Lambda-Calculus and Combinators, an Introduction

15B Syntax-free definitions 241
Conversely, if we take a syntax-free λ-model 〈D, •, Λ〉 and let e be
any representative of Λ, then 〈D, •,e〉 is a loose Scott–Meyer model. So
the loose Scott–Meyer definition of model is essentially equivalent to the
earlier definitions, 15.3 and 15.19.
However, one mapping Λ may have many representatives, so one
model in the sense of 15.19 may give rise to many loose Scott–Meyer
models. But only one of these is strict. To find it, choose
e0 = [[1]] = [[λxy.xy]], (4)
where [[ ]] is defined from Λ by 15.20(a)–(c). This e0 represents Λ, by
the end of Discussion 15.18, and we also have
e0 • e0 = [[1]]• [[ 1]] = [[1 1]] = [[1]] = e0, (5)
so 〈D, •,e0〉 is a strict model.
No other representative of Λ gives a strict model. Because if e ′ represents
Λ and 〈D, •,e ′ 〉 is a strict model, we get e ′ = e0. In detail:
e ′ = e ′ • e ′ by 15.22(d) for e ′ ,
= Λ(e ′ ) since e ′ represents Λ,
= e0 • e ′ since e0 represents Λ,
= e0 • e0 by 15.22(c) for e0 since e ′ ∼ e0,
= e0 by 15.22(d) for e0.
This discussion can be summed up as follows.
Theorem 15.24 The constructions Λ=Fun(e) and e0 =[[1]]are
mutual inverses. That is: if 〈D, •, Λ〉 is a syntax-free λ-model in the
sense of Definition 15.19, and e0 = [[ 1 ]], then 〈D, •,e0〉 is a strict
Scott–Meyer λ-model and Fun(e0) =Λ;and,conversely,if〈D, •,e〉 is a
strict Scott–Meyer λ-model and Λ=Fun(e), then 〈D, •, Λ〉 is a syntaxfree
λ-model and [[ 1]]=e.
Remark 15.25 The Scott–Meyer definition of model is clearly simpler
than both the previous definitions, in fact it is almost as simple as the
definition of ‘combinatory algebra’ in 14.9. All its clauses except (c) can
be expressed in the form
(∃x1,...,xm )(∀y1,...,yn)(P = Q) (m, n ≥ 0). (6)
But the exception is very important. If every clause had form (6), then
an analogue of the submodel theorem could be proved (Theorem 14.23),