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

  r8, r10, r14, r21, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = 0
        cons_2_A(x1,x2) = max{27, x1 + 1, x2 - 3}
        app#_A(x1,x2) = 0
        map_1_A(x1) = max{9, x1 - 22}
        filter2_3_A(x1,x2,x3) = max{0, x1 - 22, x2 - 22, x3 - 22}
        filter2_4#_A(x1,x2,x3,x4) = 0
        false_0_A = 26
        filter_2#_A(x1,x2) = 0
        filter_1_A(x1) = max{17, x1 - 22}
        app_A(x1,x2) = max{x1 - 10, x2 + 25}
        true_0_A = 26
        filter_0_A = 9
        filter2_0_A = 11
        filter2_1_A(x1) = max{9, x1 - 1}
        map_0_A = 8
        *_2_A(x1,x2) = max{x1 + 10, x2 + 13}
        |0|_0_A = 12
        map_2_A(x1,x2) = max{18, x1 - 33, x2 + 13}
        nil_0_A = 19
        filter_2_A(x1,x2) = max{x1 - 33, x2 + 6}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = max{1, x1 - 53, x2 - 7}
          cons_2_A(x1,x2) = max{x1 + 6, x2}
          app#_A(x1,x2) = max{1, x1 - 53, x2 - 1}
          map_1_A(x1) = max{8, x1 + 6}
          filter2_3_A(x1,x2,x3) = max{x1 + 6, x2 + 6, x3 + 52}
          filter2_4#_A(x1,x2,x3,x4) = max{1, x2 - 53, x4 - 7}
          false_0_A = 47
          filter_2#_A(x1,x2) = max{1, x1 - 53, x2 - 7}
          filter_1_A(x1) = max{7, x1 + 6}
          app_A(x1,x2) = max{32, x1 - 44, x2 + 7}
          true_0_A = 47
          filter_0_A = 43
          filter2_0_A = 43
          filter2_1_A(x1) = max{32, x1 + 7}
          map_0_A = 43
          *_2_A(x1,x2) = max{32, x2 - 13}
          |0|_0_A = 14
          map_2_A(x1,x2) = max{38, x1 - 38, x2 + 7}
          nil_0_A = 39
          filter_2_A(x1,x2) = max{10, x1 - 38, x2 + 1}
          map_2#!_A(x1,x2) = x2 + 5
          cons_2!_A(x1,x2) = max{x1 + 43, x2 + 4}
          app#!_A(x1,x2) = x2 + 9
          map_1!_A(x1) = max{45, x1 - 41}
          filter2_3!_A(x1,x2,x3) = max{0, x1 - 89, x2 - 46, x3 - 42}
          filter2_4#!_A(x1,x2,x3,x4) = max{9, x4 + 3}
          false_0!_A = 1
          filter_2#!_A(x1,x2) = max{9, x2 + 3}
          filter_1!_A(x1) = max{3, x1 - 93}
          app!_A(x1,x2) = max{45, x1 - 51, x2 - 41}
          true_0!_A = 1
          filter_0!_A = 1
          filter2_0!_A = 1
          filter2_1!_A(x1) = max{2, x1 - 89}
          map_0!_A = 1
          *_2!_A(x1,x2) = max{3, x1 - 9}
          |0|_0!_A = 0
          map_2!_A(x1,x2) = max{44, x2 - 40}
          nil_0!_A = 2
          filter_2!_A(x1,x2) = max{2, x1 - 93, x2 - 48}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [1]
      pi(app#) = []
      pi(map_1) = []
      pi(filter2_3) = []
      pi(filter2_4#) = []
      pi(false_0) = []
      pi(filter_2#) = []
      pi(filter_1) = []
      pi(app) = [2]
      pi(true_0) = []
      pi(filter_0) = []
      pi(filter2_0) = []
      pi(filter2_1) = []
      pi(map_0) = []
      pi(*_2) = [2]
      pi(|0|_0) = []
      pi(map_2) = []
      pi(nil_0) = []
      pi(filter_2) = []

The following rules are removed:

  p1, p2



-- 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)) -> app#(f,x)
p5: app#(filter_1(x1),y) -> filter_2#(x1,y)
p6: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs)
p7: 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:

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


-- 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

  (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) = x2 + 1
        cons_2_A(x1,x2) = max{x1 + 3, x2 + 2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = 0
          cons_2_A(x1,x2) = max{10, x1 - 16, x2 - 16}
          map_2#!_A(x1,x2) = x1 + 4
          cons_2!_A(x1,x2) = max{2, x1 - 63, x2 - 12}
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [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: 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{41, x2 + 30}
        filter2_3_A(x1,x2,x3) = max{x1, x2 - 62, x3 - 62}
        filter2_4#_A(x1,x2,x3,x4) = max{39, x4 + 21}
        true_0_A = 32
        filter_2#_A(x1,x2) = max{31, x2 + 13}
        cons_2_A(x1,x2) = max{40, x1 + 29, x2 + 22}
        app_A(x1,x2) = max{43, x1 - 45, x2 + 16}
        false_0_A = 17
        filter_1_A(x1) = max{17, x1 - 62}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{21, x1 - 17}
          filter2_3_A(x1,x2,x3) = max{37, x1 + 16, x2 + 16, x3 + 17}
          filter2_4#_A(x1,x2,x3,x4) = max{21, x2 - 17}
          true_0_A = 72
          filter_2#_A(x1,x2) = max{21, x1 - 17}
          cons_2_A(x1,x2) = max{2, x1 - 4}
          app_A(x1,x2) = max{10, x1 - 90}
          false_0_A = 0
          filter_1_A(x1) = max{37, x1 + 16}
          app#!_A(x1,x2) = max{4, x1 - 54, x2 - 16}
          filter2_3!_A(x1,x2,x3) = max{3, x2 - 20, x3 - 17}
          filter2_4#!_A(x1,x2,x3,x4) = max{36, x2 - 19, x3 - 14, x4 - 8}
          true_0!_A = 1
          filter_2#!_A(x1,x2) = max{35, x1 - 37, x2 - 11}
          cons_2!_A(x1,x2) = max{3, x1 - 15, x2 - 9}
          app!_A(x1,x2) = max{34, x1 - 36, x2 - 16}
          false_0!_A = 36
          filter_1!_A(x1) = 34
  
    partial status:
  
      pi(app#) = [2]
      pi(filter2_3) = [1]
      pi(filter2_4#) = [4]
      pi(true_0) = []
      pi(filter_2#) = [2]
      pi(cons_2) = [1, 2]
      pi(app) = [2]
      pi(false_0) = []
      pi(filter_1) = []

The following rules are removed:

  p1, p2, p3, p4, p5, p6

Then no dependency pair remains.

-- 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

  r18

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        fact_1#_A(x1) = 0
        s_1_A(x1) = max{2, x1 - 32}
        p_1_A(x1) = x1 + 33
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          fact_1#_A(x1) = max{0, x1 - 16}
          s_1_A(x1) = x1 + 15
          p_1_A(x1) = max{2, x1 - 15}
          fact_1#!_A(x1) = max{5, x1 - 9}
          s_1!_A(x1) = max{5, x1 - 38}
          p_1!_A(x1) = max{8, x1 - 28}
  
    partial status:
  
      pi(fact_1#) = []
      pi(s_1) = []
      pi(p_1) = []

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        *_2#_A(x1,x2) = max{2, x1 + 1}
        s_1_A(x1) = x1 + 5
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          *_2#_A(x1,x2) = 0
          s_1_A(x1) = max{52, x1 - 22}
          *_2#!_A(x1,x2) = x2 + 9
          s_1!_A(x1) = max{7, x1 - 91}
  
    partial status:
  
      pi(*_2#) = [1]
      pi(s_1) = [1]

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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        +_2#_A(x1,x2) = max{3, x2 + 2}
        s_1_A(x1) = x1 + 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          +_2#_A(x1,x2) = 0
          s_1_A(x1) = max{15, x1 - 8}
          +_2#!_A(x1,x2) = max{x1 + 9, x2 + 17}
          s_1!_A(x1) = max{7, x1 - 12}
  
    partial status:
  
      pi(+_2#) = [2]
      pi(s_1) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.