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, r14, r15, r18, r21, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{5, x1 - 1}
        cons_2_A(x1,x2) = max{4, x1 - 59, x2 - 59}
        app#_A(x1,x2) = max{4, x1 - 2}
        map_1_A(x1) = x1 + 7
        filter2_3_A(x1,x2,x3) = max{x1 + 6, x2 + 7, x3 + 6}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 4
        false_0_A = 1
        filter_2#_A(x1,x2) = x1 + 4
        filter_1_A(x1) = x1 + 7
        app_A(x1,x2) = max{49, x1 - 8, x2 - 8}
        true_0_A = 3
        *_0_A = 6
        *_1_A(x1) = max{7, x1 - 202}
        *_2_A(x1,x2) = max{47, x1 - 210, x2 - 71}
        +_0_A = 9
        +_1_A(x1) = max{10, x1 - 9}
        +_2_A(x1,x2) = max{47, x1 - 48, x2 - 467}
        -_0_A = 9
        -_1_A(x1) = max{10, x1 - 9}
        -_2_A(x1,x2) = max{3, x1 - 420, x2 - 69}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 9}
        div_0_A = 2
        div_1_A(x1) = max{3, x1 - 9}
        div_2_A(x1,x2) = max{4, x1 - 439, x2 - 814}
        filter2_0_A = 4
        filter2_1_A(x1) = max{5, x1 - 9}
        filter2_2_A(x1,x2) = max{6, x1 - 18, x2 - 9}
        ln_0_A = 7
        ln_1_A(x1) = max{6, x1 - 416}
        minus_0_A = 7
        minus_1_A(x1) = max{47, x1 - 397}
        pow_0_A = 6
        pow_1_A(x1) = max{7, x1 - 9}
        pow_2_A(x1,x2) = max{48, x1 - 18, x2 - 22}
        D_1_A(x1) = max{47, x1 - 397}
        t_0_A = 396
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{47, x1 - 59, x2 - 14}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = +_0 = t_0 > minus_1 = pow_2 > cons_2 = map_1 = filter2_3 = filter_1 = minus_0 = pow_1 = D_1 = constant_0 = filter_2 > false_0 = true_0 = *_2 = cons_1 = filter2_2 = ln_1 = pow_0 = |1|_0 = |0|_0 = nil_0 > filter2_4# = filter_2# = *_1 = +_2 = cons_0 = div_1 = div_2 = filter2_1 = ln_0 > app# = *_0 = +_1 = -_2 = div_0 = filter2_0 > map_2# = -_1 > -_0
      
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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

  r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{4, x1 - 2}
        cons_2_A(x1,x2) = max{3, x1 - 14, x2 - 14}
        app#_A(x1,x2) = max{3, x1 - 3}
        filter_1_A(x1) = x1 + 8
        filter2_4#_A(x1,x2,x3,x4) = max{4, x2 - 2}
        app_A(x1,x2) = max{10, x1 - 13}
        true_0_A = 2
        false_0_A = 5
        filter2_3_A(x1,x2,x3) = max{11, x1 - 15, x2 + 2}
        D_1_A(x1) = max{39, x1 - 215}
        constant_0_A = 214
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{9, x1 - 39, x2 - 19}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 = nil_0 > app > filter_1 > cons_2 = filter2_3 > true_0 = false_0 > filter_2# = filter2_4# = D_1 = constant_0 > app# = |0|_0
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app#) = []
      pi(filter_1) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(filter2_3) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
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:

  {p1, p2, p3, p4, p5}


-- 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#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p4: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p5: 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, r10, r11, r12, r14, r16, r18, r19, r21, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x1 + 31
        cons_2_A(x1,x2) = max{x1 + 12, x2 - 22}
        app#_A(x1,x2) = x1 + 30
        filter2_3_A(x1,x2,x3) = max{x1 - 6, x2 + 32, x3 - 26}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 31
        false_0_A = 9
        app_A(x1,x2) = max{9, x1 - 25, x2 - 25}
        true_0_A = 1
        *_0_A = 24
        *_1_A(x1) = max{6, x1 - 26}
        *_2_A(x1,x2) = max{7, x1 - 412, x2 - 293}
        +_0_A = 5
        +_1_A(x1) = max{6, x1 - 25}
        +_2_A(x1,x2) = max{7, x1 - 119, x2 - 26}
        -_0_A = 1
        -_1_A(x1) = max{2, x1 - 26}
        -_2_A(x1,x2) = max{3, x1 - 943, x2 - 835}
        cons_0_A = 63
        cons_1_A(x1) = max{38, x1 - 25}
        div_0_A = 7
        div_1_A(x1) = max{8, x1 - 26}
        div_2_A(x1,x2) = max{7, x1 - 285, x2 - 1120}
        filter_0_A = 8
        filter_1_A(x1) = max{9, x1 - 25}
        filter2_0_A = 36
        filter2_1_A(x1) = max{10, x1 - 26}
        filter2_2_A(x1,x2) = max{x1 + 20, x2 + 58}
        ln_0_A = 7
        ln_1_A(x1) = max{8, x1 - 234}
        map_0_A = 1
        map_1_A(x1) = max{8, x1 - 26}
        minus_0_A = 1
        minus_1_A(x1) = max{8, x1 - 884}
        pow_0_A = 1
        pow_1_A(x1) = max{7, x1 - 26}
        pow_2_A(x1,x2) = max{8, x1 - 52, x2 - 56}
        D_1_A(x1) = max{7, x1 - 886}
        t_0_A = 885
        |1|_0_A = 6
        constant_0_A = 885
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{10, x1 - 24, x2 - 36}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_2 > filter_2 = nil_0 > false_0 = true_0 > filter_2# = filter2_4# = app = map_1 > app# = filter2_3 = minus_1 = pow_2 > cons_1 = filter_1 = filter2_0 = filter2_2 = map_0 = minus_0 = pow_1 = D_1 = t_0 = constant_0 > cons_0 = filter_0 = filter2_1 = ln_1 = pow_0 = |1|_0 = |0|_0 > div_2 = ln_0 > *_2 = -_2 = div_1 > *_1 = +_2 = -_1 = div_0 > *_0 = +_1 = -_0 > +_0
      
  
    partial status:
  
      pi(filter_2#) = [1]
      pi(cons_2) = [1]
      pi(app#) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      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(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_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: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_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, r9, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = max{x3 + 3, x4 + 3}
        false_0_A = 6
        filter_2#_A(x1,x2) = x2 + 2
        cons_2_A(x1,x2) = max{8, x1 + 4, x2 + 4}
        app_A(x1,x2) = max{16, x1 - 18, x2 + 5}
        true_0_A = 6
        *_0_A = 1
        *_1_A(x1) = max{15, x1 - 1}
        *_2_A(x1,x2) = max{14, x1 - 73, x2 - 57}
        +_0_A = 11
        +_1_A(x1) = max{12, x1 - 1}
        +_2_A(x1,x2) = max{13, x2 - 15}
        -_0_A = 17
        -_1_A(x1) = max{16, x1 - 1}
        -_2_A(x1,x2) = max{10, x1 - 21, x2 - 25}
        cons_1_A(x1) = x1 + 23
        div_0_A = 9
        div_1_A(x1) = max{10, x1 - 1}
        div_2_A(x1,x2) = max{11, x1 - 74, x2 - 177}
        filter_0_A = 19
        filter_1_A(x1) = max{4, x1 - 57}
        filter2_0_A = 12
        filter2_1_A(x1) = max{13, x1 - 1}
        filter2_2_A(x1,x2) = max{14, x1 - 20, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{15, x1 - 38, x2 - 19, x3 - 1}
        ln_0_A = 1
        ln_1_A(x1) = max{12, x1 - 63}
        map_0_A = 13
        map_1_A(x1) = max{14, x1 - 1}
        minus_0_A = 10
        minus_1_A(x1) = max{11, x1 - 114}
        pow_0_A = 1
        pow_1_A(x1) = max{6, x1 - 1}
        pow_2_A(x1,x2) = max{15, x1 - 41, x2 - 20}
        D_1_A(x1) = max{12, x1 - 114}
        t_0_A = 113
        |1|_0_A = 4
        constant_0_A = 113
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{7, x1 - 76, x2 - 57}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_0 = t_0 = filter_2 = nil_0 > app > filter_1 = filter2_3 = minus_1 = pow_2 > cons_2 = *_2 = filter_0 = filter2_2 = map_1 = minus_0 = pow_1 > filter2_4# = filter_2# = *_1 = +_2 = cons_1 = filter2_1 = map_0 = pow_0 > false_0 = true_0 = *_0 = +_1 = -_2 = filter2_0 = D_1 = constant_0 > -_1 = ln_1 = |1|_0 = |0|_0 > -_0 = div_2 = ln_0 > div_1 > div_0
      
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      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) = []
      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(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_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, r6, r8, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x2 + 34
        cons_2_A(x1,x2) = max{x1 - 32, x2 + 35}
        app_A(x1,x2) = max{31, x1 - 37, x2 - 70}
        *_0_A = 5
        *_1_A(x1) = max{6, x1 - 71}
        *_2_A(x1,x2) = max{7, x1 - 413, x2 - 284}
        +_0_A = 36
        +_1_A(x1) = max{30, x1 - 71}
        +_2_A(x1,x2) = max{11, x1 - 131, x2 - 120}
        -_0_A = 30
        -_1_A(x1) = max{31, x1 - 70}
        -_2_A(x1,x2) = max{8, x1 - 256, x2 - 117}
        cons_0_A = 30
        cons_1_A(x1) = max{31, x1 - 70}
        div_0_A = 7
        div_1_A(x1) = max{8, x1 - 71}
        div_2_A(x1,x2) = max{9, x1 - 109, x2 - 252}
        filter_0_A = 29
        filter_1_A(x1) = max{30, x1 - 71}
        filter2_0_A = 36
        filter2_1_A(x1) = max{29, x1 - 71}
        filter2_2_A(x1,x2) = max{30, x1 - 109, x2 - 71}
        filter2_3_A(x1,x2,x3) = max{31, x1 - 146, x2 - 108, x3 - 70}
        ln_0_A = 29
        ln_1_A(x1) = max{30, x1 - 100}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 71}
        minus_0_A = 30
        minus_1_A(x1) = max{31, x1 - 71}
        pow_0_A = 36
        pow_1_A(x1) = max{11, x1 - 71}
        pow_2_A(x1,x2) = max{12, x1 - 130, x2 - 125}
        D_1_A(x1) = max{10, x1 - 130}
        t_0_A = 129
        |1|_0_A = 9
        constant_0_A = 129
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{32, x1 - 36, x2 - 37}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2 > cons_2 = app = nil_0 > filter_0 = ln_1 = minus_0 > map_2# = *_0 = -_0 = cons_1 = filter_1 = filter2_3 = ln_0 = map_1 = minus_1 = pow_2 = t_0 > cons_0 = filter2_2 = map_0 = pow_1 = D_1 = constant_0 > +_2 = div_2 = filter2_1 = pow_0 = |1|_0 = |0|_0 > *_2 = +_1 = -_2 = div_1 = filter2_0 > *_1 = +_0 = -_1 = div_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = []
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_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(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: 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

  r1, r3, r5, r8, r9, r10, r12, r14, r15, r16, r19, r22, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{2, x1}
        +_2_A(x1,x2) = max{131, x1 + 14, x2 + 3}
        pow_2_A(x1,x2) = max{x1 + 126, x2 + 130}
        ln_1_A(x1) = max{266, x1 + 131}
        div_2_A(x1,x2) = max{133, x1 + 4, x2 + 34}
        minus_1_A(x1) = max{2, x1 + 1}
        -_2_A(x1,x2) = max{x1 + 129, x2 + 65}
        *_2_A(x1,x2) = max{117, x1 + 5, x2 + 71}
        app_A(x1,x2) = max{116, x1 - 108, x2 - 100}
        *_0_A = 106
        *_1_A(x1) = max{107, x1 - 101}
        +_0_A = 106
        +_1_A(x1) = max{107, x1 - 101}
        -_0_A = 62
        -_1_A(x1) = max{63, x1 - 101}
        cons_0_A = 107
        cons_1_A(x1) = max{1, x1 - 101}
        cons_2_A(x1,x2) = max{4, x1 - 210, x2 - 101}
        div_0_A = 107
        div_1_A(x1) = max{1, x1 - 100}
        filter_0_A = 101
        filter_1_A(x1) = max{102, x1 - 101}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 101}
        filter2_2_A(x1,x2) = max{3, x1 - 210, x2 - 101}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 319, x2 - 210, x3 - 101}
        map_0_A = 106
        map_1_A(x1) = max{107, x1 - 319}
        pow_0_A = 107
        pow_1_A(x1) = max{34, x1 - 101}
        D_1_A(x1) = max{63, x1 - 132}
        t_0_A = 63
        |1|_0_A = 62
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{103, x1 - 210, x2 - 101}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > *_2 > pow_2 = ln_1 = *_0 = filter_0 = t_0 > -_2 > div_2 = D_1 = constant_0 > minus_1 = |1|_0 = |0|_0 = filter_2 = nil_0 > D_1# = app > *_1 = +_1 = -_1 = cons_2 = div_1 = filter_1 = filter2_3 = map_1 = pow_1 > +_0 = -_0 = cons_1 = div_0 = filter2_2 = map_0 = pow_0 > cons_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(pow_2) = [2]
      pi(ln_1) = []
      pi(div_2) = [2]
      pi(minus_1) = [1]
      pi(-_2) = []
      pi(*_2) = []
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p12



-- 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#(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)

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#(+_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)
p10: D_1#(pow_2(x,y)) -> D_1#(x)
p11: D_1#(pow_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, r5, r8, r9, 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:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{9, x1}
        +_2_A(x1,x2) = max{x1 + 8, x2 + 27}
        *_2_A(x1,x2) = max{64, x1 + 11, x2 + 31}
        -_2_A(x1,x2) = max{x1 + 29, x2 + 19}
        minus_1_A(x1) = x1 + 11
        div_2_A(x1,x2) = max{x1 + 10, x2 + 11}
        ln_1_A(x1) = x1 + 9
        pow_2_A(x1,x2) = max{x1 + 79, x2 + 30}
        app_A(x1,x2) = max{405, x1 - 66, x2 + 12}
        *_0_A = 64
        *_1_A(x1) = max{65, x1 - 1}
        +_0_A = 5
        +_1_A(x1) = max{6, x1 - 1}
        -_0_A = 65
        -_1_A(x1) = max{16, x1 - 1}
        cons_0_A = 65
        cons_1_A(x1) = max{67, x1 - 1}
        cons_2_A(x1,x2) = max{203, x1 - 403, x2 - 205}
        div_0_A = 12
        div_1_A(x1) = max{13, x1 - 1}
        filter_0_A = 64
        filter_1_A(x1) = max{65, x1 - 201}
        filter2_0_A = 12
        filter2_1_A(x1) = max{13, x1 - 1}
        filter2_2_A(x1,x2) = max{10, x1 - 68, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{11, x1 - 135, x2 - 68, x3 - 1}
        ln_0_A = 7
        map_0_A = 65
        map_1_A(x1) = max{404, x1 - 404}
        minus_0_A = 9
        pow_0_A = 65
        pow_1_A(x1) = max{68, x1 - 1}
        D_1_A(x1) = max{76, x1 - 63}
        t_0_A = 62
        |1|_0_A = 61
        constant_0_A = 62
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{203, x1 - 268, x2 - 202}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > *_2 > app = +_0 = -_0 = t_0 > -_2 = *_1 = +_1 = -_1 > div_2 = *_0 = cons_1 = cons_2 = filter2_3 = map_1 = filter_2 = nil_0 > pow_2 = cons_0 = div_1 = filter_1 = filter2_2 = map_0 > minus_1 = ln_1 = div_0 = filter_0 = filter2_1 = pow_1 = D_1 = constant_0 > D_1# = filter2_0 = ln_0 = minus_0 = pow_0 = |1|_0 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(*_2) = []
      pi(-_2) = []
      pi(minus_1) = []
      pi(div_2) = []
      pi(ln_1) = []
      pi(pow_2) = []
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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) = []
      pi(nil_0) = []

The following rules are removed:

  p11



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(*_2(x,y)) -> D_1#(x)
p3: D_1#(*_2(x,y)) -> D_1#(y)
p4: D_1#(-_2(x,y)) -> D_1#(x)
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#(x)
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, r5, r8, r9, r12, r14, r15, r16, r19, r22, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{15, x1}
        +_2_A(x1,x2) = max{x1 + 15, x2 + 8}
        pow_2_A(x1,x2) = max{200, x1 + 138, x2 + 133}
        ln_1_A(x1) = max{199, x1 + 49}
        div_2_A(x1,x2) = max{x1 + 26, x2 + 48}
        minus_1_A(x1) = max{256, x1 + 249}
        -_2_A(x1,x2) = max{x1 + 202, x2 + 209}
        *_2_A(x1,x2) = max{124, x1 + 67, x2 + 21}
        app_A(x1,x2) = max{6, x1 - 8, x2 - 257}
        *_0_A = 4
        *_1_A(x1) = max{5, x1 - 258}
        +_0_A = 5
        +_1_A(x1) = max{6, x1 - 258}
        -_0_A = 1
        -_1_A(x1) = max{2, x1 - 258}
        cons_0_A = 7
        cons_1_A(x1) = max{1, x1 - 258}
        cons_2_A(x1,x2) = max{2, x1 - 543, x2 - 278}
        filter_0_A = 1
        filter_1_A(x1) = max{3, x1 - 257}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 258}
        filter2_2_A(x1,x2) = max{3, x1 - 267, x2 - 258}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 276, x2 - 267, x3 - 258}
        map_0_A = 9
        map_1_A(x1) = 5
        pow_0_A = 4
        pow_1_A(x1) = max{5, x1 - 258}
        D_1_A(x1) = max{6, x1 - 257}
        constant_0_A = 256
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{4, x1 - 294, x2 - 282}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 > -_2 = *_2 > pow_2 = minus_1 = *_0 = cons_0 > ln_1 > div_2 > D_1# = filter_2 = nil_0 > app = map_0 > *_1 = +_1 = -_1 = cons_2 = filter_1 = filter2_3 = map_1 = pow_1 = D_1 = constant_0 > +_0 = -_0 = cons_1 = filter_0 = filter2_2 = pow_0 = |0|_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1, 2]
      pi(pow_2) = [2]
      pi(ln_1) = []
      pi(div_2) = [2]
      pi(minus_1) = [1]
      pi(-_2) = [2]
      pi(*_2) = []
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p10



-- 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#(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)

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#(+_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)
p9: 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, r5, r8, r9, r10, r14, r15, r16, r19, r22, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{11, x1}
        +_2_A(x1,x2) = max{11, x1 + 4, x2 + 8}
        *_2_A(x1,x2) = max{55, x1 + 16, x2 + 51}
        -_2_A(x1,x2) = max{x1 + 53, x2 + 58}
        minus_1_A(x1) = max{162, x1 + 158}
        div_2_A(x1,x2) = max{56, x1 + 53, x2 + 40}
        ln_1_A(x1) = x1 + 41
        pow_2_A(x1,x2) = max{x1 + 53, x2 + 15}
        app_A(x1,x2) = max{15, x1 - 18, x2 - 8}
        +_0_A = 14
        +_1_A(x1) = max{15, x1 - 8}
        -_0_A = 13
        -_1_A(x1) = max{14, x1 - 9}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 56}
        cons_2_A(x1,x2) = max{3, x1 - 74, x2 - 10}
        div_0_A = 1
        div_1_A(x1) = max{15, x1 - 8}
        filter2_0_A = 10
        filter2_1_A(x1) = max{11, x1 - 9}
        filter2_2_A(x1,x2) = max{12, x1 - 27, x2 - 8}
        filter2_3_A(x1,x2,x3) = max{13, x1 - 46, x2 - 26, x3 - 8}
        map_0_A = 1
        map_1_A(x1) = max{13, x1 - 93}
        pow_0_A = 17
        pow_1_A(x1) = max{15, x1 - 8}
        D_1_A(x1) = max{3, x1 - 163}
        t_0_A = 162
        |1|_0_A = 0
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{9, x1 - 84, x2 - 54}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        map_1 > +_2 = minus_1 = filter_2 = nil_0 > app > div_2 = +_1 = cons_2 = t_0 > *_2 = +_0 = div_1 = |1|_0 > pow_2 > -_2 = ln_1 = cons_0 = pow_0 > D_1# = -_1 = cons_1 = div_0 = filter2_3 = map_0 = pow_1 = D_1 = constant_0 > -_0 = filter2_2 = |0|_0 > filter2_1 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(*_2) = []
      pi(-_2) = [2]
      pi(minus_1) = [1]
      pi(div_2) = [2]
      pi(ln_1) = []
      pi(pow_2) = [2]
      pi(app) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      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(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p9



-- 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, r5, r8, r9, r10, r12, r14, r15, r16, r19, r22, r23, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = max{5, x1}
        +_2_A(x1,x2) = max{x1 + 5, x2 + 4}
        ln_1_A(x1) = max{60, x1 + 7}
        div_2_A(x1,x2) = max{x1 + 56, x2 + 6}
        minus_1_A(x1) = max{60, x1 + 59}
        -_2_A(x1,x2) = max{195, x1 + 6, x2 + 56}
        *_2_A(x1,x2) = max{x1 + 62, x2 + 63}
        app_A(x1,x2) = max{64, x1 - 10, x2 - 15}
        *_0_A = 9
        *_1_A(x1) = max{8, x1 - 16}
        +_0_A = 8
        +_1_A(x1) = max{9, x1 - 16}
        -_0_A = 8
        -_1_A(x1) = max{9, x1 - 16}
        cons_0_A = 9
        cons_1_A(x1) = max{11, x1 - 16}
        cons_2_A(x1,x2) = max{12, x1 - 65, x2 - 17}
        div_0_A = 8
        div_1_A(x1) = max{9, x1 - 16}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 15}
        filter2_0_A = 9
        filter2_1_A(x1) = max{1, x1 - 16}
        filter2_2_A(x1,x2) = max{2, x1 - 27, x2 - 16}
        filter2_3_A(x1,x2,x3) = max{3, x1 - 38, x2 - 27, x3 - 16}
        map_0_A = 9
        map_1_A(x1) = 8
        pow_0_A = 9
        pow_1_A(x1) = max{0, x1 - 16}
        pow_2_A(x1,x2) = max{63, x2 - 195}
        D_1_A(x1) = max{0, x1 - 61}
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{3, x1 - 60, x2 - 17}
        nil_0_A = 2
  
    base order 2:
  
      precedence:
      
        ln_1 = -_2 = *_2 > div_2 = minus_1 > app = filter_0 = map_0 > +_2 = *_1 = -_1 = div_1 = map_1 = pow_2 = D_1 = constant_0 > D_1# = *_0 = +_1 = -_0 = cons_1 = cons_2 = div_0 = filter_1 = filter2_2 = filter2_3 = pow_1 = |0|_0 = filter_2 > +_0 = cons_0 = filter2_1 = pow_0 = nil_0 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(ln_1) = [1]
      pi(div_2) = []
      pi(minus_1) = [1]
      pi(-_2) = [1]
      pi(*_2) = []
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(filter2_3) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p8



-- 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#(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)

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#(+_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#(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)
p7: 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, r2, r3, r5, r8, r9, r10, r12, r14, r15, r16, r19, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 22
        +_2_A(x1,x2) = max{x1 + 150, x2 - 10}
        -_2_A(x1,x2) = max{x1 + 53, x2 + 261}
        minus_1_A(x1) = x1 + 23
        div_2_A(x1,x2) = max{40, x1 + 30, x2 + 23}
        ln_1_A(x1) = max{41, x1 + 24}
        app_A(x1,x2) = max{9, x1 - 6, x2 - 3}
        *_0_A = 5
        *_1_A(x1) = max{1, x1 - 154}
        *_2_A(x1,x2) = max{9, x1 - 160, x2 - 42}
        +_0_A = 4
        +_1_A(x1) = max{5, x1 - 4}
        -_0_A = 5
        -_1_A(x1) = max{4, x1 - 4}
        cons_0_A = 5
        cons_1_A(x1) = max{0, x1 - 4}
        cons_2_A(x1,x2) = max{2, x2 - 4}
        div_0_A = 4
        div_1_A(x1) = max{5, x1 - 4}
        filter_0_A = 5
        filter_1_A(x1) = max{2, x1 - 13}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 4}
        filter2_2_A(x1,x2) = max{4, x1 - 11, x2 - 4}
        filter2_3_A(x1,x2,x3) = max{5, x1 - 17, x2 - 11, x3 - 3}
        map_0_A = 5
        map_1_A(x1) = max{7, x1 - 3}
        pow_0_A = 4
        pow_1_A(x1) = max{5, x1 - 453}
        pow_2_A(x1,x2) = max{6, x1 - 460, x2 - 319}
        D_1_A(x1) = max{8, x1 - 262}
        t_0_A = 6
        |1|_0_A = 5
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{3, x1 - 19, x2 - 3}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        -_2 = ln_1 > +_2 = minus_1 = div_2 > D_1# = app > div_0 = filter2_3 = D_1 = constant_0 > *_2 = +_1 = -_1 = cons_2 = div_1 = filter2_2 = map_1 = pow_2 = t_0 = |0|_0 = filter_2 > *_1 = +_0 = -_0 = cons_1 = filter_1 = filter2_1 = map_0 = pow_1 = |1|_0 = nil_0 > *_0 = cons_0 = filter_0 = filter2_0 = pow_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1]
      pi(-_2) = [2]
      pi(minus_1) = [1]
      pi(div_2) = []
      pi(ln_1) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(map_0) = []
      pi(map_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p7



-- SCC decomposition.

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

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

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, r5, r8, r9, r10, r12, r14, r15, r16, r18, r19, r21, r22, r23, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 6
        +_2_A(x1,x2) = max{x1 + 65, x2 - 1}
        div_2_A(x1,x2) = max{x1 + 11, x2 + 54}
        minus_1_A(x1) = x1 + 7
        -_2_A(x1,x2) = max{x1 + 29, x2 + 68}
        app_A(x1,x2) = max{634, x1 - 69, x2 + 8}
        *_0_A = 6
        *_1_A(x1) = max{7, x1 - 64}
        *_2_A(x1,x2) = max{43, x1 - 134, x2 - 191}
        +_0_A = 72
        +_1_A(x1) = max{2, x1 - 1}
        -_0_A = 6
        -_1_A(x1) = max{7, x1 - 1}
        cons_0_A = 70
        cons_1_A(x1) = max{71, x1 - 1}
        cons_2_A(x1,x2) = max{213, x1 - 204, x2 - 215}
        div_0_A = 1
        div_1_A(x1) = max{2, x1 - 1}
        filter_0_A = 68
        filter_1_A(x1) = max{67, x1 - 282}
        filter2_0_A = 68
        filter2_1_A(x1) = max{9, x1 - 1}
        filter2_2_A(x1,x2) = max{67, x1 - 71, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{68, x1 - 141, x2 - 71, x3 - 1}
        ln_0_A = 1
        ln_1_A(x1) = max{9, x1 - 165}
        map_0_A = 1
        map_1_A(x1) = max{68, x1 - 1}
        minus_0_A = 1
        pow_0_A = 68
        pow_1_A(x1) = max{9, x1 - 53}
        pow_2_A(x1,x2) = max{12, x1 - 123, x2 - 327}
        D_1_A(x1) = max{67, x1 - 41}
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{635, x1 - 281, x2 - 147}
        nil_0_A = 148
  
    base order 2:
  
      precedence:
      
        -_2 = div_1 > div_2 > app > minus_1 = ln_0 = pow_2 = D_1 = constant_0 > +_2 = +_0 = cons_2 = ln_1 = |0|_0 > D_1# = *_2 = +_1 = -_1 = cons_1 = div_0 = filter_0 = filter2_3 = map_1 = minus_0 = pow_1 = filter_2 = nil_0 > *_1 = -_0 = cons_0 = filter_1 = filter2_0 = filter2_2 = map_0 = pow_0 > *_0 = filter2_1
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(div_2) = [2]
      pi(minus_1) = [1]
      pi(-_2) = []
      pi(app) = [2]
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(minus_0) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(D_1) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

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

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#(+_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)
p5: 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, r2, r3, r5, r8, r9, r10, r14, r15, r16, r18, r19, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 35
        +_2_A(x1,x2) = max{x1 + 197, x2 - 6}
        -_2_A(x1,x2) = max{x1 - 18, x2 + 48}
        minus_1_A(x1) = max{297, x1 + 35}
        div_2_A(x1,x2) = max{x1 + 36, x2 + 126}
        app_A(x1,x2) = max{14, x1 - 4, x2 - 6}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 257}
        *_2_A(x1,x2) = max{1, x1 - 262, x2 - 214}
        +_0_A = 3
        +_1_A(x1) = max{13, x1 - 7}
        -_0_A = 2
        -_1_A(x1) = max{3, x1 - 14}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 7}
        cons_2_A(x1,x2) = max{4, x1 - 15, x2 - 8}
        div_0_A = 2
        div_1_A(x1) = max{3, x1 - 7}
        filter2_0_A = 3
        filter2_1_A(x1) = max{1, x1 - 7}
        filter2_2_A(x1,x2) = max{2, x1 - 12, x2 - 7}
        filter2_3_A(x1,x2,x3) = max{3, x1 - 17, x2 - 12, x3 - 15}
        ln_0_A = 1
        ln_1_A(x1) = max{2, x1 - 474}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 18}
        pow_0_A = 2
        pow_1_A(x1) = max{3, x1 - 170}
        pow_2_A(x1,x2) = max{4, x1 - 175, x2 - 519}
        D_1_A(x1) = max{5, x1 - 298}
        t_0_A = 4
        |1|_0_A = 3
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{10, x1 - 13, x2 - 8}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        -_2 > minus_1 = div_2 = filter_2 = nil_0 > app = -_0 > -_1 = cons_0 = cons_2 = div_1 = filter2_3 = map_1 = pow_2 = D_1 = t_0 = constant_0 > +_2 = cons_1 = div_0 = filter2_2 = ln_1 = map_0 = pow_1 = |1|_0 = |0|_0 > D_1# = *_1 = *_2 = +_1 = filter2_1 = ln_0 = pow_0 > *_0 = +_0 = filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1]
      pi(-_2) = []
      pi(minus_1) = [1]
      pi(div_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(filter2_0) = []
      pi(filter2_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(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: 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, r5, r8, r9, r10, r12, r14, r15, r16, r18, r19, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 37
        +_2_A(x1,x2) = max{x1 + 38, x2 - 32}
        div_2_A(x1,x2) = max{163, x1 + 38, x2 - 32}
        minus_1_A(x1) = max{198, x1 + 99}
        -_2_A(x1,x2) = max{192, x1 + 1, x2 + 57}
        app_A(x1,x2) = max{58, x1 - 4, x2 - 33}
        *_0_A = 2
        *_1_A(x1) = max{3, x1 - 33}
        *_2_A(x1,x2) = max{57, x1 - 96, x2 - 74}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 34}
        -_0_A = 2
        -_1_A(x1) = max{3, x1 - 34}
        cons_0_A = 5
        cons_1_A(x1) = max{2, x1 - 34}
        cons_2_A(x1,x2) = max{3, x1 - 59, x2 - 34}
        div_0_A = 2
        div_1_A(x1) = max{3, x1 - 34}
        filter_0_A = 2
        filter_1_A(x1) = max{3, x1 - 34}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 34}
        filter2_2_A(x1,x2) = max{3, x1 - 46, x2 - 34}
        filter2_3_A(x1,x2,x3) = max{1, x1 - 50, x2 - 38, x3 - 65}
        ln_0_A = 3
        ln_1_A(x1) = max{23, x1 - 131}
        map_0_A = 1
        map_1_A(x1) = max{2, x1 - 59}
        pow_0_A = 1
        pow_1_A(x1) = max{3, x1 - 71}
        pow_2_A(x1,x2) = max{19, x1 - 76, x2 - 193}
        D_1_A(x1) = x1 + 98
        t_0_A = 19
        |1|_0_A = 18
        constant_0_A = 31
        |0|_0_A = 30
        filter_2_A(x1,x2) = max{5, x1 - 35, x2 - 4}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        -_2 > div_2 = minus_1 > app = -_0 = filter_2 > *_2 = -_1 = div_1 = filter_1 = ln_1 = D_1 = t_0 = constant_0 > +_2 = *_1 = cons_2 = div_0 = filter_0 = ln_0 = map_1 = |0|_0 = nil_0 > D_1# = *_0 = +_1 = cons_1 = filter2_3 = map_0 = pow_2 > +_0 = cons_0 = filter2_2 = pow_1 = |1|_0 > filter2_1 = pow_0 > filter2_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(div_2) = []
      pi(minus_1) = [1]
      pi(-_2) = [2]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      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) = []
      pi(nil_0) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(div_2(x,y)) -> D_1#(x)
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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(minus_1(x)) -> D_1#(x)
p3: 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, r2, r3, r5, r6, r8, r9, r10, r12, r14, r15, r16, r18, r19, r22, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 98
        +_2_A(x1,x2) = max{x1 + 99, x2 - 44}
        minus_1_A(x1) = x1 + 110
        div_2_A(x1,x2) = max{x1 + 99, x2 - 52}
        app_A(x1,x2) = max{97, x1 - 5, x2 - 75}
        *_0_A = 4
        *_1_A(x1) = max{1, x1 - 76}
        *_2_A(x1,x2) = max{34, x1 - 206, x2 - 143}
        +_0_A = 4
        +_1_A(x1) = max{97, x1 - 75}
        -_0_A = 3
        -_1_A(x1) = max{4, x1 - 300}
        -_2_A(x1,x2) = max{33, x1 - 306, x2 - 149}
        cons_0_A = 4
        cons_1_A(x1) = max{8, x1 - 76}
        cons_2_A(x1,x2) = max{2, x1 - 175, x2 - 77}
        div_0_A = 3
        div_1_A(x1) = max{4, x1 - 76}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 93}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 76}
        filter2_2_A(x1,x2) = max{3, x1 - 82, x2 - 76}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 87, x2 - 81, x3 - 76}
        ln_0_A = 4
        ln_1_A(x1) = max{36, x1 - 75}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 175}
        pow_0_A = 1
        pow_1_A(x1) = max{2, x1 - 76}
        D_1_A(x1) = max{43, x1 - 111}
        t_0_A = 43
        |1|_0_A = 42
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{3, x1 - 99, x2 - 77}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        minus_1 = div_2 = filter_2 = nil_0 > app > +_2 = filter_0 = map_0 > D_1# = +_1 = -_2 = cons_1 = cons_2 = div_1 = filter_1 = filter2_3 = map_1 = pow_0 = D_1 = t_0 = constant_0 > +_0 = -_1 = cons_0 = div_0 = filter2_0 = filter2_2 = |1|_0 = |0|_0 > *_2 = -_0 = filter2_1 > *_1 = ln_1 = pow_1 > *_0 = ln_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = []
      pi(minus_1) = [1]
      pi(div_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      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(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(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: D_1#(+_2(x,y)) -> D_1#(x)
p2: D_1#(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, r5, r6, r9, r11, r12, r14, r15, r16, r18, r19, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 21
        +_2_A(x1,x2) = max{x1 + 22, x2 - 15}
        minus_1_A(x1) = max{60, x1 + 57}
        app_A(x1,x2) = max{29, x1 - 5, x2 - 36}
        *_0_A = 1
        *_1_A(x1) = max{2, x1 - 51}
        *_2_A(x1,x2) = max{3, x1 - 57, x2 - 42}
        +_0_A = 2
        +_1_A(x1) = max{3, x1 - 37}
        -_0_A = 4
        -_1_A(x1) = max{3, x1 - 37}
        -_2_A(x1,x2) = max{27, x1 - 42, x2 - 57}
        cons_1_A(x1) = max{3, x1 - 1}
        cons_2_A(x1,x2) = max{4, x1 - 30, x2 - 37}
        div_1_A(x1) = max{4, x1 - 9}
        div_2_A(x1,x2) = max{28, x1 - 15, x2 - 57}
        filter_0_A = 4
        filter_1_A(x1) = max{1, x1 - 37}
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 36}
        filter2_2_A(x1,x2) = max{3, x1 - 41, x2 - 37}
        filter2_3_A(x1,x2,x3) = max{4, x1 - 46, x2 - 42, x3 - 37}
        ln_0_A = 4
        ln_1_A(x1) = max{28, x1 - 57}
        map_0_A = 3
        map_1_A(x1) = max{4, x1 - 37}
        pow_0_A = 3
        pow_1_A(x1) = max{4, x1 - 51}
        pow_2_A(x1,x2) = max{26, x1 - 57, x2 - 57}
        D_1_A(x1) = 2
        t_0_A = 2
        |1|_0_A = 1
        constant_0_A = 2
        |0|_0_A = 1
        filter_2_A(x1,x2) = max{39, x1 - 4, x2 - 5}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 = minus_1 = filter_2 = nil_0 > app > cons_2 = filter_1 = ln_1 > div_2 > -_2 = div_1 > pow_2 > *_2 = filter2_3 > D_1# = *_1 = +_1 = -_1 = cons_1 = filter_0 = filter2_0 = filter2_2 = ln_0 = map_1 = pow_1 = D_1 = t_0 = constant_0 > *_0 = +_0 = -_0 = filter2_1 = map_0 = pow_0 = |1|_0 = |0|_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1]
      pi(minus_1) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_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(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

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

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(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#(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, r5, r6, r8, r9, r10, r11, r12, r14, r15, r16, r18, r19, r21, r22, r23, r24, r25, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = x1 + 62
        +_2_A(x1,x2) = max{x1 + 63, x2 - 17}
        app_A(x1,x2) = max{128, x1 - 82, x2 - 4}
        *_0_A = 81
        *_1_A(x1) = max{1, x1 - 5}
        *_2_A(x1,x2) = max{15, x1 - 120, x2 - 92}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 5}
        -_0_A = 15
        -_1_A(x1) = max{16, x1 - 5}
        -_2_A(x1,x2) = max{17, x1 - 88, x2 - 17}
        cons_0_A = 81
        cons_1_A(x1) = max{1, x1 - 5}
        cons_2_A(x1,x2) = max{2, x1 - 259, x2 - 128}
        div_0_A = 80
        div_1_A(x1) = max{81, x1 - 5}
        div_2_A(x1,x2) = max{8, x1 - 195, x2 - 273}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 254}
        filter2_0_A = 78
        filter2_1_A(x1) = max{79, x1 - 5}
        filter2_2_A(x1,x2) = max{80, x1 - 88, x2 - 5}
        filter2_3_A(x1,x2,x3) = max{81, x1 - 171, x2 - 88, x3 - 5}
        ln_0_A = 14
        ln_1_A(x1) = max{15, x1 - 180}
        map_0_A = 1
        map_1_A(x1) = max{81, x1 - 259}
        minus_0_A = 81
        minus_1_A(x1) = max{88, x1 - 17}
        pow_0_A = 81
        pow_1_A(x1) = max{80, x1 - 5}
        pow_2_A(x1,x2) = max{79, x1 - 89, x2 - 18}
        D_1_A(x1) = max{16, x1 - 90}
        t_0_A = 16
        |1|_0_A = 15
        constant_0_A = 1
        |0|_0_A = 0
        filter_2_A(x1,x2) = max{3, x1 - 337, x2 - 129}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        +_2 = filter_2 = nil_0 > app > pow_2 > +_0 = -_1 = -_2 = cons_2 = filter2_3 = minus_1 = pow_0 > D_1# = *_0 = +_1 = -_0 = cons_1 = filter_1 = filter2_2 = map_1 = minus_0 = pow_1 = D_1 = t_0 = constant_0 > *_1 = *_2 = cons_0 = filter_0 = filter2_1 = ln_1 = map_0 = |1|_0 = |0|_0 > div_2 = filter2_0 = ln_0 > div_1 > div_0
      
  
    partial status:
  
      pi(D_1#) = [1]
      pi(+_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(cons_2) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_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(pow_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.