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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ β{Notes on Lee Blaine's thesis ␈↓↓Programs for Structured Proofs␈↓

␈↓ α∧␈↓␈↓ αTMy␈αprimary␈α
impression␈αis␈αthat␈α
Blaine␈αhasn't␈α
faced␈αthe␈αfact␈α
(or␈αis␈αglossing␈α
it␈αover)␈α
that␈αthe
␈↓ α∧␈↓style␈αof␈α
proof␈αused␈αin␈α
the␈αthesis␈α
and␈αby␈αEXCHECK␈α
generally␈αis␈αsuitable␈α
only␈αfor␈α
relatively␈αeasy
␈↓ α∧␈↓theorems.

␈↓ α∧␈↓␈↓ αTA␈αmajor␈α
trouble␈αis␈αthat␈α
there␈αare␈αno␈α
examples␈αin␈αwhich␈α
a␈αnew␈αconcept␈α
has␈αto␈αbe␈α
introduced
␈↓ α∧␈↓that␈αisn't␈α
present␈αin␈α
the␈αstatement␈αof␈α
the␈αtheorem␈α
or␈αthe␈αprevious␈α
theorems,␈αe.g.␈α
the␈αcolors␈α
of␈αthe
␈↓ α∧␈↓squares␈α⊂in␈α⊃the␈α⊂theorem␈α⊃about␈α⊂not␈α⊃being␈α⊂able␈α⊂to␈α⊃tile␈α⊂a␈α⊃checkerboard␈α⊂with␈α⊃diagonally␈α⊂opposite
␈↓ α∧␈↓squares␈α
removed␈αor␈α
the␈α
pairings␈αof␈α
integers␈αwith␈α
their␈α
inverses␈α␈↓↓(mod p)␈↓␈α
in␈αthe␈α
proof␈α
of␈αWilson's
␈↓ α∧␈↓theorem.␈α⊃ I␈α⊃would␈α⊃like␈α⊃to␈α⊃see␈α⊃some␈α⊃discussion␈α⊃of␈α⊃this␈α⊃issue␈α⊃even␈α⊃if␈α⊃it␈α⊃is␈α⊃too␈α⊃late␈α⊃to␈α⊂improve
␈↓ α∧␈↓EXCHECK␈α
or␈αto␈α
make␈α
more␈αexperiments.␈α
 Otherwise,␈αthe␈α
thesis␈α
is␈αsimply␈α
misleading␈α
about␈αthe
␈↓ α∧␈↓progress that has been made in proof-checking.

␈↓ α∧␈↓␈↓ αTIt␈α
also␈α
seems␈α
to␈α
me␈α
that␈α
the␈α
content␈α
of␈α
the␈α
set␈α
theory␈α
course␈α
has␈α
been␈α
oversimplified␈αin␈α
order
␈↓ α∧␈↓to␈αfit␈αEXCHECK's␈αlimitations.␈α For␈αexample,␈αthere␈αis␈αno␈αmention␈αof␈αmetatheorems␈αsuch␈α
as␈αthose
␈↓ α∧␈↓concerning␈α
the␈α
relations␈α∞of␈α
the␈α
various␈α∞systems␈α
of␈α
set␈α∞theory␈α
or␈α
independence␈α∞or␈α
incompleteness.
␈↓ α∧␈↓While␈α
I␈α
agree␈αthat␈α
this␈α
is␈α
beyond␈αthe␈α
state␈α
of␈α
the␈αpresent␈α
art,␈α
and␈α
a␈αuseful␈α
body␈α
of␈α
material␈αcan␈α
be
␈↓ α∧␈↓taught that doesn't involve it, it is again misleading to ignore the issue.

␈↓ α∧␈↓␈↓ αTThere are a few specific errors that I have noticed.

␈↓ α∧␈↓47␈α⊃-␈α⊃The␈α⊃rewrite␈α⊃scheme␈α⊃for␈α⊃finite␈α⊃universal␈α⊃quantifiers␈α⊃should␈α⊃have␈α⊃∧␈α⊃instead␈α⊃of␈α⊃∨␈α⊃in␈α⊂the
␈↓ α∧␈↓conclusion.

␈↓ α∧␈↓73␈α-␈αThe␈αstatement␈αof␈αRussell's␈αtheorem␈αin␈αthe␈αtext␈αis␈αgarbled.␈α I␈αmean␈αthe␈αEnglish␈αform␈α
not␈αthe
␈↓ α∧␈↓symbolic form.

␈↓ α∧␈↓␈↓ αTI␈α
am␈α
prepared␈α
to␈αsign,␈α
but␈α
I␈α
would␈αlike␈α
some␈α
consideration␈α
of␈αthe␈α
above␈α
criticisms␈α
if␈αthere␈α
is
␈↓ α∧␈↓time.

␈↓ α∧␈↓John McCarthy, Computer Science Department