YES
We show the termination of the TRS R:

  app(app(filter(),f),nil()) -> nil()
  app(app(filter(),f),app(app(cons(),y),ys)) -> app(app(app(filtersub(),app(f,y)),f),app(app(cons(),y),ys))
  app(app(app(filtersub(),true()),f),app(app(cons(),y),ys)) -> app(app(cons(),y),app(app(filter(),f),ys))
  app(app(app(filtersub(),false()),f),app(app(cons(),y),ys)) -> app(app(filter(),f),ys)

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(filter_0(),y) -> filter_1(y)
  app(filter_1(x1),y) -> filter_2(x1,y)
  app(filtersub_0(),y) -> filtersub_1(y)
  app(filtersub_1(x1),y) -> filtersub_2(x1,y)
  app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
  filter_2(f,nil_0()) -> nil_0()
  filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
  filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
  filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)


-- SCC decomposition.

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

p1: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p3: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p4: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p5: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p6: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

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: app#(filter_1(x1),y) -> filter_2#(x1,y)
p2: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p3: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p4: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p5: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p6: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The set of usable rules consists of

  r1, r5, r8

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 28
        filter_1_A(x1) = x1 + 30
        filter_2#_A(x1,x2) = x1 + 29
        cons_2_A(x1,x2) = max{3, x1 - 3, x2 - 9}
        filtersub_2_A(x1,x2) = max{x1 - 24, x2 + 30}
        filtersub_3#_A(x1,x2,x3) = x2 + 29
        false_0_A = 1
        app_A(x1,x2) = max{4, x1 - 37, x2 - 22}
        true_0_A = 1
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 23}
        filtersub_0_A = 2
        filtersub_1_A(x1) = max{3, x1 - 23}
        filter_2_A(x1,x2) = max{5, x1 - 6, x2 - 18}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        filter_2# = cons_2 = filtersub_3# = app = filter_2 > app# = filter_1 = filtersub_2 = false_0 = true_0 = cons_1 = nil_0 > cons_0 = filtersub_1 > filtersub_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(filter_1) = []
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(filtersub_2) = [2]
      pi(filtersub_3#) = [2]
      pi(false_0) = []
      pi(app) = []
      pi(true_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p3: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p4: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p5: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p2: app#(filtersub_2(x1,x2),y) -> filtersub_3#(x1,x2,y)
p3: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p4: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p5: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The set of usable rules consists of

  r1, r5, r8

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = x1 + 21
        cons_2_A(x1,x2) = max{5, x1 - 2, x2 - 3}
        app#_A(x1,x2) = x1 + 20
        filtersub_2_A(x1,x2) = max{x1 - 16, x2 + 22}
        filtersub_3#_A(x1,x2,x3) = x2 + 21
        true_0_A = 1
        app_A(x1,x2) = max{0, x1 - 7, x2 - 7}
        false_0_A = 5
        cons_0_A = 6
        cons_1_A(x1) = max{0, x1 - 7}
        filtersub_0_A = 9
        filtersub_1_A(x1) = max{1, x1 - 8}
        filter_2_A(x1,x2) = max{0, x1 - 7}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        app# > filter_2# = cons_2 = filtersub_3# = filter_2 > true_0 = false_0 = nil_0 > app > filtersub_2 = cons_1 > cons_0 = filtersub_1 > filtersub_0
      
  
    partial status:
  
      pi(filter_2#) = [1]
      pi(cons_2) = []
      pi(app#) = [1]
      pi(filtersub_2) = [2]
      pi(filtersub_3#) = [2]
      pi(true_0) = []
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filter_2) = []
      pi(nil_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: filter_2#(f,cons_2(y,ys)) -> app#(f,y)
p2: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p3: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p4: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

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: filtersub_3#(true_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)
p2: filter_2#(f,cons_2(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p3: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filtersub_3#_A(x1,x2,x3) = max{2, x1, x3 - 2}
        true_0_A = 7
        cons_2_A(x1,x2) = 1
        filter_2#_A(x1,x2) = 6
        app_A(x1,x2) = 5
        false_0_A = 7
        cons_0_A = 0
        cons_1_A(x1) = 0
        filter_0_A = 0
        filter_1_A(x1) = 0
        filter_2_A(x1,x2) = 4
        filtersub_0_A = 0
        filtersub_1_A(x1) = 1
        filtersub_2_A(x1,x2) = 0
        filtersub_3_A(x1,x2,x3) = 4
        nil_0_A = 3
  
    base order 2:
  
      precedence:
      
        true_0 = false_0 > filter_2# > filtersub_3# > cons_2 > filter_2 = filtersub_3 > app = nil_0 > cons_1 = filter_1 = filtersub_2 > cons_0 = filter_0 = filtersub_1 > filtersub_0
      
  
    partial status:
  
      pi(filtersub_3#) = []
      pi(true_0) = []
      pi(cons_2) = []
      pi(filter_2#) = []
      pi(app) = []
      pi(false_0) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(filter_0) = []
      pi(filter_1) = []
      pi(filter_2) = []
      pi(filtersub_0) = []
      pi(filtersub_1) = []
      pi(filtersub_2) = []
      pi(filtersub_3) = []
      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(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p2: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

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(y,ys)) -> filtersub_3#(app(f,y),f,cons_2(y,ys))
p2: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        filter_2#_A(x1,x2) = max{2, x2 - 7}
        cons_2_A(x1,x2) = max{x1 + 15, x2 + 11}
        filtersub_3#_A(x1,x2,x3) = max{1, x3 - 12}
        app_A(x1,x2) = 7
        false_0_A = 17
  
    base order 2:
  
      precedence:
      
        false_0 > cons_2 > filtersub_3# > filter_2# > app
      
  
    partial status:
  
      pi(filter_2#) = []
      pi(cons_2) = [1, 2]
      pi(filtersub_3#) = []
      pi(app) = []
      pi(false_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: filtersub_3#(false_0(),f,cons_2(y,ys)) -> filter_2#(f,ys)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(filter_0(),y) -> filter_1(y)
r4: app(filter_1(x1),y) -> filter_2(x1,y)
r5: app(filtersub_0(),y) -> filtersub_1(y)
r6: app(filtersub_1(x1),y) -> filtersub_2(x1,y)
r7: app(filtersub_2(x1,x2),y) -> filtersub_3(x1,x2,y)
r8: filter_2(f,nil_0()) -> nil_0()
r9: filter_2(f,cons_2(y,ys)) -> filtersub_3(app(f,y),f,cons_2(y,ys))
r10: filtersub_3(true_0(),f,cons_2(y,ys)) -> cons_2(y,filter_2(f,ys))
r11: filtersub_3(false_0(),f,cons_2(y,ys)) -> filter_2(f,ys)

The estimated dependency graph contains the following SCCs:

  (no SCCs)