YES
We show the termination of the TRS R:

  app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
  app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
  app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
  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)


-- SCC decomposition.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)
p3: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(|:|(),y)
p4: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
p5: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(+(),app(app(|:|(),x),z))
p6: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),x),z)
p7: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(|:|(),x)
p8: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),y),z)
p9: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(|:|(),y)
p10: app#(app(|:|(),z),app(app(+(),x),app(f,y))) -> app#(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
p11: app#(app(|:|(),z),app(app(+(),x),app(f,y))) -> app#(|:|(),app(app(g(),z),y))
p12: app#(app(|:|(),z),app(app(+(),x),app(f,y))) -> app#(app(g(),z),y)
p13: app#(app(|:|(),z),app(app(+(),x),app(f,y))) -> app#(g(),z)
p14: app#(app(|:|(),z),app(app(+(),x),app(f,y))) -> app#(app(+(),x),a())
p15: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
p16: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
p17: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
p18: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
p19: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
p20: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
p21: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
p22: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
p23: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
p24: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
p25: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
p26: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
p27: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
p28: app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
p29: app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The estimated dependency graph contains the following SCCs:

  {p17, p18, p19, p23, p26, p28}
  {p1, p2, p6, p8}


-- Reduction pair.

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

p1: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
p2: app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
p3: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
p4: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
p5: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
p6: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x2 + 5
          app_A(x1,x2) = x1 + x2 + 5
          filter_A() = 3
          cons_A() = 1
          filter2_A() = 2
          false_A() = 4
          true_A() = 4
          map_A() = 2
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        app# = app > false = true = map > filter > filter2 = nil > cons
      
  
    partial status:
  
      pi(app#) = [2]
      pi(app) = []
      pi(filter) = []
      pi(cons) = []
      pi(filter2) = []
      pi(false) = []
      pi(true) = []
      pi(map) = []
      pi(nil) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
p2: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
p3: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
p4: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
p5: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
p2: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r3, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1
          app_A(x1,x2) = x2 + 4
          map_A() = 3
          cons_A() = 1
          |:|_A() = 15
          +_A() = 10
          g_A() = 3
          a_A() = 1
          filter_A() = 2
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        |:| = nil > app# = app = + > map = g = a > cons = filter
      
  
    partial status:
  
      pi(app#) = []
      pi(app) = []
      pi(map) = []
      pi(cons) = []
      pi(|:|) = []
      pi(+) = []
      pi(g) = []
      pi(a) = []
      pi(filter) = []
      pi(nil) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r3, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x2 + 1
          app_A(x1,x2) = x2 + 4
          map_A() = 3
          cons_A() = 2
          |:|_A() = 2
          +_A() = 3
          g_A() = 1
          a_A() = 0
          filter_A() = 2
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        app# = app > map = + > cons = |:| = filter > g = a = nil
      
  
    partial status:
  
      pi(app#) = [2]
      pi(app) = []
      pi(map) = []
      pi(cons) = []
      pi(|:|) = []
      pi(+) = []
      pi(g) = []
      pi(a) = []
      pi(filter) = []
      pi(nil) = []

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#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
p2: app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
p3: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r3, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x2 + 10
          app_A(x1,x2) = x2 + 5
          filter2_A() = 4
          false_A() = 3
          filter_A() = 2
          cons_A() = 3
          true_A() = 4
          |:|_A() = 2
          +_A() = 3
          g_A() = 1
          a_A() = 4
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        a > app > app# = filter2 = + > false = true = |:| > filter = cons = g > nil
      
  
    partial status:
  
      pi(app#) = [2]
      pi(app) = []
      pi(filter2) = []
      pi(false) = []
      pi(filter) = []
      pi(cons) = []
      pi(true) = []
      pi(|:|) = []
      pi(+) = []
      pi(g) = []
      pi(a) = []
      pi(nil) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
p2: app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),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: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),y),z)
p3: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),x),z)
p4: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1
          app_A(x1,x2) = x1 + x2 + 5
          |:|_A() = 3
          +_A() = 4
          filter_A() = 23
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        filter > app = |:| > + > app# > nil
      
  
    partial status:
  
      pi(app#) = []
      pi(app) = [1, 2]
      pi(|:|) = []
      pi(+) = []
      pi(filter) = []
      pi(nil) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),x),z)
p3: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)
p3: app#(app(|:|(),app(app(+(),x),y)),z) -> app#(app(|:|(),x),z)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r4, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + 1
          app_A(x1,x2) = x1 + x2 + 7
          |:|_A() = 9
          +_A() = 11
          map_A() = 4
          nil_A() = 1
          filter_A() = 3
  
    base order 2:
  
      precedence:
      
        app = + = map = filter > app# = |:| = nil
      
  
    partial status:
  
      pi(app#) = [1]
      pi(app) = [1]
      pi(|:|) = []
      pi(+) = []
      pi(map) = []
      pi(nil) = []
      pi(filter) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),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: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))
p2: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),y),z)

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r4, r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1
          app_A(x1,x2) = x1 + x2 + 6
          |:|_A() = 20
          map_A() = 2
          nil_A() = 9
          filter_A() = 1
  
    base order 2:
  
      precedence:
      
        app# = app = map = nil = filter > |:|
      
  
    partial status:
  
      pi(app#) = []
      pi(app) = [1]
      pi(|:|) = []
      pi(map) = []
      pi(nil) = []
      pi(filter) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: app#(app(|:|(),app(app(|:|(),x),y)),z) -> app#(app(|:|(),x),app(app(|:|(),y),z))

and R consists of:

r1: app(app(|:|(),app(app(|:|(),x),y)),z) -> app(app(|:|(),x),app(app(|:|(),y),z))
r2: app(app(|:|(),app(app(+(),x),y)),z) -> app(app(+(),app(app(|:|(),x),z)),app(app(|:|(),y),z))
r3: app(app(|:|(),z),app(app(+(),x),app(f,y))) -> app(app(|:|(),app(app(g(),z),y)),app(app(+(),x),a()))
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r6: app(app(filter(),f),nil()) -> nil()
r7: app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
r8: app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
r9: app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

The set of usable rules consists of

  r6

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + 22
          app_A(x1,x2) = x1 + x2 + 9
          |:|_A() = 3
          filter_A() = 2
          nil_A() = 1
  
    base order 2:
  
      precedence:
      
        app# > app > |:| > nil > filter
      
  
    partial status:
  
      pi(app#) = []
      pi(app) = [2]
      pi(|:|) = []
      pi(filter) = []
      pi(nil) = []

The following rules are removed:

  p1

Then no dependency pair remains.