We consider the system 459.

  Alphabet:

    and : [form * form] --> form 
    exists : [term -> form] --> form 
    forall : [term -> form] --> form 
    neg : [form] --> form 
    or : [form * form] --> form 

  Rules:

    neg(neg(X)) => X 
    neg(and(X, Y)) => or(neg(X), neg(Y)) 
    neg(or(X, Y)) => and(neg(X), neg(Y)) 
    neg(forall(/\x.X(x))) => exists(/\y.neg(X(y))) 
    neg(exists(/\x.X(x))) => forall(/\y.neg(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]):

  neg(neg(X)) >? X 
  neg(and(X, Y)) >? or(neg(X), neg(Y)) 
  neg(or(X, Y)) >? and(neg(X), neg(Y)) 
  neg(forall(/\x.X(x))) >? exists(/\y.neg(X(y))) 
  neg(exists(/\x.X(x))) >? forall(/\y.neg(X(y))) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.y0 + y1 
  exists = \G0.1 + 2G0(0) 
  forall = \G0.2 + 2G0(0) 
  neg = \y0.2y0 
  or = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[neg(neg(_x0))]] = 4x0 >= x0 = [[_x0]] 
  [[neg(and(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[or(neg(_x0), neg(_x1))]] 
  [[neg(or(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[and(neg(_x0), neg(_x1))]] 
  [[neg(forall(/\x._x0(x)))]] = 4 + 4F0(0) > 1 + 4F0(0) = [[exists(/\x.neg(_x0(x)))]] 
  [[neg(exists(/\x._x0(x)))]] = 2 + 4F0(0) >= 2 + 4F0(0) = [[forall(/\x.neg(_x0(x)))]] 

We can thus remove the following rules:

  neg(forall(/\x.X(x))) => exists(/\y.neg(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]):

  neg(neg(X)) >? X 
  neg(and(X, Y)) >? or(neg(X), neg(Y)) 
  neg(or(X, Y)) >? and(neg(X), neg(Y)) 
  neg(exists(/\x.X(x))) >? forall(/\y.neg(X(y))) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.y0 + y1 
  exists = \G0.3 + 3G0(0) 
  forall = \G0.G0(0) 
  neg = \y0.y0 
  or = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[neg(neg(_x0))]] = x0 >= x0 = [[_x0]] 
  [[neg(and(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[or(neg(_x0), neg(_x1))]] 
  [[neg(or(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[and(neg(_x0), neg(_x1))]] 
  [[neg(exists(/\x._x0(x)))]] = 3 + 3F0(0) > F0(0) = [[forall(/\x.neg(_x0(x)))]] 

We can thus remove the following rules:

  neg(exists(/\x.X(x))) => forall(/\y.neg(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]):

  neg(neg(X)) >? X 
  neg(and(X, Y)) >? or(neg(X), neg(Y)) 
  neg(or(X, Y)) >? and(neg(X), neg(Y)) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.3 + y0 + y1 
  neg = \y0.3y0 
  or = \y0y1.3 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[neg(neg(_x0))]] = 9x0 >= x0 = [[_x0]] 
  [[neg(and(_x0, _x1))]] = 9 + 3x0 + 3x1 > 3 + 3x0 + 3x1 = [[or(neg(_x0), neg(_x1))]] 
  [[neg(or(_x0, _x1))]] = 9 + 3x0 + 3x1 > 3 + 3x0 + 3x1 = [[and(neg(_x0), neg(_x1))]] 

We can thus remove the following rules:

  neg(and(X, Y)) => or(neg(X), neg(Y)) 
  neg(or(X, Y)) => and(neg(X), neg(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]):

  neg(neg(X)) >? X 

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

The following interpretation satisfies the requirements:

  neg = \y0.3 + y0 

Using this interpretation, the requirements translate to:

  [[neg(neg(_x0))]] = 6 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  neg(neg(X)) => X 

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.