 0
        constant_0_A = 26
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{30, x1 - 43, x2 - 16}
        nil_0_A = 1
  
    base order 2:
  
      precedence:
      
        app = *_1 = *_2 = -_0 > filter_2# = *_0 = -_2 > cons_2 = -_1 = filter2_3 = D_1 = t_0 = constant_0 = filter_2 > filter2_4# = false_0 = +_2 = cons_1 = filter_1 = filter2_1 = filter2_2 = |1|_0 = |0|_0 = nil_0 > +_1 = filter_0 = filter2_0 > +_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(false_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

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)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(-_2(x,y)) -> D_1#(y)
p3: D_1#(-_2(x,y)) -> D_1#(x)
p4: D_1#(*_2(x,y)) -> D_1#(y)
p5: D_1#(*_2(x,y)) -> D_1#(x)
p6: D_1#(+_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r9, r10, r12, r13, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{8, x1 + 4}
        +_2_A(x1,x2) = max{18, x1 + 5, x2 + 7}
        -_2_A(x1,x2) = max{39, x1 + 26, x2 + 13}
        *_2_A(x1,x2) = max{17, x1 + 10, x2 + 9}
        app_A(x1,x2) = max{16, x1 - 3, x2 - 9}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 9}
        +_0_A = 2
        +_1_A(x1) = max{1, x1 - 10}
        -_0_A = 2
        -_1_A(x1) = max{1, x1 - 10}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 10}
        cons_2_A(x1,x2) = max{7, x1 - 17, x2 - 10}
        filter_0_A = 2
        filter_1_A(x1) = max{4, x1 - 10}
        filter2_0_A = 2
        filter2_1_A(x1) = max{0, x1 - 10}
        filter2_2_A(x1,x2) = max{0, x1 - 14, x2 - 10}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 21}
        D_1_A(x1) = max{12, x1 - 40}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{5, x1 - 14, x2 - 10}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > -_2 = *_2 > D_1# = app = -_0 > *_1 = +_1 = -_1 = cons_2 = map_1 = D_1 = t_0 = constant_0 > *_0 = +_0 = cons_1 = map_0 = |1|_0 = |0|_0 = filter_2 > cons_0 = filter_1 = filter2_0 = filter2_2 = nil_0 > filter_0 = filter2_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(-_2) = [2]
      pi(*_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(-_2(x,y)) -> D_1#(y)
p3: D_1#(-_2(x,y)) -> D_1#(x)
p4: D_1#(*_2(x,y)) -> D_1#(y)
p5: D_1#(*_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(x)
p3: D_1#(*_2(x,y)) -> D_1#(y)
p4: D_1#(-_2(x,y)) -> D_1#(x)
p5: D_1#(-_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r9, r10, r12, r14, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 28
        +_2_A(x1,x2) = max{x1 + 34, x2 - 14}
        *_2_A(x1,x2) = max{x1 + 31, x2 + 65}
        -_2_A(x1,x2) = max{x1 + 61, x2 + 29}
        app_A(x1,x2) = max{31, x1 - 5, x2 - 14}
        *_0_A = 3
        *_1_A(x1) = max{4, x1 - 15}
        +_0_A = 3
        +_1_A(x1) = max{4, x1 - 15}
        -_0_A = 4
        -_1_A(x1) = max{3, x1 - 14}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 15}
        cons_2_A(x1,x2) = max{14, x1 - 32, x2 - 16}
        filter_0_A = 3
        filter_1_A(x1) = max{4, x1 - 19}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 15}
        filter2_2_A(x1,x2) = max{3, x1 - 1, x2 + 1}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 6, x2 - 14, x3 - 14}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 15}
        D_1_A(x1) = max{30, x1 - 67}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 66
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{18, x1 - 18, x2 + 1}
        nil_0_A = 14
  
    base order 2:
  
      precedence:
      
        +_2 = *_2 = -_2 > D_1# = app > *_1 = +_1 = -_1 = cons_0 = map_1 = D_1 = t_0 = constant_0 = filter_2 > *_0 = +_0 = -_0 = cons_2 = filter_1 = filter2_3 = map_0 = |1|_0 = |0|_0 = nil_0 > cons_1 = filter_0 = filter2_2 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(*_2) = [2]
      pi(-_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(x)
p3: D_1#(*_2(x,y)) -> D_1#(y)
p4: D_1#(-_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(-_2(x,y)) -> D_1#(x)
p3: D_1#(*_2(x,y)) -> D_1#(y)
p4: D_1#(*_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r9, r10, r12, r13, r14, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 20
        +_2_A(x1,x2) = max{x1 + 23, x2 - 9}
        -_2_A(x1,x2) = max{x1 + 21, x2 - 9}
        *_2_A(x1,x2) = max{x1 + 28, x2 + 21}
        app_A(x1,x2) = max{57, x1 - 24, x2 + 3}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 1}
        +_0_A = 22
        +_1_A(x1) = max{23, x1 - 1}
        -_0_A = 23
        -_1_A(x1) = max{10, x1 - 1}
        cons_0_A = 4
        cons_1_A(x1) = max{5, x1 - 1}
        cons_2_A(x1,x2) = max{4, x1 - 58, x2 - 64}
        filter_0_A = 1
        filter_1_A(x1) = max{4, x1 - 1}
        filter2_0_A = 21
        filter2_1_A(x1) = max{22, x1 - 1}
        filter2_2_A(x1,x2) = max{23, x1 - 26, x2 - 27}
        filter2_3_A(x1,x2,x3) = max{22, x1 - 51, x2 - 52}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 3}
        D_1_A(x1) = max{8, x1 - 45}
        t_0_A = 1
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{14, x1 - 77, x2 - 52}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 = -_2 = *_2 > D_1# = app = *_0 = map_0 > *_1 = +_1 = -_1 = cons_2 = filter2_1 = filter2_2 = map_1 = D_1 = t_0 = constant_0 = filter_2 > +_0 = -_0 = cons_1 = filter_1 = filter2_0 = filter2_3 = |1|_0 = |0|_0 = nil_0 > cons_0 = filter_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(-_2) = [1]
      pi(*_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(-_2(x,y)) -> D_1#(x)
p3: D_1#(*_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(y)
p3: D_1#(-_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r10, r12, r13, r14, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 11
        +_2_A(x1,x2) = max{52, x1 + 26, x2 - 9}
        *_2_A(x1,x2) = max{x1 - 1, x2 + 51}
        -_2_A(x1,x2) = max{x1 + 25, x2 - 6}
        app_A(x1,x2) = max{32, x1 - 5, x2 - 10}
        *_0_A = 4
        *_1_A(x1) = max{3, x1 - 11}
        +_0_A = 3
        +_1_A(x1) = max{4, x1 - 10}
        -_0_A = 3
        -_1_A(x1) = max{4, x1 - 10}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 32}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 28}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 11}
        filter2_2_A(x1,x2) = max{3, x1 - 17, x2 - 11}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 23, x2 - 16, x3 - 11}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 43}
        D_1_A(x1) = max{5, x1 - 27}
        t_0_A = 26
        |1|_0_A = 0
        constant_0_A = 26
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{3, x1 - 34, x2 - 11}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > *_2 = -_2 = filter2_2 > D_1# = app = +_0 = -_0 > *_1 = +_1 = -_1 = cons_1 = filter_1 = filter2_1 = map_1 = D_1 = t_0 = constant_0 = filter_2 > *_0 = cons_0 = filter_0 = filter2_0 = filter2_3 = map_0 = |1|_0 = |0|_0 = nil_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(*_2) = [2]
      pi(-_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_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(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r8, r9, r10, r12, r14, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 69
        +_2_A(x1,x2) = max{x1 + 107, x2 - 34}
        *_2_A(x1,x2) = max{x1 - 34, x2 + 70}
        app_A(x1,x2) = max{32, x1 - 5, x2 - 12}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 35}
        +_0_A = 3
        +_1_A(x1) = max{4, x1 - 13}
        -_0_A = 4
        -_1_A(x1) = max{3, x1 - 56}
        -_2_A(x1,x2) = max{31, x1 - 62, x2 - 31}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 12}
        cons_2_A(x1,x2) = max{3, x1 - 45, x2 - 12}
        filter_0_A = 3
        filter_1_A(x1) = max{4, x1 - 12}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 13}
        filter2_2_A(x1,x2) = max{x1 - 1, x2 + 10}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 7, x2 - 1, x3 - 13}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 13}
        D_1_A(x1) = max{30, x1 - 76}
        t_0_A = 75
        |1|_0_A = 29
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{18, x1 - 18, x2 - 14}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > app = -_0 > +_1 = -_2 = map_1 > +_0 = -_1 = cons_2 = filter2_3 = map_0 = D_1 = t_0 = constant_0 = filter_2 > *_2 = cons_1 = filter_1 = filter2_2 = |1|_0 = |0|_0 = nil_0 > D_1# = *_1 = cons_0 = filter_0 = filter2_1 > *_0 = filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(*_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: 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: D_1#(+_2(x,y)) -> D_1#(x)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(filter_0(),y) -> filter_1(y)
r11: app(filter_1(x1),y) -> filter_2(x1,y)
r12: app(filter2_0(),y) -> filter2_1(y)
r13: app(filter2_1(x1),y) -> filter2_2(x1,y)
r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r16: app(map_0(),y) -> map_1(y)
r17: app(map_1(x1),y) -> map_2(x1,y)
r18: D_1(t_0()) -> |1|_0()
r19: D_1(constant_0()) -> |0|_0()
r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r23: map_2(f,nil_0()) -> nil_0()
r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r25: filter_2(f,nil_0()) -> nil_0()
r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r8, r9, r10, r12, r13, r14, r16, r18, r19, r25

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 13
        +_2_A(x1,x2) = max{x1 + 14, x2 - 4}
        app_A(x1,x2) = max{6, x1 - 7, x2 - 5}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 6}
        *_2_A(x1,x2) = max{3, x1 - 17, x2 - 17}
        +_0_A = 4
        +_1_A(x1) = max{5, x1 - 6}
        cons_0_A = 6
        cons_1_A(x1) = max{1, x1 - 28}
        cons_2_A(x1,x2) = max{2, x1 - 35, x2 - 7}
        filter_0_A = 2
        filter_1_A(x1) = max{3, x1 - 29}
        filter2_0_A = 3
        filter2_1_A(x1) = max{4, x1 - 6}
        filter2_2_A(x1,x2) = max{5, x1 - 14, x2 - 6}
        filter2_3_A(x1,x2,x3) = max{6, x1 - 21, x2 - 14, x3 - 5}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 6}
        D_1_A(x1) = max{2, x1 - 23}
        t_0_A = 22
        |1|_0_A = 0
        constant_0_A = 22
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{4, x1 - 37, x2 - 26}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_0 > +_2 = filter_1 = filter_2 = nil_0 > app > *_1 = *_2 = filter2_3 > D_1# = *_0 = +_1 = cons_0 = cons_2 = filter2_2 = map_1 = D_1 = t_0 = constant_0 > +_0 = cons_1 = filter2_1 = map_0 = |1|_0 = |0|_0 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.