o< is well-founded on o-ps
Major Section: MISCELLANEOUS
The soundness of ACL2 rests in part on the well-foundedness of o< on
o-ps. This can be taken as obvious if one is willing to grant that
those concepts are simply encodings of the standard mathematical notions of
the ordinals below epsilon-0 and its natural ordering relation. But it
is possible to prove that o< is well-founded on o-ps without
having to assert any connection to the ordinals and that is what we do here.
The book books/ordinals/proof-of-well-foundedness carries out the proof
outlined below in ACL2, using only that the natural numbers are
well-founded.
Before outlining the above mentioned proof, we note that in the analogous
documentation page of ACL2 Version_2.7, there is a proof of the
well-foundedness of e0-ord-< on e0-ordinalps, the less-than relation
and recognizer for the old ordinals (that is, for the ordinals appearing in
ACL2 up through that version). Manolios and Vroon have given a proof in ACL2
Version_2.7 that the current ordinals (based on o< and o-p) are
order-isomorphic to the old ordinals (based on e0-ord-< and
e0-ordinalp). Their proof establishes that switching from the old
ordinals to the current ordinals preserves the soundness of ACL2. For
details see their 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 URL http://www.cs.utexas.edu/users/moore/acl2/workshop-2003/.
We now give an outline of the above mentioned proof of well-foundedness. We
first observe three facts about o< on ordinals that have been proved by
ACL2 using only structural induction on lists. These theorems can be proved
by hand.
(defthm transitivity-of-o<
(implies (and (o< x y)
(o< y z))
(o< x z))
:rule-classes nil)
(defthm non-circularity-of-o<
(implies (o< x y)
(not (o< y x)))
:rule-classes nil)
(defthm trichotomy-of-o<
(implies (and (o-p x)
(o-p y))
(or (equal x y)
(o< x y)
(o< y x)))
:rule-classes nil)
These three properties establish that o< orders the
o-ps. To put such a statement in the most standard
mathematical nomenclature, we can define the macro:
(defmacro o<= (x y) `(not (o< ,y ,x)))and then establish that
o<= is a relation that is a simple,
complete (i.e., total) order on ordinals by the following three
lemmas, which have been proved:
(defthm antisymmetry-of-o<=
(implies (and (o-p x)
(o-p y)
(o<= x y)
(o<= y x))
(equal x y))
:rule-classes nil
:hints (("Goal" :use non-circularity-of-o<)))
(defthm transitivity-of-o<=
(implies (and (o-p x)
(o-p y)
(o<= x y)
(o<= y z))
(o<= x z))
:rule-classes nil
:hints (("Goal" :use transitivity-of-o<)))
(defthm trichotomy-of-o<=
(implies (and (o-p x)
(o-p y))
(or (o<= x y)
(o<= y x)))
:rule-classes nil
:hints (("Goal" :use trichotomy-of-o<)))
Crucially important to the proof of the well-foundedness of
o< on o-ps is the concept of ordinal-depth,
abbreviated od:
(defun od (l)
(if (o-finp l)
0
(1+ (od (o-first-expt l)))))
If the od of an o-p x is smaller than that of an
o-p y, then x is o< y:
(defun od-1 (x y)
(if (o-finp x)
(list x y)
(od-1 (o-first-expt x) (o-first-expt y))))
(defthm od-implies-ordlessp
(implies (and (o-p x)
(< (od x) (od y)))
(o< x y))
:hints (("Goal"
:induct (od-1 x y))))
Remark. A consequence of this lemma is the fact that if s = s(1),
s(2), ... is an infinite, o< descending sequence of o-ps, then
od(s(1)), od(s(2)), ... is a ``weakly'' descending sequence of
non-negative integers: od(s(i)) is greater than or equal to
od(s(i+1)).
Lemma Main. For each non-negative integer n, o< well-orders
the set of o-ps with od less than or equal to n .
Base Case. n = 0. The o-ps with 0 od are the non-negative integers. On the non-negative integers, o< is the same as <.Theorem.Induction Step. n > 0. We assume that o< well-orders the o-ps with od less than n.
If o< does not well-order the o-ps with od less than or equal to n, consider, D, the set of infinite, o< descending sequences of o-ps of od less than or equal to n. The first element of a sequence in D has od n. Therefore, the o-first-expt of the first element of a sequence in D has od n-1. Since o<, by IH, well-orders the o-ps with od less than n, the set of o-first-expts of first elements of the sequences in D has a minimal element, which we denote by B and which has od of n-1.
Let k be the minimum integer such that for some infinite, o< descending sequence s of o-ps with od less than or equal to n, the first element of s has an o-first-expt of B and an o-first-coeff of k. Notice that k is positive.
Having fixed B and k, let s = s(1), s(2), ... be an infinite, o< descending sequence of o-ps with od less than or equal to n such that s(1) has a o-first-expt of B and an o-first-coeff of k.
We show that each s(i) has a o-first-expt of B and an o-first-coeff of k. For suppose that s(j) is the first member of s either with o-first-expt B and o-first-coeff m (m neq k) or with o-first-expt B' and o-first-coeff B' (B' neq B). If (o-first-expt s(j)) = B', then B' has od n-1 (otherwise, by IH, s would not be infinite) and B' is o< B, contradicting the minimality of B. If 0 < m < k, then the fact that the sequence beginning at s(j) is infinitely descending contradicts the minimality of k. If m > k, then s(j) is greater than its predecessor; but this contradicts the fact that s is descending.
Thus, by the definition of o<, for s to be a decreasing sequence of o-ps, (o-rst s(1)), (o-rst s(2)), ... must be a decreasing sequence. We end by showing this cannot be the case. Let t = t(1), t(2), ... be an infinite sequence of o-ps such that t(i) = (o-rst s(i)). Then t is infinitely descending. Furthermore, t(1) begins with an o-p B' that is o< B. Since t is in D, t(1) has od n, therefore, B' has od n-1. But this contradicts the minimality of B. Q.E.D.
o< well-orders the o-ps. Proof. Every
infinite, o< descending sequence of o-ps has the
property that each member has od less than or equal to the
od, n, of the first member of the sequence. This
contradicts Lemma Main.
Q.E.D.
