YES
We show the termination of the TRS R:

  app(p(),app(s(),x)) -> x
  app(fact(),|0|()) -> app(s(),|0|())
  app(fact(),app(s(),x)) -> app(app(*(),app(s(),x)),app(fact(),app(p(),app(s(),x))))
  app(app(*(),|0|()),y) -> |0|()
  app(app(*(),app(s(),x)),y) -> app(app(+(),app(app(*(),x),y)),y)
  app(app(+(),x),|0|()) -> x
  app(app(+(),x),app(s(),y)) -> app(s(),app(app(+(),x),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(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(fact_0(),y) -> fact_1(y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filter2_0(),y) -> filter2_1(y)
  app(filter2_1(x1),y) -> filter2_2(x1,y)
  app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
  app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(p_0(),y) -> p_1(y)
  app(s_0(),y) -> s_1(y)
  p_1(s_1(x)) -> x
  fact_1(|0|_0()) -> s_1(|0|_0())
  fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
  *_2(|0|_0(),y) -> |0|_0()
  *_2(s_1(x),y) -> +_2(*_2(x,y),y)
  +_2(x,|0|_0()) -> x
  +_2(x,s_1(y)) -> s_1(+_2(x,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#(*_1(x1),y) -> *_2#(x1,y)
p2: app#(+_1(x1),y) -> +_2#(x1,y)
p3: app#(fact_0(),y) -> fact_1#(y)
p4: app#(filter_1(x1),y) -> filter_2#(x1,y)
p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p6: app#(map_1(x1),y) -> map_2#(x1,y)
p7: app#(p_0(),y) -> p_1#(y)
p8: fact_1#(s_1(x)) -> *_2#(s_1(x),fact_1(p_1(s_1(x))))
p9: fact_1#(s_1(x)) -> fact_1#(p_1(s_1(x)))
p10: fact_1#(s_1(x)) -> p_1#(s_1(x))
p11: *_2#(s_1(x),y) -> +_2#(*_2(x,y),y)
p12: *_2#(s_1(x),y) -> *_2#(x,y)
p13: +_2#(x,s_1(y)) -> +_2#(x,y)
p14: map_2#(f,cons_2(x,xs)) -> app#(f,x)
p15: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)
p16: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p17: filter_2#(f,cons_2(x,xs)) -> app#(f,x)
p18: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p19: 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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

  {p4, p5, p6, p14, p15, p16, p17, p18, p19}
  {p9}
  {p12}
  {p13}


-- 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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r17, r19, r27

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{15, x1 + 14}
        cons_2_A(x1,x2) = max{16, x1 - 21, x2 - 21}
        app#_A(x1,x2) = x1 + 13
        map_1_A(x1) = max{15, x1 + 14}
        filter2_3_A(x1,x2,x3) = max{x1 - 6, x2 + 15, x3 - 3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 14
        false_0_A = 4
        filter_2#_A(x1,x2) = x1 + 14
        filter_1_A(x1) = max{4, x1 + 2}
        app_A(x1,x2) = max{27, x1, x2 + 3}
        true_0_A = 4
        *_0_A = 0
        *_1_A(x1) = max{0, x1 - 1}
        +_0_A = 1
        +_1_A(x1) = 0
        cons_0_A = 0
        cons_1_A(x1) = max{0, x1 - 21}
        filter_0_A = 0
        filter2_0_A = 0
        filter2_1_A(x1) = max{0, x1 - 1}
        filter2_2_A(x1,x2) = max{0, x1 - 2, x2 - 1}
        s_0_A = 1
        s_1_A(x1) = max{4, x1 - 31}
        fact_1_A(x1) = max{26, x1 - 6}
        |0|_0_A = 5
        filter_2_A(x1,x2) = max{20, x1 - 7, x2 - 17}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app = +_0 = filter2_0 > filter2_3 = fact_1 > cons_2 = map_1 = filter2_2 = filter_2 = nil_0 > false_0 = filter_1 = true_0 = *_1 = cons_1 = filter2_1 = |0|_0 > map_2# = filter2_4# = filter_2# = *_0 = +_1 = cons_0 = filter_0 = s_1 > app# = s_0
      
  
    partial status:
  
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(app#) = []
      pi(map_1) = [1]
      pi(filter2_3) = [2]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(filter_1) = [1]
      pi(app) = [1]
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(fact_1) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

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

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs)

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r8, r10, r11, r12, r14, r17, r19, r27

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{7, x2}
        cons_2_A(x1,x2) = max{x1 + 7, x2 + 6}
        app_A(x1,x2) = max{17, x1 - 19, x2 - 8}
        *_0_A = 18
        *_1_A(x1) = max{16, x1 - 9}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 13}
        filter_0_A = 16
        filter_1_A(x1) = max{17, x1 - 17}
        filter2_0_A = 14
        filter2_1_A(x1) = max{15, x1 - 9}
        filter2_2_A(x1,x2) = max{16, x1 - 29, x2 - 9}
        filter2_3_A(x1,x2,x3) = max{17, x1 - 48, x2 - 28, x3 - 8}
        map_0_A = 38
        map_1_A(x1) = max{18, x1 - 9}
        s_0_A = 6
        s_1_A(x1) = max{7, x1 - 31}
        fact_1_A(x1) = max{17, x1 - 9}
        |0|_0_A = 8
        filter_2_A(x1,x2) = max{18, x1 - 16, x2 - 13}
        nil_0_A = 8
  
    base order 2:
  
      precedence:
      
        filter_2 > app = filter_0 = nil_0 > cons_2 = +_0 = filter_1 = filter2_3 = map_1 = fact_1 > map_2# = filter2_2 = map_0 > *_1 = filter2_1 = |0|_0 > *_0 = +_1 = filter2_0 = s_1 > s_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1]
      pi(app) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_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(s_0) = []
      pi(s_1) = []
      pi(fact_1) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: app(*_0(),y) -> *_1(y)
r2: app(*_1(x1),y) -> *_2(x1,y)
r3: app(+_0(),y) -> +_1(y)
r4: app(+_1(x1),y) -> +_2(x1,y)
r5: app(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r10, r14, r17, r19, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 8
        filter2_3_A(x1,x2,x3) = max{x2 + 10, x3 - 3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 9
        true_0_A = 4
        filter_2#_A(x1,x2) = x1 + 9
        cons_2_A(x1,x2) = max{6, x1 - 28, x2 - 27}
        app_A(x1,x2) = max{23, x1 - 13, x2 - 13}
        false_0_A = 2
        filter_1_A(x1) = x1 + 10
        *_0_A = 11
        *_1_A(x1) = max{12, x1 - 22}
        +_0_A = 12
        +_1_A(x1) = max{11, x1 - 57}
        cons_0_A = 12
        cons_1_A(x1) = max{5, x1 - 14}
        filter2_0_A = 5
        filter2_1_A(x1) = max{6, x1 - 14}
        map_0_A = 11
        map_1_A(x1) = max{12, x1 - 14}
        s_0_A = 1
        s_1_A(x1) = max{2, x1 - 24}
        fact_1_A(x1) = max{2, x1 - 17}
        |0|_0_A = 20
        filter_2_A(x1,x2) = max{26, x1 - 66, x2 - 36}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_2 > filter_2 > app = false_0 = nil_0 > app# = filter2_3 = filter_1 = cons_1 = map_1 > filter2_4# = filter_2# = *_1 = cons_0 = map_0 = |0|_0 > true_0 = *_0 = +_1 = filter2_1 = s_1 > +_0 = filter2_0 = s_0 = fact_1
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(true_0) = []
      pi(filter_2#) = []
      pi(cons_2) = []
      pi(app) = []
      pi(false_0) = []
      pi(filter_1) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(fact_1) = []
      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: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y)
p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs)
p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs)
p5: filter_2#(f,cons_2(x,xs)) -> app#(f,x)

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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: 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: 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)) -> app#(f,x)
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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r8, r10, r11, r14, r17, r19, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 14
        filter2_3_A(x1,x2,x3) = max{18, x1 - 1, x2 + 15, x3}
        filter2_4#_A(x1,x2,x3,x4) = x2 + 15
        false_0_A = 1
        filter_2#_A(x1,x2) = x1 + 15
        cons_2_A(x1,x2) = max{19, x1 - 40, x2 - 40}
        app_A(x1,x2) = max{17, x1 - 21, x2 - 21}
        true_0_A = 16
        *_0_A = 15
        *_1_A(x1) = max{16, x1 - 22}
        +_0_A = 20
        +_1_A(x1) = max{16, x1 - 84}
        filter_0_A = 15
        filter_1_A(x1) = max{16, x1 - 22}
        filter2_0_A = 20
        filter2_1_A(x1) = max{16, x1 - 22}
        filter2_2_A(x1,x2) = max{17, x1 - 44, x2 - 22}
        map_0_A = 15
        map_1_A(x1) = max{16, x1 - 40}
        s_0_A = 1
        s_1_A(x1) = max{2, x1 - 44}
        fact_1_A(x1) = max{19, x1 - 14}
        |0|_0_A = 13
        filter_2_A(x1,x2) = max{39, x1 - 44, x2 - 22}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        true_0 > filter_2 > app = *_0 = filter2_2 > filter2_3 = filter_1 = filter2_1 = fact_1 > cons_2 = filter_0 = filter2_0 = map_1 = nil_0 > filter2_4# = filter_2# = *_1 = map_0 = |0|_0 > app# = false_0 = +_1 = s_1 > +_0 = s_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter2_3) = [2, 3]
      pi(filter2_4#) = [2]
      pi(false_0) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(filter2_2) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(fact_1) = []
      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: 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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: 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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r8, r9, r10, r14, r16, r17, r18, r27, r28, r29, r30

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 + 12, x4 + 2}
        false_0_A = 24
        filter_2#_A(x1,x2) = max{1, x2 - 1}
        cons_2_A(x1,x2) = max{x1 + 14, x2 + 4}
        app_A(x1,x2) = x2 + 23
        true_0_A = 3
        *_0_A = 1
        *_1_A(x1) = max{14, x1 - 23}
        +_0_A = 15
        +_1_A(x1) = max{14, x1 - 23}
        cons_0_A = 1
        cons_1_A(x1) = max{16, x1 - 1}
        filter_0_A = 1
        filter_1_A(x1) = max{22, x1 - 23}
        filter_2_A(x1,x2) = x2
        filter2_0_A = 1
        filter2_1_A(x1) = max{2, x1 - 23}
        map_0_A = 1
        map_1_A(x1) = max{14, x1 - 23}
        p_0_A = 1
        p_1_A(x1) = max{23, x1 + 20}
        s_0_A = 1
        s_1_A(x1) = max{16, x1 - 19}
        nil_0_A = 0
        filter2_4_A(x1,x2,x3,x4) = max{x3 + 14, x4 + 4}
  
    base order 2:
  
      precedence:
      
        filter2_1 > app = +_0 = filter_0 = filter2_0 > filter2_4# = filter_1 = filter_2 = map_1 = filter2_4 > false_0 = cons_2 = +_1 = map_0 = nil_0 > filter_2# = *_1 = cons_1 > true_0 = *_0 = cons_0 > s_1 > p_1 = s_0 > p_0
      
  
    partial status:
  
      pi(filter2_4#) = [3]
      pi(false_0) = []
      pi(filter_2#) = []
      pi(cons_2) = [2]
      pi(app) = []
      pi(true_0) = []
      pi(*_0) = []
      pi(*_1) = []
      pi(+_0) = []
      pi(+_1) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(p_0) = []
      pi(p_1) = []
      pi(s_0) = []
      pi(s_1) = []
      pi(nil_0) = []
      pi(filter2_4) = []

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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: 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: fact_1#(s_1(x)) -> fact_1#(p_1(s_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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r12, r14, r16, r17, r18, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        fact_1#_A(x1) = max{3, x1 + 2}
        s_1_A(x1) = x1 + 10
        p_1_A(x1) = max{7, x1 - 5}
        app_A(x1,x2) = max{59, x1 - 14, x2 + 10}
        *_0_A = 12
        *_1_A(x1) = max{13, x1 - 1}
        +_0_A = 10
        +_1_A(x1) = max{11, x1 - 46}
        cons_0_A = 10
        cons_1_A(x1) = max{11, x1 - 91}
        cons_2_A(x1,x2) = max{12, x1 - 106, x2 - 46}
        filter_0_A = 13
        filter_1_A(x1) = max{11, x1 - 46}
        filter2_0_A = 10
        filter2_1_A(x1) = max{11, x1 - 1}
        filter2_2_A(x1,x2) = max{12, x1 - 16, x2 - 1}
        filter2_3_A(x1,x2,x3) = max{13, x1 - 31, x2 - 16, x3 - 90}
        map_0_A = 1
        map_1_A(x1) = max{0, x1 - 1}
        p_0_A = 6
        s_0_A = 1
        filter_2_A(x1,x2) = max{12, x1 - 61, x2 - 36}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter2_2 > app = filter2_1 > filter2_0 > *_1 = cons_2 = filter2_3 > s_1 = *_0 = +_1 = cons_1 = map_1 > fact_1# = +_0 = cons_0 = filter_1 = map_0 = s_0 = filter_2 = nil_0 > p_1 = filter_0 > p_0
      
  
    partial status:
  
      pi(fact_1#) = [1]
      pi(s_1) = [1]
      pi(p_1) = []
      pi(app) = [2]
      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(p_0) = []
      pi(s_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: *_2#(s_1(x),y) -> *_2#(x,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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r12, r14, r19, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        *_2#_A(x1,x2) = x1 + 1
        s_1_A(x1) = max{14, x1 + 2}
        app_A(x1,x2) = max{16, x1 - 4, x2 - 1}
        *_0_A = 3
        *_1_A(x1) = max{2, x1 - 2}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 2}
        cons_0_A = 1
        cons_1_A(x1) = max{0, x1 - 2}
        cons_2_A(x1,x2) = max{2, x2 - 5}
        filter_0_A = 1
        filter_1_A(x1) = max{2, x1 - 18}
        filter2_0_A = 2
        filter2_1_A(x1) = max{3, x1 - 2}
        filter2_2_A(x1,x2) = max{5, x1 - 7, x2 - 2}
        filter2_3_A(x1,x2,x3) = max{3, x1 - 12, x2 - 7, x3 - 2}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 2}
        fact_1_A(x1) = max{15, x1 - 8}
        |0|_0_A = 4
        filter_2_A(x1,x2) = max{3, x1 - 22, x2 - 4}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        *_2# > s_1 = *_0 = +_0 > app > *_1 = +_1 = fact_1 > cons_2 = filter2_3 = map_1 = |0|_0 = filter_2 > cons_1 = filter_1 = filter2_2 = map_0 = nil_0 > cons_0 = filter_0 = filter2_1 > filter2_0
      
  
    partial status:
  
      pi(*_2#) = [1]
      pi(s_1) = [1]
      pi(app) = []
      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(fact_1) = []
      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: +_2#(x,s_1(y)) -> +_2#(x,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(cons_0(),y) -> cons_1(y)
r6: app(cons_1(x1),y) -> cons_2(x1,y)
r7: app(fact_0(),y) -> fact_1(y)
r8: app(filter_0(),y) -> filter_1(y)
r9: app(filter_1(x1),y) -> filter_2(x1,y)
r10: app(filter2_0(),y) -> filter2_1(y)
r11: app(filter2_1(x1),y) -> filter2_2(x1,y)
r12: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y)
r13: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y)
r14: app(map_0(),y) -> map_1(y)
r15: app(map_1(x1),y) -> map_2(x1,y)
r16: app(p_0(),y) -> p_1(y)
r17: app(s_0(),y) -> s_1(y)
r18: p_1(s_1(x)) -> x
r19: fact_1(|0|_0()) -> s_1(|0|_0())
r20: fact_1(s_1(x)) -> *_2(s_1(x),fact_1(p_1(s_1(x))))
r21: *_2(|0|_0(),y) -> |0|_0()
r22: *_2(s_1(x),y) -> +_2(*_2(x,y),y)
r23: +_2(x,|0|_0()) -> x
r24: +_2(x,s_1(y)) -> s_1(+_2(x,y))
r25: map_2(f,nil_0()) -> nil_0()
r26: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs))
r27: filter_2(f,nil_0()) -> nil_0()
r28: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs)
r29: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs))
r30: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs)

The set of usable rules consists of

  r1, r3, r5, r6, r8, r10, r11, r12, r14, r19, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        +_2#_A(x1,x2) = max{7, x2}
        s_1_A(x1) = max{7, x1}
        app_A(x1,x2) = max{12, x1 - 4, x2 - 7}
        *_0_A = 3
        *_1_A(x1) = max{2, x1 - 8}
        +_0_A = 1
        +_1_A(x1) = max{2, x1 - 8}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 8}
        cons_2_A(x1,x2) = max{4, x1 - 13, x2 - 9}
        filter_0_A = 3
        filter_1_A(x1) = max{5, x1 - 7}
        filter2_0_A = 3
        filter2_1_A(x1) = max{1, x1 - 8}
        filter2_2_A(x1,x2) = max{2, x1 - 13, x2 - 8}
        filter2_3_A(x1,x2,x3) = max{3, x1 - 18, x2 - 13, x3 - 8}
        map_0_A = 3
        map_1_A(x1) = max{1, x1 - 7}
        fact_1_A(x1) = max{8, x1 - 9}
        |0|_0_A = 2
        filter_2_A(x1,x2) = max{6, x1 - 26, x2 - 16}
        nil_0_A = 1
  
    base order 2:
  
      precedence:
      
        s_1 = filter_2 > +_2# = app = *_0 = +_0 = cons_1 = cons_2 = filter2_0 > *_1 = +_1 = cons_0 = filter_1 = filter2_2 = map_1 = fact_1 = nil_0 > filter_0 = filter2_1 = filter2_3 = map_0 = |0|_0
      
  
    partial status:
  
      pi(+_2#) = [2]
      pi(s_1) = [1]
      pi(app) = []
      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(fact_1) = []
      pi(|0|_0) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.