![](bg8.png)
of the canonical terms. The proof of Theorem 4 in-
volves details of general canonical forms and induc-
tion steps. It demonstrates what part of a canonical
form is reduced, by inessential divergence from the
strictest referential synonymy of the entire term. It
demonstrates that the replacements preserve the de-
notation of the entire term.
The presented η-rule has applications to computa-
tional semantics of human languages and to semantics
of programs and compilers. Replacements based on
the η-rule are not always possible because there are
intervening (algorithmic) structures. The effect of a
general β-replacement would have been like reversing
iteration to recursive program interpreter. Something
like a compiler that translates a program with recur-
sion (i.e. a L
λ
ar
term) to a program with induction,
and then finds functional components that compute
constant functions (not depending on inputs), and in
attempts to provide the constant value, without using
the “vacuous” function applications, is reversing the
tail recursion into recursion. The presented η-rule
avoids this.
In the analyses of certain classes of human lan-
guage expressions, at least those that we have covered
in recent work, the η-rule provides simplification of
canonical terms that are otherwise irreducible.
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