The Communications of the TEX Users Group Volume 30 ... - TUG

192 TUGboat, Volume 30 (2009), No. 2
of definitions is taken to include a standard default
set of infix operators as well as quantifiers. The default
symbols may be easily replaced and with a little
additional effort, default forms can be replaced. The
resulting language is unambiguous and possesses the
property that no term or formula can begin another.
ProofCheck minimizes the loss of syntactical
freedom by striving to keep the syntactical restrictions
as close as possible to the absolute requirements
of logic and consistency.
5 Proof language
The following TEX macros suffice to mark up a proof
for checking. We are grateful to Karl Berry for his
help in streamlining the first two of these. The general
proof structure which these macros implement
is very similar to that defined in [1].
\prop Any proposition, whether a theorem, a definition
or an axiom, must be introduced by this
macro. Its use requires ProofCheck to look for
a proof. Its syntax is \prop, followed by the
enumeration, followed by the proposition. For
example:
\prop 14.7 $(1 < 2)$
The \prop macro must be at the beginning
of a line. The enumeration is of the form n.m
where n and m are positive integers with n representing
the section or chapter number. The
proposition must be enclosed within TEX dollar
signs. References in a proof to other theorems
use the same n.m style and not L ATEX labels.
When re-arrangement of theorems necessitates
renumbering, this is handled by a ProofCheck
program called renum. This program recognizes
L ATEX section macros so that unless the section
counter is reset manually, consistency with
L ATEX sectioning is maintained.
\note This macro is used to introduce an assertion
within the proof which can be referred to later
in the proof. The assertion may be the result of
telescoping multiple lines of proof, each having
an optional justification. The ProofCheck syntax
is \note, followed by the note enumeration,
followed by the (possibly multi-line) assertion.
Here the enumeration consists of a positive integer.
\By This macro is used at the end of a line to introduce
justification for the step. A note with a
single line of assertion may be justified by one
of the following:
\By G (Given) signals an assertion which is to
be accepted locally as a hypothesis. It
Bob Neveln and Bob Alps
initiates a block of the proof within which
this hypothesis is in effect.
\By S (Set) indicates that the note is used to
locally define a variable for use in the proof.
\By .n H .m (Hence) where n is the number
of a note established using note m as a
Given, ends the block initiated by note
m. Such blocks are called Given-Hence
blocks. A Hence justification is typically
used to establish an assertion such as (p →
q) where p is note m and q is note n.
Notes with either a single line or more than
one line of assertion may be justified using other
notes and a .m enumeration or other propositions
and n.m enumeration with punctuation as
shown in the sample proof. More discussion on
this syntax may be found in [4].
\Bye in addition to introducing a justification, signals
the end of a proof and prints “Q.E.D.”.
\linea, \lineb, ... These begin a new line with
increasing degrees of indentation. They are not
proof macros per se but are used for any mathematical
expressions that need to go beyond one
line and need indentation. At present the parser
allows only these.
In section 7 the use of these macros in a sample
proof is shown.
6 Sample proof: Reader view
The simple proof in figure 1 is taken from a development
of the von Neumann model of the natural
numbers, ω, in which each natural number is the
set of the preceding natural numbers. The theorem
asserts that if y is an element of a natural number x
then y is a subset of x.
The first line of the proof defines a set A. This
note should be easy to read except for possibly the
quantifier notation and the classifier notation. Notational
changes are discussed in section 9. The
second note translates the definition in note 1 into
a bi-conditional which is much more useful deductively.
We often refer to steps that turn a definition
into one or more implications as “unwrapping” steps.
Explicit inclusion of such unwrapping steps is often
key in getting a proof to check. The stage is set for
an induction proof.
The theorem 4.7 referred to is just the standard
induction theorem:
(∅ ∈ A∧ � x ∈ A(x ∈ A → scsrx ∈ A) → ω ⊂ A)
where “scsr x” denotes the successor of x. Its two
hypotheses are the base case which in this proof is
established in note 3 and the universalization of the