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, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = x1 + 26
        cons_2_A(x1,x2) = 0
        app#_A(x1,x2) = max{24, x1 + 22}
        map_1_A(x1) = max{20, x1 + 5}
        filter2_3_A(x1,x2,x3) = max{25, x1 - 23, x2 + 23, x3 - 19}
        filter2_4#_A(x1,x2,x3,x4) = max{47, x2 + 23}
        false_0_A = 27
        filter_2#_A(x1,x2) = max{47, x1 + 23}
        filter_1_A(x1) = max{27, x1 + 9}
        app_A(x1,x2) = max{22, x1 + 18, x2 + 10}
        true_0_A = 19
        *_0_A = 6
        *_1_A(x1) = max{23, x1 - 19}
        *_2_A(x1,x2) = max{39, x2 + 1}
        +_0_A = 6
        +_1_A(x1) = max{23, x1 + 3}
        +_2_A(x1,x2) = max{2, x1 - 39}
        -_0_A = 1
        -_1_A(x1) = max{9, x1 - 19}
        -_2_A(x1,x2) = max{24, x1 - 25, x2 - 26}
        D_0_A = 25
        D_1_A(x1) = 24
        cons_0_A = 6
        cons_1_A(x1) = max{23, x1 - 1}
        div_0_A = 1
        div_1_A(x1) = max{0, x1 - 8}
        div_2_A(x1,x2) = max{11, x1}
        filter_0_A = 10
        filter_2_A(x1,x2) = 26
        filter2_0_A = 1
        filter2_1_A(x1) = max{20, x1 - 57}
        filter2_2_A(x1,x2) = max{21, x1 - 40, x2 + 6}
        filter2_4_A(x1,x2,x3,x4) = 26
        ln_0_A = 40
        ln_1_A(x1) = max{39, x1 - 12}
        map_0_A = 1
        map_2_A(x1,x2) = max{24, x2 + 3}
        minus_0_A = 24
        minus_1_A(x1) = 23
        pow_0_A = 1
        pow_1_A(x1) = max{21, x1 - 1}
        pow_2_A(x1,x2) = max{38, x1, x2}
        t_0_A = 24
        |1|_0_A = 23
        constant_0_A = 2
        |0|_0_A = 1
        |2|_0_A = 12
        nil_0_A = 25
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x1 - 35, x2 - 55}
          cons_2_A(x1,x2) = max{11, x1 + 5, x2}
          app#_A(x1,x2) = max{0, x1 - 40, x2 - 50}
          map_1_A(x1) = max{28, x1 + 5}
          filter2_3_A(x1,x2,x3) = max{28, x1 - 30, x2 + 27, x3 - 24}
          filter2_4#_A(x1,x2,x3,x4) = max{0, x1 - 70, x2 - 40, x3 - 64, x4 - 55}
          false_0_A = 69
          filter_2#_A(x1,x2) = max{0, x1 - 40, x2 - 55}
          filter_1_A(x1) = max{38, x1 + 5}
          app_A(x1,x2) = max{x1 + 29, x2 + 5}
          true_0_A = 69
          *_0_A = 47
          *_1_A(x1) = max{76, x1 - 30}
          *_2_A(x1,x2) = max{105, x1 - 1, x2 - 16}
          +_0_A = 29
          +_1_A(x1) = max{58, x1 - 30}
          +_2_A(x1,x2) = max{87, x1 - 339, x2 - 412}
          -_0_A = 29
          -_1_A(x1) = max{58, x1 - 30}
          -_2_A(x1,x2) = max{87, x1 - 338, x2 - 410}
          D_0_A = 105
          D_1_A(x1) = max{134, x1 - 292}
          cons_0_A = 0
          cons_1_A(x1) = max{29, x1 - 19}
          div_0_A = 0
          div_1_A(x1) = max{28, x1 - 30}
          div_2_A(x1,x2) = max{28, x1 - 135, x2 - 442}
          filter_0_A = 9
          filter_2_A(x1,x2) = max{57, x1 + 1, x2 + 1}
          filter2_0_A = 0
          filter2_1_A(x1) = max{29, x1 - 30}
          filter2_2_A(x1,x2) = max{58, x1 - 30, x2 - 2}
          filter2_4_A(x1,x2,x3,x4) = max{57, x1 - 28, x2 + 1, x3 + 5, x4 + 1}
          ln_0_A = 261
          ln_1_A(x1) = max{290, x1 - 77}
          map_0_A = 0
          map_2_A(x1,x2) = max{54, x1 + 34, x2 + 5}
          minus_0_A = 0
          minus_1_A(x1) = max{29, x1 - 135}
          pow_0_A = 85
          pow_1_A(x1) = max{114, x1 - 30}
          pow_2_A(x1,x2) = max{143, x1 - 2, x2 - 24}
          t_0_A = 0
          |1|_0_A = 0
          constant_0_A = 0
          |0|_0_A = 0
          |2|_0_A = 27
          nil_0_A = 2
          map_2#!_A(x1,x2) = max{12, x2 + 1}
          cons_2!_A(x1,x2) = max{1, x1 - 86, x2 - 58}
          app#!_A(x1,x2) = max{x1 - 29, x2}
          map_1!_A(x1) = max{13, x1 - 6}
          filter2_3!_A(x1,x2,x3) = max{12, x1 - 72, x2 - 3, x3 - 53}
          filter2_4#!_A(x1,x2,x3,x4) = max{x1 - 59, x2 - 24, x4}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{10, x1 - 24, x2}
          filter_1!_A(x1) = max{9, x1 - 25}
          app!_A(x1,x2) = 14
          true_0!_A = 12
          *_0!_A = 16
          *_1!_A(x1) = 15
          *_2!_A(x1,x2) = max{16, x1 - 218, x2 - 237}
          +_0!_A = 1
          +_1!_A(x1) = max{2, x1 - 238}
          +_2!_A(x1,x2) = max{16, x1 - 237}
          -_0!_A = 1
          -_1!_A(x1) = max{12, x1 - 560}
          -_2!_A(x1,x2) = max{13, x1 - 559, x2 - 424}
          D_0!_A = 54
          D_1!_A(x1) = max{53, x1 - 88}
          cons_0!_A = 1
          cons_1!_A(x1) = 14
          div_0!_A = 1
          div_1!_A(x1) = max{11, x1 - 988}
          div_2!_A(x1,x2) = max{12, x1 - 987, x2 - 1296}
          filter_0!_A = 1
          filter_2!_A(x1,x2) = max{13, x1 - 2, x2 - 56}
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{15, x1 - 74}
          filter2_2!_A(x1,x2) = max{11, x1 - 73, x2 - 4}
          filter2_4!_A(x1,x2,x3,x4) = max{13, x1 - 71, x2 - 2, x3 - 52, x4 - 56}
          ln_0!_A = 1
          ln_1!_A(x1) = max{15, x1 - 89}
          map_0!_A = 1
          map_2!_A(x1,x2) = max{14, x1 - 5, x2 - 3}
          minus_0!_A = 16
          minus_1!_A(x1) = max{15, x1 - 358}
          pow_0!_A = 1
          pow_1!_A(x1) = max{2, x1 - 1}
          pow_2!_A(x1,x2) = max{x1 + 53, x2 - 88}
          t_0!_A = 55
          |1|_0!_A = 54
          constant_0!_A = 1
          |0|_0!_A = 0
          |2|_0!_A = 54
          nil_0!_A = 15
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = [1]
      pi(map_1) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = [1]
      pi(app) = [2]
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(*_2) = [2]
      pi(+_0) = []
      pi(+_1) = [1]
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = []
      pi(-_2) = []
      pi(D_0) = []
      pi(D_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [2]
      pi(filter2_4) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_2) = [2]
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(|2|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2, p6, p7



-- SCC decomposition.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p2: 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 estimated dependency graph contains the following SCCs:

  {p1}
  {p3, p4, p5}


-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{4, x1 - 1}
        cons_2_A(x1,x2) = max{3, x1 - 334, x2 - 340}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{0, x2 - 2}
          cons_2_A(x1,x2) = max{7, x1 - 9, x2 + 1}
          map_2#!_A(x1,x2) = max{3, x2 - 2}
          cons_2!_A(x1,x2) = max{2, x1 - 1280, x2 - 10}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_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, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter2_4#_A(x1,x2,x3,x4) = x2 + 80
        false_0_A = 66
        filter_2#_A(x1,x2) = x1 + 80
        cons_2_A(x1,x2) = max{x1 - 5, x2}
        app_A(x1,x2) = max{78, x1 - 3, x2 + 76}
        true_0_A = 64
        *_0_A = 4
        *_1_A(x1) = max{75, x1 + 4}
        *_2_A(x1,x2) = max{74, x1 - 80, x2 + 45}
        +_0_A = 8
        +_1_A(x1) = 4
        +_2_A(x1,x2) = 0
        -_0_A = 76
        -_1_A(x1) = max{77, x1 + 30}
        -_2_A(x1,x2) = max{72, x1 + 27}
        D_0_A = 36
        D_1_A(x1) = max{32, x1 + 30}
        cons_0_A = 81
        cons_1_A(x1) = max{77, x1 - 1}
        div_0_A = 83
        div_1_A(x1) = max{79, x1 - 1}
        div_2_A(x1,x2) = max{43, x2 - 28}
        filter_0_A = 73
        filter_1_A(x1) = x1 + 71
        filter_2_A(x1,x2) = max{x1 + 67, x2 + 16}
        filter2_0_A = 83
        filter2_1_A(x1) = max{79, x1 + 8}
        filter2_2_A(x1,x2) = max{x1 + 4, x2 + 75}
        filter2_3_A(x1,x2,x3) = max{77, x1 - 1, x2 + 71, x3 + 15}
        filter2_4_A(x1,x2,x3,x4) = max{x1 - 65, x2 + 67, x3 + 11, x4 + 16}
        ln_0_A = 4
        ln_1_A(x1) = max{68, x1 - 57}
        map_0_A = 4
        map_1_A(x1) = max{77, x1 + 74}
        map_2_A(x1,x2) = max{x1, x2 + 76}
        minus_0_A = 4
        minus_1_A(x1) = 0
        pow_0_A = 4
        pow_1_A(x1) = max{5, x1 - 151}
        pow_2_A(x1,x2) = max{34, x1 - 155, x2 - 191}
        t_0_A = 178
        |1|_0_A = 177
        constant_0_A = 0
        |0|_0_A = 31
        |2|_0_A = 1
        nil_0_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 1}
          false_0_A = 303
          filter_2#_A(x1,x2) = max{1, x1 - 1}
          cons_2_A(x1,x2) = 0
          app_A(x1,x2) = 147
          true_0_A = 301
          *_0_A = 71
          *_1_A(x1) = 56
          *_2_A(x1,x2) = 0
          +_0_A = 144
          +_1_A(x1) = 147
          +_2_A(x1,x2) = 1
          -_0_A = 0
          -_1_A(x1) = 56
          -_2_A(x1,x2) = 128
          D_0_A = 56
          D_1_A(x1) = 145
          cons_0_A = 202
          cons_1_A(x1) = 147
          div_0_A = 56
          div_1_A(x1) = 56
          div_2_A(x1,x2) = 17
          filter_0_A = 56
          filter_1_A(x1) = 56
          filter_2_A(x1,x2) = 1
          filter2_0_A = 207
          filter2_1_A(x1) = 56
          filter2_2_A(x1,x2) = 56
          filter2_3_A(x1,x2,x3) = 56
          filter2_4_A(x1,x2,x3,x4) = 1
          ln_0_A = 56
          ln_1_A(x1) = 19
          map_0_A = 56
          map_1_A(x1) = 147
          map_2_A(x1,x2) = 1
          minus_0_A = 220
          minus_1_A(x1) = 19
          pow_0_A = 0
          pow_1_A(x1) = 70
          pow_2_A(x1,x2) = 1
          t_0_A = 17
          |1|_0_A = 31
          constant_0_A = 19
          |0|_0_A = 1
          |2|_0_A = 102
          nil_0_A = 0
          filter2_4#!_A(x1,x2,x3,x4) = max{302, x1}
          false_0!_A = 303
          filter_2#!_A(x1,x2) = 302
          cons_2!_A(x1,x2) = max{147, x2 + 146}
          app!_A(x1,x2) = max{150, x1 - 55, x2 + 148}
          true_0!_A = 148
          *_0!_A = 14
          *_1!_A(x1) = x1 + 16
          *_2!_A(x1,x2) = max{16, x2 - 4}
          +_0!_A = 91
          +_1!_A(x1) = 90
          +_2!_A(x1,x2) = max{91, x1 - 54, x2 - 78}
          -_0!_A = 1
          -_1!_A(x1) = max{0, x1 - 1}
          -_2!_A(x1,x2) = 0
          D_0!_A = 0
          D_1!_A(x1) = 0
          cons_0!_A = 1
          cons_1!_A(x1) = max{146, x1 - 1}
          div_0!_A = 13
          div_1!_A(x1) = max{12, x1 - 1}
          div_2!_A(x1,x2) = max{13, x1 + 11, x2 + 2}
          filter_0!_A = 154
          filter_1!_A(x1) = x1 + 153
          filter_2!_A(x1,x2) = 152
          filter2_0!_A = 1
          filter2_1!_A(x1) = 151
          filter2_2!_A(x1,x2) = x2 + 149
          filter2_3!_A(x1,x2,x3) = max{0, x1 - 149, x2 - 1, x3 - 1}
          filter2_4!_A(x1,x2,x3,x4) = max{151, x1 - 148}
          ln_0!_A = 1
          ln_1!_A(x1) = max{15, x1 - 1}
          map_0!_A = 1
          map_1!_A(x1) = max{2, x1 - 2}
          map_2!_A(x1,x2) = 147
          minus_0!_A = 1
          minus_1!_A(x1) = max{164, x1 + 18}
          pow_0!_A = 1
          pow_1!_A(x1) = max{149, x1 - 1}
          pow_2!_A(x1,x2) = max{x1 + 1, x2 + 16}
          t_0!_A = 18
          |1|_0!_A = 17
          constant_0!_A = 2
          |0|_0!_A = 1
          |2|_0!_A = 12
          nil_0!_A = 148
  
    partial status:
  
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = [1]
      pi(*_2) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(-_0) = []
      pi(-_1) = [1]
      pi(-_2) = []
      pi(D_0) = []
      pi(D_1) = [1]
      pi(cons_0) = []
      pi(cons_1) = []
      pi(div_0) = []
      pi(div_1) = []
      pi(div_2) = []
      pi(filter_0) = []
      pi(filter_1) = [1]
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = [1]
      pi(filter2_3) = []
      pi(filter2_4) = []
      pi(ln_0) = []
      pi(ln_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(map_2) = []
      pi(minus_0) = []
      pi(minus_1) = []
      pi(pow_0) = []
      pi(pow_1) = []
      pi(pow_2) = []
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(|2|_0) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(div_0(),y) -> div_1(y)
r11: app(div_1(x1),y) -> div_2(x1,y)
r12: app(filter_0(),y) -> filter_1(y)
r13: app(filter_1(x1),y) -> filter_2(x1,y)
r14: app(filter2_0(),y) -> filter2_1(y)
r15: app(filter2_1(x1),y) -> filter2_2(x1,y)
r16: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r17: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r18: app(ln_0(),y) -> ln_1(y)
r19: app(map_0(),y) -> map_1(y)
r20: app(map_1(x1),y) -> map_2(x1,y)
r21: app(minus_0(),y) -> minus_1(y)
r22: app(pow_0(),y) -> pow_1(y)
r23: app(pow_1(x1),y) -> pow_2(x1,y)
r24: D_1(t_0()) -> |1|_0()
r25: D_1(constant_0()) -> |0|_0()
r26: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y))
r27: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y)))
r28: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y))
r29: D_1(minus_1(x)) -> minus_1(D_1(x))
r30: D_1(div_2(x,y)) -> -_2(div_2(D_1(x),y),div_2(*_2(x,D_1(y)),pow_2(y,|2|_0())))
r31: D_1(ln_1(x)) -> div_2(D_1(x),x)
r32: D_1(pow_2(x,y)) -> +_2(*_2(*_2(y,pow_2(x,-_2(y,|1|_0()))),D_1(x)),*_2(*_2(pow_2(x,y),ln_1(x)),D_1(y)))
r33: map_2(f,nil_0()) -> nil_0()
r34: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r35: filter_2(f,nil_0()) -> nil_0()
r36: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r37: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r38: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r4, r14, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r35

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{2, x2 - 2}
        cons_2_A(x1,x2) = max{25, x1 + 22, x2 + 20}
        filter2_4#_A(x1,x2,x3,x4) = max{21, x4 + 1}
        app_A(x1,x2) = 27
        true_0_A = 1
        *_0_A = 20
        *_1_A(x1) = 0
        +_1_A(x1) = max{0, x1 - 21}
        +_2_A(x1,x2) = 20
        filter2_0_A = 22
        filter2_1_A(x1) = 23
        D_1_A(x1) = x1 + 66
        t_0_A = 83
        |1|_0_A = 82
        constant_0_A = 21
        |0|_0_A = 20
        *_2_A(x1,x2) = 33
        -_2_A(x1,x2) = max{x1 - 50, x2 + 65}
        minus_1_A(x1) = max{18, x1 - 67}
        div_2_A(x1,x2) = max{29, x1 - 34}
        pow_2_A(x1,x2) = max{x1 + 15, x2 - 163}
        |2|_0_A = 230
        ln_1_A(x1) = x1 + 30
        map_2_A(x1,x2) = max{27, x1 + 26, x2}
        nil_0_A = 28
        filter_2_A(x1,x2) = max{24, x2 + 1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          filter_2#_A(x1,x2) = max{1, x1 - 1, x2 - 71}
          cons_2_A(x1,x2) = max{70, x1 + 14, x2}
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 1, x3 - 57, x4 - 71}
          app_A(x1,x2) = max{112, x1 - 27, x2 + 56}
          true_0_A = 47
          *_0_A = 2
          *_1_A(x1) = max{2, x1 - 1}
          +_1_A(x1) = max{2, x1 - 92}
          +_2_A(x1,x2) = max{98, x1 - 119, x2 - 127}
          filter2_0_A = 7
          filter2_1_A(x1) = max{4, x1 - 1}
          D_1_A(x1) = max{118, x1 - 127}
          t_0_A = 98
          |1|_0_A = 118
          constant_0_A = 98
          |0|_0_A = 0
          *_2_A(x1,x2) = max{112, x1 - 64, x2 - 1}
          -_2_A(x1,x2) = max{113, x1 - 131, x2 - 119}
          minus_1_A(x1) = max{98, x1 - 119}
          div_2_A(x1,x2) = max{0, x1 - 119, x2 - 158}
          pow_2_A(x1,x2) = max{32, x1 - 32, x2 - 71}
          |2|_0_A = 308
          ln_1_A(x1) = max{98, x1 - 31}
          map_2_A(x1,x2) = max{x1 - 1, x2 + 56}
          nil_0_A = 58
          filter_2_A(x1,x2) = max{x1 - 55, x2 + 57}
          filter_2#!_A(x1,x2) = max{x1, x2 - 5}
          cons_2!_A(x1,x2) = 0
          filter2_4#!_A(x1,x2,x3,x4) = max{x2, x3 + 10, x4 - 5}
          app!_A(x1,x2) = max{10, x1 - 3, x2 + 9}
          true_0!_A = 1
          *_0!_A = 1
          *_1!_A(x1) = max{0, x1 - 1}
          +_1!_A(x1) = max{11, x1 - 116}
          +_2!_A(x1,x2) = max{8, x1 - 116, x2 - 111}
          filter2_0!_A = 4
          filter2_1!_A(x1) = max{3, x1 - 4}
          D_1!_A(x1) = max{12, x1 - 99}
          t_0!_A = 13
          |1|_0!_A = 12
          constant_0!_A = 1
          |0|_0!_A = 0
          *_2!_A(x1,x2) = 0
          -_2!_A(x1,x2) = max{11, x1 - 108, x2 - 337}
          minus_1!_A(x1) = max{12, x1 - 337}
          div_2!_A(x1,x2) = max{0, x1 - 704, x2 - 585}
          pow_2!_A(x1,x2) = 0
          |2|_0!_A = 13
          ln_1!_A(x1) = max{11, x1 - 832}
          map_2!_A(x1,x2) = max{x1 + 55, x2 - 1}
          nil_0!_A = 0
          filter_2!_A(x1,x2) = 0
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(filter2_4#) = []
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_1) = []
      pi(+_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(D_1) = [1]
      pi(t_0) = []
      pi(|1|_0) = []
      pi(constant_0) = []
      pi(|0|_0) = []
      pi(*_2) = []
      pi(-_2) = []
      pi(minus_1) = []
      pi(div_2) = []
      pi(pow_2) = []
      pi(|2|_0) = []
      pi(ln_1) = []
      pi(map_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2

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

  r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = 0
        +_2_A(x1,x2) = max{7, x1 - 7, x2 - 15}
        pow_2_A(x1,x2) = max{x1 + 13, x2 - 32}
        ln_1_A(x1) = max{4, x1 - 33}
        div_2_A(x1,x2) = max{4, x1 - 33, x2 - 33}
        minus_1_A(x1) = max{2, x1 - 33}
        -_2_A(x1,x2) = max{0, x1 - 32, x2 - 32}
        *_2_A(x1,x2) = max{0, x1 - 16, x2 - 33}
        D_1_A(x1) = max{3, x1 - 8}
        t_0_A = 12
        |1|_0_A = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          D_1#_A(x1) = max{0, x1 - 49}
          +_2_A(x1,x2) = max{38, x1 + 18, x2 + 36}
          pow_2_A(x1,x2) = max{x1 + 17, x2 + 1}
          ln_1_A(x1) = x1 + 44
          div_2_A(x1,x2) = max{x1 + 45, x2 + 33}
          minus_1_A(x1) = max{44, x1 + 1}
          -_2_A(x1,x2) = max{x1 + 16, x2 + 16}
          *_2_A(x1,x2) = max{x1 + 48, x2 + 31}
          D_1_A(x1) = max{65, x1 - 93}
          t_0_A = 0
          |1|_0_A = 0
          D_1#!_A(x1) = max{6, x1 - 37}
          +_2!_A(x1,x2) = max{5, x2 - 127}
          pow_2!_A(x1,x2) = max{7, x1 - 13, x2 - 233}
          ln_1!_A(x1) = max{3, x1 - 219}
          div_2!_A(x1,x2) = max{2, x1 - 283, x2 - 346}
          minus_1!_A(x1) = max{3, x1 - 192}
          -_2!_A(x1,x2) = max{4, x1 - 234, x2 - 142}
          *_2!_A(x1,x2) = max{3, x1 - 284, x2 - 190}
          D_1!_A(x1) = max{4, x1 - 47}
          t_0!_A = 1
          |1|_0!_A = 0
  
    partial status:
  
      pi(D_1#) = []
      pi(+_2) = []
      pi(pow_2) = []
      pi(ln_1) = []
      pi(div_2) = []
      pi(minus_1) = []
      pi(-_2) = []
      pi(*_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []

The following rules are removed:

  p4, p5, p6, p7, p10, 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#(pow_2(x,y)) -> D_1#(y)
p3: D_1#(pow_2(x,y)) -> D_1#(x)
p4: D_1#(-_2(x,y)) -> D_1#(y)
p5: D_1#(-_2(x,y)) -> D_1#(x)
p6: D_1#(+_2(x,y)) -> D_1#(y)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(-_0(),y) -> -_1(y)
r6: app(-_1(x1),y) -> -_2(x1,y)
r7: app(D_0(),y) -> D_1(y)
r8: app(cons_0(),y) -> cons_1(y)
r9: app(cons_1(x1),y) -> cons_2(x1,y)
r10: app(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#(+_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#(pow_2(x,y)) -> D_1#(x)
p6: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = 0
        +_2_A(x1,x2) = max{8, x1 - 22, x2 - 22}
        -_2_A(x1,x2) = max{17, x1 - 98, x2 - 28}
        pow_2_A(x1,x2) = max{19, x1 - 17, x2 - 22}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          D_1#_A(x1) = max{0, x1 - 87}
          +_2_A(x1,x2) = max{123, x1 + 5, x2 + 1}
          -_2_A(x1,x2) = max{128, x1 + 86, x2 + 46}
          pow_2_A(x1,x2) = max{126, x1 + 5, x2 + 72}
          D_1#!_A(x1) = max{1, x1 - 124}
          +_2!_A(x1,x2) = max{6, x1 - 136, x2 - 93}
          -_2!_A(x1,x2) = max{7, x1 - 137, x2 - 53}
          pow_2!_A(x1,x2) = max{8, x1 - 123, x2 - 130}
  
    partial status:
  
      pi(D_1#) = []
      pi(+_2) = []
      pi(-_2) = []
      pi(pow_2) = []

The following rules are removed:

  p3, p4, p5, p6



-- SCC decomposition.

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

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

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#(+_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

  r24

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D_1#_A(x1) = 1
        +_2_A(x1,x2) = max{2, x1 - 200, x2 - 128}
        D_1_A(x1) = max{16, x1 - 55}
        t_0_A = 54
        |1|_0_A = 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          D_1#_A(x1) = max{0, x1 - 144}
          +_2_A(x1,x2) = max{805, x1 + 143, x2 + 1}
          D_1_A(x1) = max{10, x1 - 1125}
          t_0_A = 0
          |1|_0_A = 2
          D_1#!_A(x1) = max{1, x1 - 803}
          +_2!_A(x1,x2) = max{4, x1 - 791, x2 - 802}
          D_1!_A(x1) = x1 + 19
          t_0!_A = 5
          |1|_0!_A = 4
  
    partial status:
  
      pi(D_1#) = []
      pi(+_2) = []
      pi(D_1) = []
      pi(t_0) = []
      pi(|1|_0) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.