We consider the system Applicative_05__TreeLevels.

  Alphabet:

    append : [a * a] --> a 
    combine : [a * a] --> a 
    cons : [a * a] --> a 
    levels : [] --> a -> a 
    map : [a -> a * a] --> a 
    nil : [] --> a 
    node : [a * a] --> a 
    zip : [a * a] --> a 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    append(x, nil) => x 
    append(nil, x) => x 
    append(cons(x, y), z) => cons(x, append(y, z)) 
    zip(nil, x) => x 
    zip(x, nil) => x 
    zip(cons(x, y), cons(z, u)) => cons(append(x, z), zip(y, u)) 
    combine(x, nil) => x 
    combine(x, cons(y, z)) => combine(zip(x, y), z) 
    levels node(x, y) => cons(cons(x, nil), combine(nil, map(levels, 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]):

  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
  append(X, nil) >? X 
  append(nil, X) >? X 
  append(cons(X, Y), Z) >? cons(X, append(Y, Z)) 
  zip(nil, X) >? X 
  zip(X, nil) >? X 
  zip(cons(X, Y), cons(Z, U)) >? cons(append(X, Z), zip(Y, U)) 
  combine(X, nil) >? X 
  combine(X, cons(Y, Z)) >? combine(zip(X, Y), Z) 
  levels node(X, Y) >? cons(cons(X, nil), combine(nil, map(levels, Y))) 

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

The following interpretation satisfies the requirements:

  append = Lam[y0;y1].y0 + y1 
  combine = Lam[y0;y1].y0 + y1 
  cons = Lam[y0;y1].1 + y0 + y1 
  levels = Lam[y0].0 
  map = Lam[G0;y1].y1 + 2*y1*G0(y1) + 2*G0(0) + 2*G0(y1) 
  nil = 0 
  node = Lam[y0;y1].3 + 3*y0 + 3*y1 
  zip = Lam[y0;y1].1 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[map(_F0, nil)]] = 4*F0(0) >= 0 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 1 + x1 + x2 + 2*x1*F0(1 + x1 + x2) + 2*x2*F0(1 + x1 + x2) + 2*F0(0) + 4*F0(1 + x1 + x2) >= 1 + x1 + x2 + F0(x1) + 2*x2*F0(x2) + 2*F0(0) + 2*F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[append(_x0, nil)]] = x0 >= x0 = [[_x0]] 
  [[append(nil, _x0)]] = x0 >= x0 = [[_x0]] 
  [[append(cons(_x0, _x1), _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[cons(_x0, append(_x1, _x2))]] 
  [[zip(nil, _x0)]] = 1 + x0 > x0 = [[_x0]] 
  [[zip(_x0, nil)]] = 1 + x0 > x0 = [[_x0]] 
  [[zip(cons(_x0, _x1), cons(_x2, _x3))]] = 3 + x0 + x1 + x2 + x3 > 2 + x0 + x1 + x2 + x3 = [[cons(append(_x0, _x2), zip(_x1, _x3))]] 
  [[combine(_x0, nil)]] = x0 >= x0 = [[_x0]] 
  [[combine(_x0, cons(_x1, _x2))]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[combine(zip(_x0, _x1), _x2)]] 
  [[levels node(_x0, _x1)]] = 3 + 3*x0 + 3*x1 > 2 + x0 + x1 = [[cons(cons(_x0, nil), combine(nil, map(levels, _x1)))]] 

We can thus remove the following rules:

  zip(nil, X) => X 
  zip(X, nil) => X 
  zip(cons(X, Y), cons(Z, U)) => cons(append(X, Z), zip(Y, U)) 
  levels node(X, Y) => cons(cons(X, nil), combine(nil, map(levels, 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]):

  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
  append(X, nil) >? X 
  append(nil, X) >? X 
  append(cons(X, Y), Z) >? cons(X, append(Y, Z)) 
  combine(X, nil) >? X 
  combine(X, cons(Y, Z)) >? combine(zip(X, Y), Z) 

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

The following interpretation satisfies the requirements:

  append = Lam[y0;y1].1 + y1 + 3*y0 
  combine = Lam[y0;y1].3 + y0 + y1 
  cons = Lam[y0;y1].3 + y0 + y1 
  map = Lam[G0;y1].3 + 3*y1 + G0(y1) + 3*y1*G0(y1) 
  nil = 2 
  zip = Lam[y0;y1].y0 + y1 

Using this interpretation, the requirements translate to:

  [[map(_F0, nil)]] = 9 + 7*F0(2) > 2 = [[nil]] 
  [[map(_F0, cons(_x1, _x2))]] = 12 + 3*x1 + 3*x2 + 3*x1*F0(3 + x1 + x2) + 3*x2*F0(3 + x1 + x2) + 10*F0(3 + x1 + x2) > 6 + x1 + 3*x2 + F0(x1) + F0(x2) + 3*x2*F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
  [[append(_x0, nil)]] = 3 + 3*x0 > x0 = [[_x0]] 
  [[append(nil, _x0)]] = 7 + x0 > x0 = [[_x0]] 
  [[append(cons(_x0, _x1), _x2)]] = 10 + x2 + 3*x0 + 3*x1 > 4 + x0 + x2 + 3*x1 = [[cons(_x0, append(_x1, _x2))]] 
  [[combine(_x0, nil)]] = 5 + x0 > x0 = [[_x0]] 
  [[combine(_x0, cons(_x1, _x2))]] = 6 + x0 + x1 + x2 > 3 + x0 + x1 + x2 = [[combine(zip(_x0, _x1), _x2)]] 

We can thus remove the following rules:

  map(F, nil) => nil 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
  append(X, nil) => X 
  append(nil, X) => X 
  append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
  combine(X, nil) => X 
  combine(X, cons(Y, Z)) => combine(zip(X, Y), Z) 

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.