Ordinals in ACL2

Ordinals are used in ACL2 for proving termination in the admission of recursive function definitions. For a proof that the ACL2 ordinals are well-founded, see proof-of-well-foundedness.

The representation of ordinals changed in ACL2 Version_2.8, and is due to Pete Manolios and Daron Vroon. They have also defined algorithms for ordinal arithmetic, created a library of theorems to reason about ordinal arithmetic, and written the rest of this documentation in order to explain this change. We thank them for their efforts. Although they have provided the implementation and even modified the community books as needed, we have looked over their work and are maintaining it (and this documentation); if there are any bugs, they should be considered ours (Matt Kaufmann and J Moore).

A book is included for compatibility with the representation before Version_2.8. For books that contain events relying on the previous ordinal implementation, insert the following lines before the first such event:

(include-book "ordinals/e0-ordinal" :dir :system) (set-well-founded-relation e0-ord-<)

The new ordinal representation is based on a slightly different version of Cantor Normal Form than that used by the old ordinals. An advantage of the new representation is that it is exponentially more succinct than the old representation.

While pre-Version_2.8 ACL2 versions provided built-in functions for
checking if an object is an ordinal and for comparing two ordinals, they did
not provide support for reasoning about and constructing ordinals. The
community books directory

Manolios, Panagiotis & Vroon, Daron. Algorithms for ordinal arithmetic. Baader, Franz (ed), 19th International Conference on Automated Deduction--CADE-19. Pages 243-257 of LNAI, vol. 2741. Springer-Verlag.

Second, the algorithms are mechanically verified and libraries of theorems which can be used to automate reasoning involving the ordinals are provided. For details, see the following paper.

Manolios, Panagiotis & Vroon, Daron. Ordinal arithmetic in ACL2. Kaufmann, Matt, & Moore, J Strother (eds). Fourth International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2-2003), July, 2003. See http://www.cs.utexas.edu/users/moore/acl2/workshop-2003/.

We now describe aspects of the above mentioned books in more detail.

The new ordering function is `o<` and the new ordinal recognizer is
`o-p`. See also `natp`, `posp`, `o<=`, `o>`,
`o>=`, `o-first-expt`, `o-first-coeff`, `o-rst`,
`make-ord`, `o-finp`, and `o-infp`.

The old ordinals were based on the following formulation of Cantor Normal Form:

For any ordinal,

Thus, a predicate recognizing ACL2's old ordinals is given by the following definition.

(defun e0-ordinalp (x) (if (consp x) (and (e0-ordinalp (car x)) (not (equal (car x) 0)) (e0-ordinalp (cdr x)) (or (atom (cdr x)) (not (e0-ord-< (car x) (cadr x))))) (and (integerp x) (>= x 0))))

The new representation is based on a corollary to the above theorem, which
we get by the left distributive property of ordinal multiplication over
ordinal addition. Thus,

For any ordinal,

Instead of representing an ordinal as a list of non-increasing ordinals, we represent it as a list of exponent-coefficient pairs, such that the exponents are strictly decreasing (see o-p). Note that this representation is exponentially more efficient than the old representation.

The ordinal arithmetic functions:

The community book arithmetic/natp-posp is a book for reasoning
about

If you have a good reason to use the old definitions of the ordinals (e.g.,
because of legacy code and theorems), then we provide a convenient way to do
this. The book

The ordinals books have been used to prove non-trivial theorems. For a
good example, see the books in the community books directory

Finally, many termination proofs can be carried out with weaker orderings
than the ordinals up to `defthm` and provide a macro
`ACL2-count`, the default measure chosen by ACL2.

- O-p
- A recognizer for the ordinals up to epsilon-0
- O<
- The well-founded less-than relation on ordinals up to
epsilon-0 - Proof-of-well-foundedness
- A proof that
`o<`is well-founded on`o-p`s - Two-nats-measure
- An ordinal measure for admitting functions: lexicographic ordering of two natural numbers.
- Nat-list-measure
- An ordinal measure for admitting functions: lexicographic ordering of a list of natural numbers.
- Make-ord
- A constructor for ordinals.
- O-first-coeff
- Returns the first coefficient of an ordinal
- E0-ord-<
- The old ordering function for ACL2 ordinals.
- O-first-expt
- The first exponent of an ordinal
- E0-ordinalp
- The old recognizer for ACL2 ordinals.
- O-rst
- Returns the rest of an infinite ordinal
- O-finp
- Recognizes if an ordinal is finite
- O>=
- The greater-than-or-equal relation for the ordinals
- O<=
- The less-than-or-equal relation for the ordinals
- O-infp
- Recognizes if an ordinal is infinite
- O>
- The greater-than relation for the ordinals