We consider the system filter.

  Alphabet:

    0 : [] --> nat 
    bool : [nat] --> boolean 
    cons : [nat * list] --> list 
    consif : [boolean * nat * list] --> list 
    false : [] --> boolean 
    filter : [nat -> boolean * list] --> list 
    nil : [] --> list 
    rand : [nat] --> nat 
    s : [nat] --> nat 
    true : [] --> boolean 

  Rules:

    rand(x) => x 
    rand(s(x)) => rand(x) 
    bool(0) => false 
    bool(s(0)) => true 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => consif(f x, x, filter(f, y)) 
    consif(true, x, y) => cons(x, y) 
    consif(false, x, y) => y 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  rand(X) >? X 
  rand(s(X)) >? rand(X) 
  bool(0) >? false 
  bool(s(0)) >? true 
  filter(F, nil) >? nil 
  filter(F, cons(X, Y)) >? consif(F X, X, filter(F, Y)) 
  consif(true, X, Y) >? cons(X, Y) 
  consif(false, X, Y) >? Y 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 3 
  bool = Lam[y0].3 + 3*y0 
  cons = Lam[y0;y1].2 + 2*y0 + 2*y1 
  consif = Lam[y0;y1;y2].y0 + 2*y1 + 2*y2 
  false = 0 
  filter = Lam[G0;y1].3*y1 + G0(y1) + y1*G0(y1) 
  nil = 0 
  rand = Lam[y0].3 + y0 
  s = Lam[y0].3 + 3*y0 
  true = 3 

Using this interpretation, the requirements translate to:

  [[rand(_x0)]] = 3 + x0 > x0 = [[_x0]] 
  [[rand(s(_x0))]] = 6 + 3*x0 > 3 + x0 = [[rand(_x0)]] 
  [[bool(0)]] = 12 > 0 = [[false]] 
  [[bool(s(0))]] = 39 > 3 = [[true]] 
  [[filter(_F0, nil)]] = F0(0) >= 0 = [[nil]] 
  [[filter(_F0, cons(_x1, _x2))]] = 6 + 6*x1 + 6*x2 + 2*x1*F0(2 + 2*x1 + 2*x2) + 2*x2*F0(2 + 2*x1 + 2*x2) + 3*F0(2 + 2*x1 + 2*x2) > 3*x1 + 6*x2 + F0(x1) + 2*x2*F0(x2) + 2*F0(x2) = [[consif(_F0 _x1, _x1, filter(_F0, _x2))]] 
  [[consif(true, _x0, _x1)]] = 3 + 2*x0 + 2*x1 > 2 + 2*x0 + 2*x1 = [[cons(_x0, _x1)]] 
  [[consif(false, _x0, _x1)]] = 2*x0 + 2*x1 >= x1 = [[_x1]] 

We can thus remove the following rules:

  rand(X) => X 
  rand(s(X)) => rand(X) 
  bool(0) => false 
  bool(s(0)) => true 
  filter(F, cons(X, Y)) => consif(F X, X, filter(F, Y)) 
  consif(true, X, Y) => cons(X, Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  filter(F, nil) >? nil 
  consif(false, X, Y) >? Y 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  consif = Lam[y0;y1;y2].3 + y0 + y1 + y2 
  false = 3 
  filter = Lam[G0;y1].3 + 3*y1 + G0(0) 
  nil = 2 

Using this interpretation, the requirements translate to:

  [[filter(_F0, nil)]] = 9 + F0(0) > 2 = [[nil]] 
  [[consif(false, _x0, _x1)]] = 6 + x0 + x1 > x1 = [[_x1]] 

We can thus remove the following rules:

  filter(F, nil) => nil 
  consif(false, X, Y) => Y 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.