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(D(),app(minus(),x)) -> app(minus(),app(D(),x))
  app(D(),app(app(div(),x),y)) -> app(app(-(),app(app(div(),app(D(),x)),y)),app(app(div(),app(app(*(),x),app(D(),y))),app(app(pow(),y),|2|())))
  app(D(),app(ln(),x)) -> app(app(div(),app(D(),x)),x)
  app(D(),app(app(pow(),x),y)) -> app(app(+(),app(app(*(),app(app(*(),y),app(app(pow(),x),app(app(-(),y),|1|())))),app(D(),x))),app(app(*(),app(app(*(),app(app(pow(),x),y)),app(ln(),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(div_0(),y) -> div_1(y)
  app(div_1(x1),y) -> div_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(ln_0(),y) -> ln_1(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(pow_0(),y) -> pow_1(y)
  app(pow_1(x1),y) -> pow_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))
  D_1(minus_1(x)) -> minus_1(D_1(x))
  D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
  D_1(ln_1(x)) -> div_2(D_1(x),x)
  D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_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: D_1#(minus_1(x)) -> D_1#(x)
p12: D_1#(div_2(x,y)) -> D_1#(x)
p13: D_1#(div_2(x,y)) -> D_1#(y)
p14: D_1#(ln_1(x)) -> D_1#(x)
p15: D_1#(pow_2(x,y)) -> D_1#(x)
p16: D_1#(pow_2(x,y)) -> D_1#(y)
p17: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p18: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p19: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p20: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p21: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p22: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p17, p18, p19, p20, p21, p22}
  {p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16}


-- 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r9, r10, r11, r12, r14, r16, r18, r19, r21, r22, r23, r25, r26, r27, r29, r30, r31, r32, r35

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 + 4
          cons_2_A(x1,x2) = x1 + x2 + 3
          app#_A(x1,x2) = x1 + x2 + 2
          map_1_A(x1) = x1 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 5
          filter2_4#_A(x1,x2,x3,x4) = x2 + x3 + x4 + 6
          false_0_A() = 1
          filter_2#_A(x1,x2) = x1 + x2 + 4
          filter_1_A(x1) = x1 + 3
          app_A(x1,x2) = x1 + x2 + 2
          true_0_A() = 1
          *_0_A() = 1
          *_1_A(x1) = x1 + 2
          *_2_A(x1,x2) = x2 + 3
          +_0_A() = 2
          +_1_A(x1) = x1 + 3
          +_2_A(x1,x2) = 2
          -_0_A() = 1
          -_1_A(x1) = x1 + 2
          -_2_A(x1,x2) = x1 + 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          div_0_A() = 2
          div_1_A(x1) = x1 + 3
          div_2_A(x1,x2) = 2
          filter_0_A() = 2
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 4
          ln_0_A() = 4
          ln_1_A(x1) = 3
          map_0_A() = 2
          minus_0_A() = 8
          minus_1_A(x1) = 7
          pow_0_A() = 6
          pow_1_A(x1) = x1 + 7
          pow_2_A(x1,x2) = x1 + x2 + 8
          D_1_A(x1) = 7
          constant_0_A() = 1
          |0|_0_A() = 0
          |2|_0_A() = 1
          |1|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > map_2# = filter2_4# > cons_2 = minus_0 = pow_2 > div_0 = ln_0 = minus_1 = D_1 > filter_2# = *_2 = -_2 = div_1 = ln_1 = constant_0 = filter_2 > false_0 = filter_1 = true_0 = cons_1 = div_2 = filter2_0 = |0|_0 = |1|_0 = nil_0 > app = cons_0 = filter_0 = |2|_0 > map_1 = filter2_3 = *_1 = +_1 = -_1 = pow_1 > *_0 = +_0 = +_2 = -_0 = filter2_2 = map_0 = pow_0 > filter2_1
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = []
      pi(filter2_3) = [3]
      pi(filter2_4#) = [3]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = [1]
      pi(*_2) = [2]
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(-_2) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(div_0) = []
      pi(div_1) = [1]
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1, 2]
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = [1]
      pi(pow_2) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(|2|_0) = []
      pi(|1|_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

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(*_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {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(*_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r9, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r25, r31, r35

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 + 8
          cons_2_A(x1,x2) = x1 + 6
          app#_A(x1,x2) = x1 + 7
          filter_1_A(x1) = x1 + 7
          filter2_4#_A(x1,x2,x3,x4) = x2 + 8
          app_A(x1,x2) = x1 + x2 + 9
          true_0_A() = 5
          false_0_A() = 5
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          *_0_A() = 1
          *_1_A(x1) = x1 + 3
          *_2_A(x1,x2) = 2
          +_0_A() = 1
          +_1_A(x1) = x1 + 2
          +_2_A(x1,x2) = 1
          -_0_A() = 1
          -_1_A(x1) = x1 + 2
          -_2_A(x1,x2) = 1
          cons_0_A() = 8
          cons_1_A(x1) = x1 + 7
          div_0_A() = 2
          div_1_A(x1) = x1 + 1
          div_2_A(x1,x2) = 0
          filter_0_A() = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          ln_0_A() = 2
          ln_1_A(x1) = 1
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 2
          minus_1_A(x1) = 1
          pow_0_A() = 1
          pow_1_A(x1) = x1 + 4
          pow_2_A(x1,x2) = x1 + x2 + 3
          D_1_A(x1) = 0
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > app# = false_0 > filter_2# = filter2_4# = cons_1 > cons_2 > ln_0 > app = true_0 = cons_0 = div_0 = ln_1 = minus_0 = pow_2 = nil_0 > filter_1 = filter2_3 = *_1 = +_1 = div_1 = map_1 = pow_1 > *_0 = *_2 = +_0 = filter_0 = filter2_2 = map_0 = pow_0 > +_2 = -_2 = filter2_1 > -_1 = div_2 = filter2_0 = minus_1 = D_1 = constant_0 > -_0 = |0|_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter2_4#) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = [3]
      pi(*_0) = []
      pi(*_1) = [1]
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = [1]
      pi(pow_2) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

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(*_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {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(*_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r9, r10, r11, r12, r14, r16, r18, r19, r21, r22, r23, r24, r25, r26, r27, r30, r31, r32, r35

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) = x1 + x2 + 9
          false_0_A() = 1
          true_0_A() = 3
          *_0_A() = 2
          *_1_A(x1) = x1 + 1
          *_2_A(x1,x2) = x2 + 5
          +_0_A() = 1
          +_1_A(x1) = x1 + 4
          +_2_A(x1,x2) = 3
          -_0_A() = 1
          -_1_A(x1) = x1 + 2
          -_2_A(x1,x2) = x1 + 1
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          div_0_A() = 1
          div_1_A(x1) = x1 + 3
          div_2_A(x1,x2) = 2
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 3
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = x1 + x2 + 1
          filter2_3_A(x1,x2,x3) = x1 + x3 + 2
          ln_0_A() = 7
          ln_1_A(x1) = 6
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 6
          minus_1_A(x1) = 5
          pow_0_A() = 1
          pow_1_A(x1) = x1 + 2
          pow_2_A(x1,x2) = x1 + x2 + 10
          D_1_A(x1) = 4
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          |2|_0_A() = 1
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_3 > +_1 = filter2_0 = filter2_2 > cons_2 = +_2 = div_1 = filter_1 = filter2_1 > app > ln_0 = minus_0 = pow_2 = filter_2 > true_0 = *_2 = cons_0 = ln_1 = minus_1 = t_0 > filter_2# = *_1 = +_0 = cons_1 = div_0 = filter_0 = map_1 = pow_1 = D_1 = |1|_0 = constant_0 = nil_0 > filter2_4# = false_0 = *_0 = -_2 = map_0 = pow_0 = |0|_0 > -_1 = div_2 > -_0 = |2|_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1]
      pi(filter2_4#) = []
      pi(app) = []
      pi(false_0) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = [1]
      pi(*_2) = [2]
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(-_2) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(div_0) = []
      pi(div_1) = [1]
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(filter2_3) = [1]
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = [1]
      pi(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(|2|_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p2



-- 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#(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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r8, r9, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r24, r25, r26, r27, r29, r32, r35

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 + 3
          filter2_4#_A(x1,x2,x3,x4) = x4 + 4
          app_A(x1,x2) = x1 + x2 + 2
          true_0_A() = 1
          *_0_A() = 4
          *_1_A(x1) = x1 + 5
          *_2_A(x1,x2) = x2 + 6
          +_0_A() = 1
          +_1_A(x1) = x1 + 2
          +_2_A(x1,x2) = 3
          -_0_A() = 6
          -_1_A(x1) = x1 + 7
          -_2_A(x1,x2) = x1 + 6
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          div_0_A() = 10
          div_1_A(x1) = x1 + 11
          div_2_A(x1,x2) = x2 + 12
          filter_0_A() = 1
          filter_1_A(x1) = x1 + 2
          filter2_0_A() = 4
          filter2_1_A(x1) = x1 + 3
          filter2_2_A(x1,x2) = x1 + x2 + 4
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 5
          ln_0_A() = 5
          ln_1_A(x1) = 4
          map_0_A() = 2
          map_1_A(x1) = x1 + 3
          minus_0_A() = 5
          minus_1_A(x1) = 4
          pow_0_A() = 7
          pow_1_A(x1) = x1 + 8
          pow_2_A(x1,x2) = x1 + x2 + 9
          D_1_A(x1) = 5
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 3
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter_2 > filter2_3 > +_0 = filter2_2 = ln_0 = map_1 = minus_0 = t_0 = constant_0 = nil_0 > app = map_0 > cons_2 = filter2_4# = -_1 = filter_1 = filter2_1 = pow_2 > true_0 = -_0 = -_2 = cons_1 = filter_0 = filter2_0 = pow_1 > filter_2# = cons_0 = pow_0 = D_1 > +_2 = minus_1 = |1|_0 = |0|_0 > *_2 = +_1 > *_1 = div_2 > *_0 = div_1 = ln_1 > div_0
      
  
    partial status:
  
      pi(filter_2#) = [2]
      pi(cons_2) = [1]
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = [1]
      pi(*_2) = [2]
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(div_0) = []
      pi(div_1) = [1]
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = [2]
      pi(filter2_3) = [1, 3]
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = [1]
      pi(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      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#(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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  (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(*_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r11, r12, r14, r16, r18, r19, r21, r22, r24, r25, r26, r27, r29, r32, r35

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 + 9
          app_A(x1,x2) = x2 + 8
          *_0_A() = 2
          *_1_A(x1) = 1
          *_2_A(x1,x2) = x2
          +_0_A() = 5
          +_1_A(x1) = 4
          +_2_A(x1,x2) = 3
          -_0_A() = 1
          -_1_A(x1) = x1 + 4
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          div_2_A(x1,x2) = x2 + 7
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + 5
          filter2_3_A(x1,x2,x3) = 6
          ln_0_A() = 4
          ln_1_A(x1) = 3
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 6
          minus_1_A(x1) = 5
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 6
          t_0_A() = 6
          |1|_0_A() = 5
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 4
          -_2_A(x1,x2) = x1 + 3
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        constant_0 > +_0 = div_2 = t_0 = |0|_0 > +_1 = D_1 = |1|_0 > cons_2 = *_2 = +_2 = pow_2 = filter_2 > map_2# = -_1 = filter2_2 = filter2_3 = -_2 > app = *_0 = -_0 = cons_0 = div_0 = filter_0 = filter2_0 = ln_0 = map_0 = minus_0 = pow_0 = nil_0 > *_1 = cons_1 = div_1 = filter_1 = filter2_1 = ln_1 = map_1 = minus_1 = pow_1
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = [2]
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [2]
      pi(-_2) = []
      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: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(pow_2(x,y)) -> D_1#(y)
p3: D_1#(pow_2(x,y)) -> D_1#(x)
p4: D_1#(ln_1(x)) -> D_1#(x)
p5: D_1#(div_2(x,y)) -> D_1#(y)
p6: D_1#(div_2(x,y)) -> D_1#(x)
p7: D_1#(minus_1(x)) -> 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: D_1#(*_2(x,y)) -> D_1#(x)
p12: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r3, r5, r8, r10, r14, r15, r16, r18, r19, r21, r22, r24, r25, r35

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:
          D_1#_A(x1) = x1
          +_2_A(x1,x2) = x1 + x2 + 10
          pow_2_A(x1,x2) = x1 + x2 + 6
          ln_1_A(x1) = x1 + 8
          div_2_A(x1,x2) = x1 + x2 + 4
          minus_1_A(x1) = x1 + 2
          -_2_A(x1,x2) = x1 + x2 + 4
          *_2_A(x1,x2) = x1 + x2 + 8
          app_A(x1,x2) = x2 + 9
          +_0_A() = 2
          +_1_A(x1) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          ln_0_A() = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 3
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 8
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        -_0 = div_0 = pow_0 = t_0 = constant_0 = filter_2 > +_2 = +_0 = nil_0 > app = +_1 = cons_0 = filter2_0 = map_0 > ln_1 = minus_1 = cons_1 = filter2_1 = map_1 = |0|_0 > -_2 = filter2_2 = ln_0 = minus_0 = D_1 > pow_2 = -_1 = filter2_3 > *_2 = pow_1 = |1|_0 > div_2 > D_1# = div_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(pow_2) = []
      pi(ln_1) = [1]
      pi(div_2) = []
      pi(minus_1) = []
      pi(-_2) = []
      pi(*_2) = [1]
      pi(app) = [2]
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_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:

  p1



-- SCC decomposition.

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

p1: D_1#(pow_2(x,y)) -> D_1#(y)
p2: D_1#(pow_2(x,y)) -> D_1#(x)
p3: D_1#(ln_1(x)) -> D_1#(x)
p4: D_1#(div_2(x,y)) -> D_1#(y)
p5: D_1#(div_2(x,y)) -> D_1#(x)
p6: D_1#(minus_1(x)) -> D_1#(x)
p7: D_1#(-_2(x,y)) -> D_1#(y)
p8: D_1#(-_2(x,y)) -> D_1#(x)
p9: D_1#(*_2(x,y)) -> D_1#(y)
p10: D_1#(*_2(x,y)) -> D_1#(x)
p11: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

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

p1: D_1#(pow_2(x,y)) -> D_1#(y)
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)
p7: D_1#(minus_1(x)) -> D_1#(x)
p8: D_1#(div_2(x,y)) -> D_1#(x)
p9: D_1#(div_2(x,y)) -> D_1#(y)
p10: D_1#(ln_1(x)) -> D_1#(x)
p11: D_1#(pow_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r3, r4, r5, r8, r10, r12, r14, r15, r16, r18, r19, r21, r22, r24, r25, r35

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:
          D_1#_A(x1) = x1 + 1
          pow_2_A(x1,x2) = x1 + x2 + 5
          +_2_A(x1,x2) = x2 + 5
          *_2_A(x1,x2) = x1 + x2 + 9
          -_2_A(x1,x2) = x1 + x2 + 2
          minus_1_A(x1) = x1 + 5
          div_2_A(x1,x2) = x1 + x2 + 8
          ln_1_A(x1) = x1 + 6
          app_A(x1,x2) = x2 + 7
          +_0_A() = 2
          +_1_A(x1) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 4
          filter_1_A(x1) = 3
          filter2_0_A() = 5
          filter2_1_A(x1) = 4
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          ln_0_A() = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = x1 + 4
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_0 > filter2_1 > filter2_2 > div_0 = filter2_3 > div_1 = map_0 > app = +_0 = -_0 = cons_0 = filter_0 = map_1 = pow_0 = t_0 > minus_1 = cons_1 > D_1# = *_2 = minus_0 = nil_0 > pow_2 = filter_2 > +_2 = filter_1 = pow_1 = D_1 = constant_0 > -_2 = +_1 = |1|_0 = |0|_0 > div_2 = -_1 > ln_1 > ln_0
      
  
    partial status:
  
      pi(D_1#) = []
      pi(pow_2) = [2]
      pi(+_2) = []
      pi(*_2) = []
      pi(-_2) = [1, 2]
      pi(minus_1) = [1]
      pi(div_2) = []
      pi(ln_1) = [1]
      pi(app) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = [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:

  p1



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(y)
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)
p6: D_1#(minus_1(x)) -> D_1#(x)
p7: D_1#(div_2(x,y)) -> D_1#(x)
p8: D_1#(div_2(x,y)) -> D_1#(y)
p9: D_1#(ln_1(x)) -> D_1#(x)
p10: D_1#(pow_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(y)
p2: D_1#(pow_2(x,y)) -> D_1#(x)
p3: D_1#(ln_1(x)) -> D_1#(x)
p4: D_1#(div_2(x,y)) -> D_1#(y)
p5: D_1#(div_2(x,y)) -> D_1#(x)
p6: D_1#(minus_1(x)) -> D_1#(x)
p7: D_1#(-_2(x,y)) -> D_1#(y)
p8: D_1#(-_2(x,y)) -> D_1#(x)
p9: D_1#(*_2(x,y)) -> D_1#(y)
p10: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r8, r10, r12, r14, r15, r16, r18, r19, r21, r22, r24, r25, r35

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:
          D_1#_A(x1) = x1
          +_2_A(x1,x2) = x2 + 3
          pow_2_A(x1,x2) = x1 + x2 + 8
          ln_1_A(x1) = x1 + 6
          div_2_A(x1,x2) = x1 + x2 + 3
          minus_1_A(x1) = x1 + 3
          -_2_A(x1,x2) = x1 + x2 + 8
          *_2_A(x1,x2) = x1 + x2 + 8
          app_A(x1,x2) = x2 + 7
          *_0_A() = 7
          *_1_A(x1) = 6
          +_0_A() = 2
          +_1_A(x1) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_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
          ln_0_A() = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 2
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        D_1# = t_0 > +_2 = -_2 = |1|_0 > pow_2 = *_2 = app = *_0 = +_0 = -_0 = cons_0 = div_0 = filter_0 = filter2_0 = map_0 = pow_0 = filter_2 > ln_1 = minus_1 = *_1 = +_1 = -_1 = filter_1 = filter2_1 = pow_1 = nil_0 > div_2 = div_1 = filter2_2 = ln_0 = minus_0 = D_1 = constant_0 > cons_1 = filter2_3 = map_1 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(pow_2) = [2]
      pi(ln_1) = []
      pi(div_2) = []
      pi(minus_1) = [1]
      pi(-_2) = [1, 2]
      pi(*_2) = []
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [1, 2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: D_1#(pow_2(x,y)) -> D_1#(x)
p2: D_1#(ln_1(x)) -> D_1#(x)
p3: D_1#(div_2(x,y)) -> D_1#(y)
p4: D_1#(div_2(x,y)) -> D_1#(x)
p5: D_1#(minus_1(x)) -> 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)

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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: D_1#(pow_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)
p6: D_1#(minus_1(x)) -> D_1#(x)
p7: D_1#(div_2(x,y)) -> D_1#(x)
p8: D_1#(div_2(x,y)) -> D_1#(y)
p9: D_1#(ln_1(x)) -> 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r6, r8, r9, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r25, r35

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:
          D_1#_A(x1) = x1
          pow_2_A(x1,x2) = x1 + x2 + 5
          *_2_A(x1,x2) = x1 + x2 + 4
          -_2_A(x1,x2) = x1 + x2 + 3
          minus_1_A(x1) = x1 + 2
          div_2_A(x1,x2) = x1 + x2 + 3
          ln_1_A(x1) = x1 + 4
          app_A(x1,x2) = x1 + x2 + 4
          *_0_A() = 6
          *_1_A(x1) = 5
          +_0_A() = 1
          +_1_A(x1) = x1 + 2
          +_2_A(x1,x2) = 1
          -_0_A() = 1
          -_1_A(x1) = x1 + 2
          cons_0_A() = 1
          cons_1_A(x1) = x1 + 2
          cons_2_A(x1,x2) = x1 + x2 + 1
          div_0_A() = 1
          div_1_A(x1) = x1 + 2
          filter_0_A() = 6
          filter_1_A(x1) = x1 + 5
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 2
          filter2_2_A(x1,x2) = x1 + x2 + 3
          filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4
          ln_0_A() = 1
          map_0_A() = 1
          map_1_A(x1) = x1 + 2
          minus_0_A() = 1
          pow_0_A() = 1
          pow_1_A(x1) = x1 + 2
          D_1_A(x1) = 0
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        cons_2 > *_2 = filter_2 > app > pow_2 = filter2_3 > -_2 = ln_1 = filter2_2 > minus_1 = div_2 = filter2_1 > D_1# = *_0 = +_1 > *_1 = +_0 = +_2 = -_1 = cons_1 = div_1 = filter_1 = filter2_0 = ln_0 = map_1 = minus_0 = pow_1 = D_1 = constant_0 = nil_0 > -_0 = cons_0 = div_0 = filter_0 = map_0 = pow_0 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(pow_2) = [2]
      pi(*_2) = []
      pi(-_2) = []
      pi(minus_1) = [1]
      pi(div_2) = [2]
      pi(ln_1) = []
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = [1]
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = [1]
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter2_0) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = [3]
      pi(ln_0) = []
      pi(map_0) = []
      pi(map_1) = [1]
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = [1]
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1



-- 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#(minus_1(x)) -> D_1#(x)
p6: D_1#(div_2(x,y)) -> D_1#(x)
p7: D_1#(div_2(x,y)) -> D_1#(y)
p8: D_1#(ln_1(x)) -> 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- 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#(ln_1(x)) -> D_1#(x)
p3: D_1#(div_2(x,y)) -> D_1#(y)
p4: D_1#(div_2(x,y)) -> D_1#(x)
p5: D_1#(minus_1(x)) -> 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)

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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r12, r15, r16, r18, r19, r21, r22, r23, r24, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1
          *_2_A(x1,x2) = x1 + x2 + 2
          ln_1_A(x1) = x1 + 10
          div_2_A(x1,x2) = x1 + x2 + 13
          minus_1_A(x1) = x1 + 6
          -_2_A(x1,x2) = x1 + x2 + 12
          app_A(x1,x2) = x2 + 11
          *_0_A() = 2
          *_1_A(x1) = 1
          +_0_A() = 5
          +_1_A(x1) = 4
          +_2_A(x1,x2) = 3
          -_0_A() = 2
          -_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_1_A(x1) = 12
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          ln_0_A() = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          pow_2_A(x1,x2) = 4
          D_1_A(x1) = x1 + 5
          t_0_A() = 4
          |1|_0_A() = 3
          constant_0_A() = 1
          |0|_0_A() = 0
          filter_2_A(x1,x2) = x1 + x2 + 12
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        nil_0 > filter_0 = filter_2 > app = *_0 = +_0 = -_0 = filter_1 = ln_0 > ln_1 = +_1 = filter2_1 = map_0 > D_1# = filter2_2 = map_1 > *_2 = minus_1 = *_1 = filter2_3 = D_1 = t_0 = constant_0 > minus_0 = pow_0 = pow_2 = |1|_0 = |0|_0 > div_2 = +_2 = pow_1 > -_2 > -_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(*_2) = []
      pi(ln_1) = []
      pi(div_2) = [1, 2]
      pi(minus_1) = [1]
      pi(-_2) = []
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = [1]
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = [1]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: D_1#(ln_1(x)) -> D_1#(x)
p2: D_1#(div_2(x,y)) -> D_1#(y)
p3: D_1#(div_2(x,y)) -> D_1#(x)
p4: D_1#(minus_1(x)) -> D_1#(x)
p5: D_1#(-_2(x,y)) -> D_1#(y)
p6: D_1#(-_2(x,y)) -> D_1#(x)
p7: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: D_1#(ln_1(x)) -> 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#(minus_1(x)) -> D_1#(x)
p6: D_1#(div_2(x,y)) -> D_1#(x)
p7: D_1#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r5, r8, r10, r12, r14, r16, r19, r22, r24, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1 + 4
          ln_1_A(x1) = x1 + 3
          *_2_A(x1,x2) = x2 + 7
          -_2_A(x1,x2) = x1 + x2 + 7
          minus_1_A(x1) = x1 + 5
          div_2_A(x1,x2) = x1 + x2 + 13
          app_A(x1,x2) = 6
          *_0_A() = 2
          *_1_A(x1) = 1
          +_0_A() = 3
          +_1_A(x1) = 2
          +_2_A(x1,x2) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 6
          filter_1_A(x1) = 5
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 5
          filter2_3_A(x1,x2,x3) = 2
          map_0_A() = 2
          map_1_A(x1) = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = x1 + 2
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 11
          filter_2_A(x1,x2) = x1 + x2 + 4
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_2 > div_2 = filter2_3 > ln_1 = *_2 = minus_1 = -_0 = filter_0 = map_0 = D_1 = t_0 = constant_0 = pow_2 > -_2 = |1|_0 = |0|_0 > D_1# = app = *_0 = cons_0 = div_0 = pow_0 > *_1 = +_1 = -_1 = cons_1 = div_1 = filter2_1 = pow_1 > +_0 = +_2 = filter2_0 = map_1 = nil_0 > filter_2 > filter_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(ln_1) = []
      pi(*_2) = []
      pi(-_2) = [1]
      pi(minus_1) = [1]
      pi(div_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = [1]
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: D_1#(*_2(x,y)) -> D_1#(y)
p2: D_1#(-_2(x,y)) -> D_1#(x)
p3: D_1#(-_2(x,y)) -> D_1#(y)
p4: D_1#(minus_1(x)) -> D_1#(x)
p5: D_1#(div_2(x,y)) -> D_1#(x)
p6: D_1#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: D_1#(*_2(x,y)) -> D_1#(y)
p2: D_1#(div_2(x,y)) -> D_1#(y)
p3: D_1#(div_2(x,y)) -> D_1#(x)
p4: D_1#(minus_1(x)) -> D_1#(x)
p5: D_1#(-_2(x,y)) -> D_1#(y)
p6: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r12, r14, r16, r18, r19, r21, r22, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1
          *_2_A(x1,x2) = x2 + 8
          div_2_A(x1,x2) = x1 + x2 + 11
          minus_1_A(x1) = x1 + 5
          -_2_A(x1,x2) = x1 + x2 + 2
          app_A(x1,x2) = x2 + 10
          *_0_A() = 2
          *_1_A(x1) = 1
          +_0_A() = 3
          +_1_A(x1) = 2
          +_2_A(x1,x2) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 1
          filter_1_A(x1) = 9
          filter2_0_A() = 4
          filter2_1_A(x1) = 3
          filter2_2_A(x1,x2) = x1 + 2
          filter2_3_A(x1,x2,x3) = 1
          ln_0_A() = 4
          ln_1_A(x1) = 3
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 4
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 9
          |1|_0_A() = 5
          filter_2_A(x1,x2) = x1 + x2 + 8
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        D_1# > *_2 = *_0 = +_0 = pow_0 > *_1 = +_1 = filter2_0 = pow_1 > app = -_0 = cons_0 = filter_0 = filter2_1 = ln_0 = map_0 > minus_1 = cons_1 = filter_1 = filter2_2 > +_2 = filter2_3 = map_1 = minus_0 = D_1 = constant_0 = pow_2 = |1|_0 = filter_2 > div_2 = |0|_0 = nil_0 > -_2 = div_1 = ln_1 > -_1 = div_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(*_2) = []
      pi(div_2) = []
      pi(minus_1) = [1]
      pi(-_2) = []
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [2]
      pi(|1|_0) = []
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: D_1#(div_2(x,y)) -> D_1#(y)
p2: D_1#(div_2(x,y)) -> D_1#(x)
p3: D_1#(minus_1(x)) -> 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: D_1#(div_2(x,y)) -> D_1#(y)
p2: D_1#(-_2(x,y)) -> D_1#(x)
p3: D_1#(-_2(x,y)) -> D_1#(y)
p4: D_1#(minus_1(x)) -> D_1#(x)
p5: D_1#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r4, r8, r10, r12, r15, r16, r18, r19, r22, r24, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1
          div_2_A(x1,x2) = x1 + x2 + 6
          -_2_A(x1,x2) = x1 + x2 + 6
          minus_1_A(x1) = x1 + 6
          app_A(x1,x2) = 5
          *_0_A() = 4
          *_1_A(x1) = 3
          +_0_A() = 3
          +_1_A(x1) = 2
          +_2_A(x1,x2) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_1_A(x1) = x1 + 6
          filter2_2_A(x1,x2) = 3
          filter2_3_A(x1,x2,x3) = 2
          ln_0_A() = 5
          ln_1_A(x1) = 4
          map_0_A() = 2
          map_1_A(x1) = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 3
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          *_2_A(x1,x2) = 2
          pow_2_A(x1,x2) = x1 + x2 + 8
          filter_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_3 > filter_2 > +_0 = div_0 = ln_0 = map_0 = pow_0 = t_0 > minus_1 = filter_0 > app = *_0 = cons_0 = filter_1 = nil_0 > *_1 = +_1 = cons_1 = filter2_1 = map_1 > +_2 = filter2_2 > ln_1 = *_2 = pow_2 > -_2 = pow_1 > D_1# > div_2 > div_1 = D_1 = constant_0 > |1|_0 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(div_2) = [1]
      pi(-_2) = []
      pi(minus_1) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_1) = [1]
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(*_2) = []
      pi(pow_2) = [1]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- 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#(minus_1(x)) -> D_1#(x)
p4: D_1#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: D_1#(-_2(x,y)) -> D_1#(x)
p2: D_1#(div_2(x,y)) -> D_1#(x)
p3: D_1#(minus_1(x)) -> D_1#(x)
p4: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r12, r14, r15, r18, r19, r21, r22, r24, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1 + 3
          -_2_A(x1,x2) = x1 + x2 + 9
          div_2_A(x1,x2) = x1 + 14
          minus_1_A(x1) = x1 + 2
          app_A(x1,x2) = x2 + 8
          *_0_A() = 2
          *_1_A(x1) = 1
          *_2_A(x1,x2) = x2 + 7
          +_0_A() = 8
          +_1_A(x1) = 7
          +_2_A(x1,x2) = 6
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = 2
          filter2_2_A(x1,x2) = 1
          ln_0_A() = 6
          ln_1_A(x1) = 5
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 6
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 11
          filter_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        app = +_0 = -_0 = cons_0 = div_0 = filter_0 = ln_0 = map_0 = pow_0 = t_0 = filter_2 > *_0 = *_2 = +_1 = cons_1 = filter_1 = map_1 = nil_0 > -_2 = div_2 = *_1 = +_2 = filter2_2 = D_1 = constant_0 = pow_2 > D_1# = minus_1 = -_1 = div_1 = filter2_1 = ln_1 = pow_1 = |1|_0 = |0|_0 > filter2_0 = minus_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(-_2) = []
      pi(div_2) = [1]
      pi(minus_1) = [1]
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [2]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: D_1#(div_2(x,y)) -> D_1#(x)
p2: D_1#(minus_1(x)) -> 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: D_1#(div_2(x,y)) -> D_1#(x)
p2: D_1#(-_2(x,y)) -> D_1#(y)
p3: D_1#(minus_1(x)) -> 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r12, r14, r15, r18, r19, r21, r22, r24, r25, r26, r27, r32, r35

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:
          D_1#_A(x1) = x1
          div_2_A(x1,x2) = x1 + x2 + 8
          -_2_A(x1,x2) = x1 + x2 + 3
          minus_1_A(x1) = x1 + 3
          app_A(x1,x2) = x2 + 4
          *_0_A() = 2
          *_1_A(x1) = 1
          *_2_A(x1,x2) = x2 + 3
          +_0_A() = 1
          +_1_A(x1) = x1 + 2
          +_2_A(x1,x2) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_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
          ln_0_A() = 2
          ln_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = x1 + 2
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 6
          filter_2_A(x1,x2) = x2 + 2
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        filter2_0 = ln_0 = t_0 > filter_0 = filter2_1 = ln_1 = |1|_0 = filter_2 > *_2 = filter_1 = filter2_2 = pow_2 > D_1# = app = *_0 = +_0 = cons_0 = div_0 = map_0 = nil_0 > minus_1 = *_1 = +_1 = cons_1 = map_1 = pow_1 > div_2 = -_2 = -_0 = minus_0 = pow_0 = D_1 = constant_0 > +_2 = -_1 = div_1 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(div_2) = [1, 2]
      pi(-_2) = [1, 2]
      pi(minus_1) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = [1]
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [1, 2]
      pi(filter_2) = [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#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: D_1#(div_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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r12, r14, r15, r18, r19, r22, r24, r25, r26, r27, r29, r32, r35

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:
          D_1#_A(x1) = x1
          div_2_A(x1,x2) = x1 + 9
          -_2_A(x1,x2) = x1 + x2 + 6
          app_A(x1,x2) = x2 + 5
          *_0_A() = 2
          *_1_A(x1) = 1
          *_2_A(x1,x2) = x2 + 4
          +_0_A() = 3
          +_1_A(x1) = 2
          +_2_A(x1,x2) = 1
          -_0_A() = 2
          -_1_A(x1) = 1
          cons_0_A() = 2
          cons_1_A(x1) = 1
          div_0_A() = 2
          div_1_A(x1) = 1
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 1
          filter2_1_A(x1) = x1 + 4
          filter2_2_A(x1,x2) = 3
          ln_0_A() = 4
          ln_1_A(x1) = 3
          map_0_A() = 2
          map_1_A(x1) = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 4
          t_0_A() = 2
          |1|_0_A() = 1
          constant_0_A() = 1
          |0|_0_A() = 0
          minus_1_A(x1) = 4
          pow_2_A(x1,x2) = x1 + x2 + 2
          filter_2_A(x1,x2) = x1 + x2 + 6
          nil_0_A() = 1
  
    base order 2:
  
      precedence:
      
        *_0 > *_1 = *_2 > app = +_0 = -_0 = cons_0 = div_0 = filter_0 = filter2_0 = ln_0 = map_0 = pow_0 = t_0 = filter_2 > div_2 = -_2 = +_1 = filter_1 = filter2_1 = map_1 = D_1 = constant_0 = minus_1 = nil_0 > D_1# = -_1 = cons_1 = div_1 = filter2_2 = ln_1 = |0|_0 > pow_2 > +_2 = pow_1 = |1|_0
      
  
    partial status:
  
      pi(D_1#) = []
      pi(div_2) = []
      pi(-_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = [2]
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(minus_1) = []
      pi(pow_2) = [2]
      pi(filter_2) = [2]
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r8, r10, r11, r12, r14, r16, r18, r19, r21, r22, r24, r25, r26, r27, r29, r32, r35, r37

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:
          D_1#_A(x1) = x1
          -_2_A(x1,x2) = x1 + x2 + 9
          app_A(x1,x2) = x2 + 11
          *_0_A() = 1
          *_1_A(x1) = 3
          *_2_A(x1,x2) = x2 + 2
          +_0_A() = 3
          +_1_A(x1) = 2
          +_2_A(x1,x2) = 1
          -_0_A() = 11
          -_1_A(x1) = 10
          cons_0_A() = 11
          cons_1_A(x1) = 10
          div_0_A() = 11
          div_1_A(x1) = 10
          div_2_A(x1,x2) = 9
          filter_0_A() = 2
          filter_1_A(x1) = 1
          filter2_0_A() = 2
          filter2_1_A(x1) = 1
          filter2_2_A(x1,x2) = x1 + x2 + 12
          filter2_3_A(x1,x2,x3) = 1
          ln_0_A() = 2
          ln_1_A(x1) = 1
          map_0_A() = 2
          map_1_A(x1) = 1
          minus_0_A() = 2
          minus_1_A(x1) = 1
          pow_0_A() = 2
          pow_1_A(x1) = 1
          D_1_A(x1) = 6
          t_0_A() = 6
          |1|_0_A() = 5
          constant_0_A() = 1
          |0|_0_A() = 0
          pow_2_A(x1,x2) = x1 + x2 + 4
          filter_2_A(x1,x2) = x1 + x2 + 9
          nil_0_A() = 1
          filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 10
          true_0_A() = 1
          cons_2_A(x1,x2) = x1 + 9
  
    base order 2:
  
      precedence:
      
        +_0 = div_2 > +_1 = filter2_0 = filter2_4 > D_1# = +_2 = filter2_1 = filter2_3 = D_1 = t_0 = constant_0 = filter_2 > -_2 = *_1 = filter2_2 = ln_0 = pow_0 = |1|_0 = |0|_0 = nil_0 = cons_2 > app = *_0 = -_0 = cons_0 = div_0 = filter_0 = map_0 = minus_0 = true_0 > *_2 = -_1 = cons_1 = div_1 = filter_1 = map_1 = minus_1 = pow_2 > ln_1 = pow_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(-_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1, 2]
      pi(filter2_3) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(pow_2) = [1, 2]
      pi(filter_2) = [1]
      pi(nil_0) = []
      pi(filter2_4) = [2, 4]
      pi(true_0) = []
      pi(cons_2) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.