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Refining Properties of Filter Models:
Sensibility, Approximability and Reducibility
ANONYMOUS AUTHOR(S)
In this paper, we study the tedious link between the properties of sensibility and approximability of models of untyped
λ-calculus.Approximability is known to be a slightly, but strictly stronger property that sensibility. However, we will see that so far, each and every (filter) model that have been proven sensible are in fact approximable. We explain this result as a weakness of the sole known approach of sensibility: the Tait reducibility candidates and its realizability variants.
In fact, we will reduce the approximability of a filter model
Dfor the
λ-calculus to the sensibility ofDbut for an extension of the
λ-calculus that we callλ-calculus withD-tests. Then we show that traditional proofs of sensibility ofDfor the
λ-calculusare smoothly extendable for this
λ-calculus withD-tests.ACM Reference format:
Anonymous Author(s). 2017. Refining Properties of Filter Models: Sensibility, Approximability and Reducibility. 1, 1, Article 1 (December 2017), 23 pages.
https://doi.org/10.1145/nnnnnnn.nnnnnnn
INTRODUCTION
Sensibility. It is the ability, for a model, to distinguish non terminating programms from meaningfull ones by collapsing their interpretation (Def. 1.7). Through Curry-Howard isomorphism, it also corresponds to the consistence of the internal theory of the model. This shows the importance in understanding sensibility, but also the undecidability of such a property.
Such profund but undecidable results are often targets for classification of more or less easy subclasses, serving as grinding stone for proof technics. Here we take an unorthodox approach consisting in classifying sensible models by using as discriminator a slightly stronger property called “Approximability”. To our surprise, we found out that available methodes to prove sensibility (reducibility) where not powerfull enought to distinguish sensibility from approximablity.
Approximability. The approximation theorem (Def. 1.10) is an important concept when considering denotational models of the head reduction. In order to study head reduction, λ-calculists systematically use B¨ohm trees, which are basically normal forms of a degenerated λ-calculus using an error symbol (Def. 1.9). Such objects are able to approximate terms, the same way as partial evaluations are approximating the notion of evaluation. The approximability simply says that the model reflects the fact that a term is the limit its finite B¨ohm approximations.
This notion has been extensively studied [1, Section III.17.3] and this article presents a new su ffi cient condition for approximability, the weak positivity by far encompassing any previous results on approximability (of filter models). As a property on models, approximability is supposed to be striclty stronger that sensibility. Indeed, approximability implies that the interpretation of any diverging terms (and only those) are collapsed into the interpretation of the error symbol ⊥. This inclusion is supposed to be strict as, for exemple, approximable models
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are not able to distinguish the Turring fixedpoint from the Church fixedpoint. In fact, there is a continuity of sensible but non-approximable λ-theories, it is surprising that we are not able to model any of those.
Reducibility. In this title, “Reducibility” refers to Tait reducibility methods [? ] and its modern extensions (including realisability). These methods used to prove structural properties of types systems and models such as sensibility and approximability but also more practical properties [? ]. For types systems, it consists in constructing saturated sets of terms with the wanted property by induction on types, then in proving that any typable term has been included. For denotational models, the method is more subtle due to the structure not being inductive : one must find a fixpoint to be able to apply the method, but the fixpoint does not need to be computable or constructive in any way.
In Section 4, we use the sensiblility and the approximability as a grinding stone to perform yet a new disection of those reducibiliy / realisability methodes. We try to be as general as possible until the last moment in order to get the the coarsest possible characterization, but also in order to point over the specific weaknesses of the methode.
We will discuss in the conclusion and along the paper why we were not able to fill the gap between approximability and sensibility. In particular, we insist on the link between this blockage and the difficulty to performe fixpoints on non-monotonous functions.
Filter Models. Introduced in the 80s using the notion of type as the elementary brick for their construction, filter models [10] (Def.1.1) are extracted from a type theory with simple types enlarged by intersection types and subtyping. Formally, the interpretation of a λ-term is the filter generated by the set of its types. Variations on the intersection type theory induce di ff erent filter models. The resulting class essentially corresponds to the class of Scott complete lattices.
Filter models (and domains) form one of the classes of models of untyped λ-calculus that have been the more broadly studied, but properties such as sensibility and approximability are yet to be understood perfectly.
In particular, a simple bibliographical analysis show that that the theoretically huge gap between sensible and approximable models have never been filed by any model. The best advancements toward this direction are covered by the third part of the recent book “Lambda-calculus with types” of Barendregt, Dekkers and Statman [1].
λ-calculi with tests. In order to exhibit the link between sensibility and approximability, we are using λ-calculi with tests of Section 2. These are syntactic extensions of the untyped λ-calculus with operators defining types of the underlying intersection type system. We will see (Sec. 3) that the approximability of a filter model D is equivalent to the sensibility of the same model D for the λ-calculus with D-tests Λ
τ(D)(with respect to a notion of head convergence). This theorem brings together the notions of sensibility and approximability in a very novel way!
The calculi with tests played a central role in this paper. The idea of test mechanisms as syntactic extensions of the λ-calculus was first used by Bucciarelli et al. [8] and developed further by the author in [4] and [5] for Krivine-models. The one presented in this paper is yet an other generalization to the broader (extensional and distributive) filter models. Originally inspired from Wadsworth’s labeled λ⊥-calculus [23] and Girard experiments [12, 15], they are syntactic extensions of the λ-calculus with operators defining compact elements of the given models. Since the model appears in the syntax, we are able to perform inductions (and co-inductions) directly on the reduction steps of actual terms, rather than on the construction of B¨ohm trees.
Content. Section 1 will focuss on preliminarries, with mostly standard presentations of the untyped lambda- calculus, the filder models and the B¨oms trees. In Section 2, we present the λ-calculi with tests, mostly following previous works of the author [4]; we give their syntaxe, their interpretation in filter models, and finaly their main properties. Section 3 is short but central in this paper: we present here the collapse of the notions of approximability and sensibility at the level of test extensions.
In a Section 4, we will present a standard proof of sensibility by reducibility adapted to λ-calculi with tests. Using
our new equivalence between sensibility for this calculus with tests and approximability, this a priori standard
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• proof of sensibility becomes a non-standard proof of approximability! This allows us to describe a condition for approximability that encompasses every known sensible extensional filter models, bringing these two properties closer than we believed them to be.
1 PRELIMINARIES 1.1 The λ -calculus
In this paper, we only consider the minimal untyped λ-calculus with the contextual and / or the head reduction, in the pure tradition of Barendregt book [2].
The λ-terms are defined up to α-equivalence by the following grammar using notation “`a la Barendregt” (where variables are denoted by final Latin letters x, y, z...):
(λ-terms) Λ M, N ::= x | λx.M | M N
We denote by FV(M) the set of free variables of a λ-term M. Moreover, we denote by M[N/x] the capture-free substitution of x by N.
The λ-terms are subject to the β-reduction:
(β) (λx.M) N →
βM[N/x]
A context C is a λ-term with possibly some occurrences of a hole, i.e.:
(contexts) Λ
(|.|)C :: = ( |.| ) | x | λx.C | C
1C
2The writing C( |M| ) denotes the term obtained by filling the holes of C by M. The small step reduction → is the closure of (β) by any context, and →
his the closure of (β) by the rules:
M →
hM
′λx.M →
hλx.M
′M →
hM
′M is an application M N →
hM
′N
The transitive reduction →
*(resp →
*h
) is the reflexive transitive closure of → (resp →
h).
The big step head reduction, denoted M ⇓
hN, is M →
*h
N for N in a head-normal form, i.e., of the form λx
1...x
k.y M
1· · · M
k, for M
1, ..., M
kany terms.
We write M ⇓
hfor the (head) convergence, i.e., whenever there is N such that M ⇓
hN.
Other notions of convergence exist (strong, lazy, call by value...), but our study focuses on head convergence.
1.2 Filter Models
Def of sensibil- ity
Def of sensibil- We introduce here the main semantic object of this article: distributive and extensional filter models (of DEFiM). ity
Despite corresponding to reflexive complete lattices (endowed with continuous functions), we are not using this presentation to describe filter models, but rather its dual representation by Stone duality: the sup-lattice of compact elements. The following presentation is rather standard, and the notations can be find here [9] for example.
This presentation has the advantage to match the representation of the interpretation of terms as intersection types derivations, as we will see in Proposition 1.
The models consists of a set D of “types” (or compact elements), and two operations: the intersection ∧ (characterising the induced order) and the functional arrow → (characterising the reflexive embeding). Moreover, we will consider extensionality, which means the the η-conversion is viable, which we represent by the existence of a specific function ext
D: D → 𝒫
f(D × D).
Definition 1.1 ([10]). A filter model is a triple (D, ∧, →) where:
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∙ D = (|D|, ∧) is a pointed join-semilattice; we denote ω the top element and ≥
Dthe induced order : α ∧ α = α α ∧ β = β ∧ α α ∧ (β ∧ γ) = (α ∧ β) ∧ γ α ∧ ω = α (α ≥
Dβ ⇔ α ∧ β = β)
∙ → is a binary operation on D such that γ→δ ≥
D⋀︁
i
α
i→β
i⇔ δ ≥
D⋀︁
{i|γ≤αi}
β
i, in particular, γ→δ = ω i ff δ = ω.
A filter model is extensional whenever there is a function ext
D: D→𝒫
f(D × D) that associates to each α ∈ D a finite subset ext
D(α) ⊆ D × D such that:
α = ⋀︁
(β,γ)∈extD(α)
β→γ
It is free to consider that the image of ext
D(α) by → is an anti-chain in the sens that for any pair (β, γ) ∈ ext
D(α) and any non empty subset I ⊆ ext
D(α):
⋀︁
(β′,γ′)∈I
(β
′→γ
′) < Im(→) and β→γ ̸≥
D⋀︁
(β′,γ′)∈extD(α)−(β,γ)
β
′→γ
′In particular (β, ω) ∈ ext
D(α) implies α = ω, moreover ext
D(ω) = { (β, ω) } for some arbitrary β since β→ω = ω.
Unfortunately, the function ext
Dis generally not unique or even canonical. in order to make it canonical, we restrict our study to distributive filter models. A filter model D is distributive whenever any α ≥ β ∧ γ is accessible in the sens that there exists a decomposition α = β
′∧ γ
′such that β
′≥
Dβ and γ
′≥
Dγ.
For short, we call DEFiM the distributive and extensional filter models.
By abuse of notation we may denote the quadruple (D, ∧, →, ext
D) simply by D when it is clear from the context that we are referring to a DEFiM.
Creating a DEFiM from scratch is often heavy, as they have to satisfy complex rules even forcing the model to be an infinite object. Fortunately, there is a way to automatically infer the required properties from a smaller (often finite) core object. This core object is a partial DEFiM which is a basically a subset of a DEFiM.
Definition 1.2. An partial filter model is a triple (E, ∧, →) satisfying the axioms of filter models except that → is partially defined and for any α, (β
i)
i≤n∈ E
n+1:
(∀i ≤ n, α→β
idefined ) ⇒ α→ ⋀︁
i
β
idefined as ⋀︁
i
(α→β
i)
It is a partial DEFiM if ext
Eis defined and E satisfies the other axioms of distributive extensional filter models.
Definition 1.3. The completion of a partial DEFiM (E, ∧ , → , ext
E) is the union E ¯ : =
⎛
⎜ ⎜
⎜ ⎜
⎜ ⎝
⋃︁
n∈N
E
n, ⋃︁
n∈N
(∧
n) , ⋃︁
n∈N
(→
n) , ⋃︁
n∈N
ext
En⎞
⎟ ⎟
⎟ ⎟
⎟ ⎠
of partial completions (E
n, ∧
n, →
n, ext
En) that are partial DEFiM defined by induction on n:
The initialization (E
0, ∧
0, →
0, ext
E0) : = (E, ∧, →, ext
E) is performed by the partial DEFiM, and we continue by completing:
∙ |E
′n+1| : = 𝒫
f(|E
n| ⊎ (|E
n|
2−Dom(→
n))), for readability, use a, b.. for elements of |E
′n+1| and we write α→
*β for (α, β) in the second component,
∙ →
′n+1
is defined only over | E
n|
2⊆ | E
′n+1|
2by { α }→
′n+1
{ β } : = { α →
nβ } whenever (α, β) ∈ Dom( →
n) and by {α}→
′n+1{β} := {α→
*β} whenever (α, β) ∈ | E
n|
2− Dom( →
n),
∙ ext
′n+1is defined over | E
n′+1| by ext
′n+1(a) = {ext
n(α) | α ∈ E
n∩ a } ∧ {(α, β) | α→
*β ∈ a }.
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•
⟦ x
i⟧
⃗xD= {(⃗α, β) | β ≥ α
i} ⟦ λy.M ⟧
⃗xD= {(⃗α, ⋀︀
i
(β
i→γ
i)) | ∀i, (⃗αβ
i, γ
i) ∈ ⟦ M ⟧
⃗xyD}
⟦ M N ⟧
⃗xD= { (⃗α, ⋀︀
i
β
i) | ∃⃗γ
i, (⃗α, ⋀︀
i
(γ
i→β
i)) ∈ ⟦ M ⟧
⃗xD∧ (⃗α, ⋀︀
i
γ
i) ∈ ⟦ N ⟧
⃗xD}
Fig. 1. Direct interpretation ofΛin the modelD∙ |E
n+1| : = |E
n′+1|/
≡is the quotient of |E
n′+1| by the equivalence a ≡ b whenever:
∀ (α, γ) ∈ ext
′n+1(a), γ ≥ ⋀︁
{(β,δ)∈extn+1(b)|α≤β}
δ ∀ (β, δ) ∈ ext
′n+1(b), δ ≥ ⋀︁
{(α,γ)∈extn+1(a)|β≤α}
γ
∙ ∧
n+1, →
n+1and ext
n+1are the quotients of ∧, →
′n+1and ext
′n+1by ≡ (notice that →
′n+1only need to be defined for one element equivalent class for →
′n+1to be defined).
We consider that E
n⊆ E
n+1since for each α ∈ |E
n|, {α} is in a di ff erent equivalence class.
R emark 1. The completion of a partial filter model (E, →, ext
E) is well defined and corresponds to the coarsest DEFiM E containing E. In particular, any DEFiM model D is the completion of itself: D ¯ = D. ¯
Example 1.4. Most filter models found in the literature can in fact be given as extensional completions of extremely simple partial filter models. Here are some example, the three first one are from the literatures and the two last one are fully expressing the power of the extensional completion:
(1) Scott’s D
∞[21] is the completion of
|D| : = {ω, *}, * ∧ ω : = *, ω→* : = * ext
D(*) : = {(ω, *)}.
Notice, that *→* is undefined in D so that we need the completion.
(2) Park’s P
∞[20] is the completion of
|P| : = {ω, *}, ω ∧ * : = *, *→* : = * ext
P(*) : = {(*, *)}.
(3) Norm or D
*∞[11] is the completion of
|D
*| : = {ω, p, q}, ω ∧ p : = p ω ∧ q : = q p ∧ q : = q p → q := q q → p := p ext
D*(q) := { (p, q) } ext
D*(p) := { (q, p) }, (4) Z
∞is the completion of
|Z| : = {n | n ≥ 0}, n ∧ ω : = n, ω→n + 1 : = n ext
D(n) : = {(ω, n + 1)}.
(5) U
∞is the completion of
|U| : = {n | n ≥ 0}, n ∧ ω : = n, n + 1→n + 1 : = n ext
D(n) : = {(n + 1, n + 1)}.
R emark 2. The completion of a partial filter model is in fact the free completion in the sens that for any partial DEFiM E ⊆ D contains in a DEFiM D, there is a function φ : ¯ E → D stable in E such that ⟦ . ⟧
E¯⊆ ⟦ . ⟧
D, where
⟦ . ⟧
E¯(resp. ⟦ . ⟧
D) is the interpretation of the λ-calculus into E (resp. D) as defined below. ¯
It is folklore that the interpretation of the λ-calculus into a given D can be equivalently characterized by a specific intersection type system whose types are elements α ∈ D, with ∧ modeling the intersection and → the logical implication.
Definition 1.5 (Interpretation of λ-terms). In Figure 1, we give the interpretation of M into a filter model D.
The interpretation ⟦ M ⟧
xD1...xnof M is suppose to be a morphism (Scott-continuous function) from D
⃗xto D where
⃗ x is a superset of the free variables of M. Concretely, we use the Cartesian closeness do define ⟦ M ⟧
xD1...xnas a
downward-close subsets of (D
op)
⃗x× D.
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x : α ⊢ x : α Γ ⊢ M : α
Γ , x : β ⊢ M : α Γ ⊢ M : β α ≥ β Γ ⊢ M : α Γ , x : α ⊢ M : β
Γ ⊢ λx.M : α→β
Γ ⊢ M : α→β Γ ⊢ N : α Γ ⊢ M N : β
Γ ⊢ M : α Γ ⊢ M : β Γ ⊢ M : α ∧ β
Fig. 2. Intersection types for theλ-calculus inDIn Figure 7a, we give the intersection-type assignment corresponding to D. Notice that we can infer typing sequents for the form Γ ⊢ M : α for Γ = (x
1: α
1, ..., x
n: α
n) an environment defined (at least) over all free variables of M.
Example 1.6.
⟦ λx.y ⟧
yD=
⎧
⎪ ⎪
⎨
⎪ ⎪
⎩ ((α), ⋀︁
i
(β
i→α
′i))
⃒ ⃒
⃒ ⃒
⃒ ⃒
⃒
∀i, α
′i≥
Dα
⎫
⎪ ⎪
⎬
⎪ ⎪
⎭ ,
⟦ λx.x ⟧
yD=
⎧
⎪ ⎪
⎨
⎪ ⎪
⎩ ((α), ⋀︁
i
(β
i→β
′i))
⃒
⃒ ⃒
⃒ ⃒
⃒ ⃒
∀i, β
′i≥
Dβ
i⎫
⎪ ⎪
⎬
⎪ ⎪
⎭ ,
⟦ I ⟧
D=
⎧
⎪ ⎪
⎨
⎪ ⎪
⎩
⋀︁
i
(α
i→α
′i)
⃒
⃒ ⃒
⃒ ⃒
⃒ ⃒
∀i, α
′i≥
Dα
i⎫
⎪ ⎪
⎬
⎪ ⎪
⎭ ,
⟦ 1 ⟧
D=
⎧
⎪ ⎪
⎨
⎪ ⎪
⎩
⋀︁
i
(α
i→α
′i)
⃒ ⃒
⃒ ⃒
⃒ ⃒
⃒
∃⃗β
′, ⃗γ
′, ⃗β, ⃗γ, ⋀︀
i
α
′i= ⋀︀
j
(β
′j→γ
′j), ⋀︀
j
γ
′j= ⋀︀
k
γ
k′,
⋀︀
i
α
i≤ ⋀︀
k
(β
k→γ
k), ⋀︀
j
β
′j≤ ⋀︀
k
β
k,
⎫
⎪ ⎪
⎬
⎪ ⎪
⎭ .
In the last two cases, terms are interpreted in an empty environment. We, then, omit the empty sequence associated with the empty environment, e.g., α→β→α stands for ((), α→β→α).
We can verify that extensionality holds, indeed ⟦ 1 ⟧
D= ⟦ I ⟧
D. To prove it we use ext
Das the witness function for both existential.
P roposition 1. Let M be a term of Λ and D a filter model, the following statements are equivalent representations of the interpretation of M in D:
∙ (⃗α, β) ∈ ⟦ M ⟧
⃗xDfor the interpretation defined in Figure 1,
∙ the type judgment ⃗ x : α ⊢ M : β is derivable by the rules of Figure 7a.
P roof . By structural induction on the grammar of Λ .
Definition 1.7 (Sensibility). A filter model D is sensible for the untyped λ-calculus if diverging terms corresponds exactly to those of empty interpretation:
M⇓
h⇔ ⟦ M ⟧
⃗x, ∅ . 1.3 B ¨ohm Approximants
The B¨ohm approximants (or finite B¨ohm trees) are the normal forms of a λ-calculus extended with a constant ⊥ and an additional reduction →
⊥.
A λ-term M is a λ-term possibly containing occurrences of the constant ⊥. The set λ of all λ-terms is generated by the grammar:
λ : M, N :: = x | λx.M | MN | ⊥
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• Similarly a (single hole) λ-context is a (single hole) context C( || )− possibly containing occurrences of ⊥. The
⊥-reduction →
⊥is defined as the λ⊥-contextual closure of the rules:
(⊥) λx.⊥ → ⊥ ⊥M → ⊥
The β-reduction is extended to λ-terms in the obvious way. We write B for the set of λ-terms in β⊥-normal forms and we denote its elements by s, t, u, . . .
The following characterization of β ⊥ -normal forms is well known.
L emma 1.8. Let M ∈ λ. We have M ∈ B if and only if either M = ⊥ or M has shape λx
1. . . x
n.x
iM
1· · · M
k(for some n, k ≥ 0) and each M
iis β⊥ -normal.
The set of all B¨ohm approximants of M can be obtained by calculating the direct approximants of all λ-terms β-convertible with M.
Definition 1.9. Let M ∈ λ.
(1) The direct approximant of M, written a p(M), is the λ-term defined as:
∙ a p(M) = ⊥ if M = λx
1. . . x
k.(λy.M
′)N M
1· · · M
k,
∙ a p(M) = λx
1. . . x
n.x
ia p(M
1) · · · a p(M
k) if M = λx
1. . . x
n.x
iM
1· · · M
k, (2) The set of finite approximants of M is defined by:
B(M) = {︀ a p(M
′) | M →
*hM
′}︀
We can now fully describe the property of approximability for a filter model:
Definition 1.10. A filter model is approximable i ff :
⟦ M ⟧
⃗x= ⋃︁
N∈B(M)
⟦ N ⟧
⃗x,
where the interpretation of B¨ohm approximants is the immediate extension of the interpretation of terms plus the minimal interpretation given to the bottom: ⟦ ⊥ ⟧ : = {(⃗α, ⊥) | ∀⃗α}.
2 λ -CALCULI WITH D-TESTS 2.1 Syntax
The original idea of using tests to recover full abstraction (via a theorem of definability) is due to Bucciarelli et al.
[8]. Here we define variants of Bucciarelli et al.’s calculus adapted to DEFiMs.
Directly dependent on a given DEFiM D, the λ-calculus with D-tests Λ
τ(D)is, to some extent, an internal calculus for D. In fact, we will see that, for D to be fully abstract for Λ
τ(D), it is su ffi cient to be sensible (Th. 2.13).
The idea is to introduce tests as a new kind in the syntax. Tests Q ∈ T
τ(D)are sort of co-terms, in the sense that their interpretations are maps from the context to the dualizing object of the linear category S cott L (ω = {*}):
⟦ Q ⟧
x1...xn∈ D
n⇒ ω
The type ω is the unit type, having only one value representing the convergence of the evaluation, seen as a success.
1The interaction between terms and tests is carried out by two groups of operations indexed by the elements α ∈ D:
τ
α: Λ
τ(D)→ T
τ(D)and τ ¯
α: T
τ(D)→ Λ
τ(D).
1We will see in Remark 3 that in a polarized context, the behavior of test does not correspond to co-term (or stack), but to commands (or processes),i.e., to interactions between usual terms and fictive co-terms extracted from the semantics.
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(term) Λ
τ(D)M, N ::= x | λx.M | M N | ∑︀
i≤n
τ ¯
αi(Q
i) , ∀ (α
i)
i∈ (D − ω)
n, n ≥ 0 (test) T
τ(D)P , Q :: = ∑︀
i≤n
P
i| ∏︀
i≤n
P
i| τ
α(M) , ∀α ∈ (D − ω), n ≥ 0
Fig. 3. Grammar of the calculus withD-tests
The first operation, τ
α, will verify that its argument M ∈ Λ
τ(D)has the point α in its interpretation. Intuitively, this is performed by recursively unfolding the B¨ohm tree of M and succeeding (i.e., converging) when α is in the interpretation of the finite unfolded B¨ohm tree. If α < ⟦ M ⟧ , the test τ
α(M) will either diverge or refute (raising a 0 considered as an error). Concretely, it is an infinite application that feeds its argument with empty ¯ τ operators.
The second operator, ¯ τ
α, simply constructs a term of interpretation ↓α if its argument succeeds and diverges otherwise. Concretely, it is an infinite abstraction that runs its test argument, but also tests each of its applicants using τ operators.
In addition to these operators, we use sums and products as ways to introduce may (for the addition) and must (for the multiplication) non-determinism; in the spirit of the λ + ||-calculus [13]. Indeed, these two forms of non-determinism are necessary to explore the branching of B¨ohm trees.
The idea of these two operators is to use the parametricity of our terms toward their intersection types. As a result, ¯ τ
α(ϵ) (further on denoted ϵ ¯
α), that transfers the always succeeding test ϵ into a term of interpretation ↓α, constitutes the canonical term of type α; its behavior is exactly the common behavior of every term of type α.
Symmetrically, the test τ
α(M) will verify whether M behaves like a term of type α.
Hereafter, D denotes a fixed DEFiM.
Definition 2.1. The λ-calculus with D-tests, for short Λ
τ(D), is given by the grammar in Figure 3. We denote the empty sum by 0, and the empty product by ϵ. Binary sums (resp. products) can be written with infix notation, e.g.
P + Q (resp P·Q).
Moreover, we use the notation ϵ ¯
α: = τ ¯
α(ϵ) and ϵ ¯
a: = ∑︀
α∈a
ϵ ¯
α; which are terms.
Sums and products are considered as multisets, in particular we suppose associativity, commutativity and neutrality with, respectively, 0 and ϵ.
In the following, an abstraction can refer either to a λ-abstraction or to a sum of ¯ τ operators. This notation is justified by the behavior of Σ
iτ ¯
αi(Q
i) that mimics an infinite abstraction.
The operational semantics is given by three sets of rules in Figure 4. The main rules of Figure 5a are the e ff ective rewriting rules. The distributive rules of Figure 5b implement the distribution of the sum over the test-operators and the product. The small step semantics → is the free contextual closure (i.e., by the rules of Figure 5d) of the rules of Figures 5a and 5b. The contextual rules of Figure 5c implement the head reduction →
hthat is the specific contextual extension we are considering.
Example 2.2. The operational behavior of D-tests depends on D. Recall the DEFiMs of Example 1.4. In the case of Scott’s D
∞we have in Λ
τ(D∞):
τ
*((λxy.x y) ϵ ¯
*) →
βhτ
*(λy. ϵ ¯
*y) →
τhτ
*(¯ ϵ
*0) →
τ¯ hτ
*(¯ ϵ
*) = τ
*(¯ τ
*(ϵ)) →
τ¯τhϵ,
τ
*((λxy.y x) ϵ ¯
*) →
βhτ
*(λy.y ϵ ¯
*) →
τhτ
*(0 ¯ ϵ
*) →
τ¯ hτ
*(0) →
τ¯τh0.
In the case of Park P
∞:
τ
*(λx.xx) →
τhτ
*(¯ ϵ
*ϵ ¯
*) →
τ¯hτ
*(¯ τ
*(τ
*(¯ ϵ
*))) →
τ¯τh→
τ¯τhϵ.
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•
(β) (λx.M) N → M[N/x]
(¯ τ) ( ∑︀
i
τ ¯
αi(Q
i)) N → ∑︀
i
∑︀
(β,γ)∈extD(αi)
τ ¯
γ(Q
i· τ
β(N))
(τ) τ
α(λx.M) → ∏︀
(β,γ)∈extD(α)
τ
γ(M[¯ ϵ
β/x]) (τ¯ τ) τ
α( ∑︀
i∈I
τ ¯
βi(Q
i)) → ∑︀
{I′⊆I|α≥⋀︀
i∈I′βi}
∏︀
i∈I′
Q
i (a) Main rules(· + ) Π
i≤nΣ
j≤kiQ
i,j→ Σ
j1≤k1,...,jn≤knΠ
i≤nQ
i,ji(¯ τ + ) τ ¯
α( Σ
iQ
i) → Σ
iτ ¯
α(Q
i)
(b) Distribution of the sum
M →
hM
′(h-cλ) λx.M →
hλx.M
′M →
hM
′M is an application
(h-c@) M N →
hM
′N
M →
hM
′M is an application (h-cτ) τ
α(M) →
hτ
α(M
′)
Q →
hQ
′Q is not a sum (h-c¯ τ) τ ¯
α(Q) →
hτ ¯
α(Q
′)
M →
hM
′(h-cs) M + N →
hM
′+ N
Q →
hQ
′(h-c + ) Q + P →
hQ
′+ P
Q →
hQ
′Q is not a sum (h-c·) Q·P →
hQ
′·P
(c) Contextual rules for the head reduction
M → M
′(cλ)
λx.M → λx.M
′M → M
′(c@L)
M N → M
′N
N → N
′(c@R) M N → M N
′M → M
′(cτ)
τ
α(M) → τ
α(M
′)
Q → Q
′(c¯ τ) τ ¯
α(Q) → τ ¯
α(Q
′) M → M
′(cs)
M + N → M
′+ N
Q → Q
′(c + ) Q + P → Q
′+ P
Q → Q
′Q·P → Q
′·P (c·)
(d) Contextual rules for the full reductionFig. 4. Operational semantics of the calculus withD-tests.
In rulesτ
¯
andτ, notice that we use the notationsτω(
M) :=ϵandϵ¯ω:=
0in order to keep the rule simpler.424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470
In the case of Norm:
τ
p(λx.x) →
τhτ
p(¯ ϵ
q) →
τ¯τhϵ, and τ
q(λx.x) →
τhτ
q(¯ ϵ
p) →
τ¯τh0.
In the case of Z
∞:
τ
n+2( ϵ ¯
nM) →
τ¯ hτ
n+2(¯ ϵ
n+1) →
τ¯τh0 , τ
n+2(¯ ϵ
nM N) →
¯τh2τ
n+2(¯ ϵ
n+2) →
τ¯τhϵ , τ
n+2(¯ ϵ
nM N L) →
τ¯ h3τ
n+2( ϵ ¯
n+3) →
τ¯τh0 R emark 3. In a polarized (or classical) framework with explicit co-terms (or stacks) as the framework presented in [19], tests would correspond to commands (or processes), or, more exactly, to conjunctions and disjunctions of commands. Indeed, a test τ
α(M) is nothing else than the command ⟨ M | π
α⟩ where π
αwould be the canonical co-term of interpretation ↑α, the same way that ϵ ¯
αis the canonical term of interpretation ↓α. Similarly, the term τ(Q) ¯ can be seen as the canonical term ϵ ¯
αendowed with a parallel composition referring to the set of commands Q. To resume, we have:
τ
α(M) ≃ ⟨M | ↑α⟩ ⟨ τ ¯
α(Q) | π⟩ ≃ ⟨↓α | π⟩·Q Definition 2.3. A test is in may-head-normal form i ff it has the following shape:
Π
iτ
αi(x
iM
i1· · · M
in) + Q, with i ≥ 0 and M
ikany term.
A term is in may-head-normal form if it has one of the following shapes:
λx
1....x
n.y M
1· · · M
m, or λx
1...x
n.¯ τ
α(Q) + N, where m, n ≥ 0, α ∈ (D − ω), M
iand N any terms, and Q any test in head-normal form without sums.
Coherently with the head convergence in λ-calculus, the convergence will be denoted by ⇓
hand the divergence by ⇑
h.
Example 2.4. For any n ∈ N, the term n (λx.¯ τ
α(τ
α(x) + τ
β(x))) ϵ ¯
αmay-head-converges.
Let us notice that this calculus enjoys the properties of confluence and standardization as shown in the Breuvart’s thesis [5, Thm. 2.3.1.26 and 2.3.1.29]. It also enjoys a very nice property stating that tests-reductions can always be postponed until the very end:
T heorem 2.5. Let D a DEFiM and M, N ∈ Λ
τ(D).
For any reduction M →
*N, there exists M
′, N
′∈ Λ
τ(D)such that M →
*βM
′with only β-reductions, M
′→
*∖βwith only tests reductions, and N →
*N.
M →
*N
β
→
*→
*M
′→
β*∖N
′In particular, M is may-head converging i ff there is a sequence of β-reductions M →
*βL with L that is may-head converging without any β-reduction.
Definition 2.6. Grammars of term-contexts Λ
(|·|τ(D))and test-contexts T
(|·|τ(D))are given in Figure 5.
Definition 2.7. The observational preorder ⊑
τ(D)of Λ
τ(D)is defined by:
M ⊑
τ(D)N iff ( ∀ K ∈T
(|·|τ(D)), K( | M | ) ⇓
himplies K( | N | ) ⇓
h).
We denote by ≡
τ(D)the observational equivalence, i.e., the equivalence induced by ⊑
τ(D).
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• (term-context) Λ
(|·|τ(D))C :: = x | ( |.| ) | C C
′| λx.C | ∑︀
i≤n
τ ¯
αi(K
i) , ∀(α
i)
i∈ D
n, n ≥ 0 (test-context) T
(|·|τ(D))K :: = ∑︀
i≤n
K
i| ∏︀
i≤n
K
i| τ
α(C) , ∀α ∈ D, n ≥ 0
Fig. 5. Grammar of the contexts in a calculus withD-tests
⟦ x
i⟧
⃗Dx= {(⃗α, β) | β ≥ α
i} ⟦ λy.M ⟧
⃗xD= {(⃗α, ⋀︀
i
(β
i→γ
i)) | ∀i, (⃗αβ
i, γ
i) ∈ ⟦ M ⟧
⃗xyD}
⟦ M N ⟧
⃗Dx= {(⃗α, ⋀︀
i
β
i) | ∃⃗γ
i, (⃗α, ⋀︀
i
(γ
i→β
i)) ∈ ⟦ M ⟧
⃗Dx∧ (⃗α, ⋀︀
i
γ
i) ∈ ⟦ N ⟧
⃗Dx}
(a) Interpretation ofΛ⟦Σ
i∈Jτ ¯
αi(Q
i) ⟧
⃗xD= {(⃗β, γ) | ∃I ⊆ J , ⃗β ∈ ⋂︀
i∈I
⟦ Q
i⟧
⃗xD∧ γ ≥
D⋀︀
i∈I
α
i}
⟦ 0 ⟧
⃗xD= { (⃗α, ω) } ⟦ τ
α(M) ⟧
⃗xD= {⃗β | (⃗β, α) ∈ ⟦ M ⟧
⃗xD}
⟦Π
i≤kQ
i⟧
⃗xD= ⋂︀
i≤k
⟦ Q
i⟧
⃗Dx⟦ ϵ ⟧
⃗xD= D
⃗x⟦Σ
i≤kQ
i⟧
⃗xD= ⋃︀
i≤k
⟦ Q
i⟧
⃗xD⟦ 0 ⟧
⃗xD= ∅
(b) Interpretation of tests extensionsFig. 6. Direct interpretation inD
x : α ⊢ x : α Γ ⊢ M : α
Γ , x : β ⊢ M : α Γ ⊢ M : β α ≥ β Γ ⊢ M : α Γ , x : α ⊢ M : β
Γ ⊢ λx.M : α→β Γ ⊢ M : α→β Γ ⊢ N : α
Γ ⊢ M N : β Γ ⊢ M : α Γ ⊢ M : β Γ ⊢ M : α ∧ β
(a) Intersection types for theλ-calculus inDΓ ⊢ M : α
Γ ⊢ τ
α(M)
Γ ⊢ Q
jΓ ⊢ ∑︀
i∈I
τ ¯
αi(Q
i) : α
jΓ ⊢ Q
jΓ ⊢ ∑︀
i∈I
Q
i∀i ∈ I, Γ ⊢ Q
iΓ ⊢ ∏︀
i∈I
Q
i (b) Intersection types for theD-tests extension inDFig. 7. Intersection type system computing the interpretation inD
R emark 4. The observational preorder could have been defined using term-contexts rather than test-contexts, but this appears to be equivalent and test-contexts are easier to manipulate (because normal forms for tests are simpler).
can be re- moved can be re- moved 2.2 Semantics
The standard interpretation of Λ into D can be extended to Λ
τ(D)(Fig. 6b).
Definition 2.8. A term M with n free variables is interpreted as a morphism (Scott-continuous function) from D
nto D and a test Q with n free variables as a morphism from D
nto the dualizing object {*} . Concretely, we use the Cartesian closeness to define ⟦ M ⟧
⃗xDas a downward-close sets of (D
op)
⃗x× D and ⟦ Q ⟧
⃗xDas a downward-close subsets of (D
op)
⃗x.
This interpretation is given in Figure 6 by structural induction.
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P roposition 2. Any DEFiM D is a model for its own test extension (the λ-calculus with D-tests), in the sens that the interpretation is contextual and invariant under reduction.
P roof . The invariance under β-reduction is obtained, as usual, by the Cartesian closedness of S cott L
!. The other
rules are easy to check directly.
The idea of intersection types can be generalized to to tests as shown in Figure 7b. Notice that tests have no type:
a test does not carry any behavior, and under a specific environment it can only be succeeding (and typable) or diverging (untypable).
T heorem 2.9 (I ntersection types ). Let D be a DEFiM and M a term of Λ
τ(D)(resp. Q a test of T
τ(D)), the following statements are equivalent:
∙ (⃗α, β) ∈ ⟦ M ⟧
⃗xD(resp. ⃗α ∈ ⟦ Q ⟧
⃗xD) in the interpretation of Figure 6,
∙ the type judgment ⃗ x : ⃗α ⊢ M : β (resp. ⃗ x : ⃗α ⊢ Q) is derivable by the rules of Figure 7.
P roof . By structural induction on the grammar of Λ
τ(D).
ref to a forth- coming article?
ref to a forth- coming article?
2.2.1 Full abstraction and sensibility for tests.
The main interest of the full abstraction with tests is to be fully abstract as soon as it is sensible (Theorem 2.13).
The sensibility is a very commune property saying that diverging terms are collapsed together and separated from non-diverging terms. In other worlds, such a model is able to give meaning to terminating terms and those only.
The full abstraction, however, is a much stronger property stating that the equality in the model corresponds exactly to the observational equality (for the head-convergence). Collapsing those two properties gives the real meaning of tests: they are syntactical representation of “reasonable” domains. Where “reasonable” means extensional and (as we will see later on) approximable domains.
Definition 2.10. A DEFiM D is sensible for Λ
τ(D)whenever diverging terms (resp. tests) correspond exactly to the terms (resp. tests) having empty interpretation, i.e., for all M ∈ Λ
τ(D)and Q ∈ T
τ(D):
M⇑
h⇔ ⟦ M ⟧
⃗xD= {(⃗α, ω) | ∀⃗α} Q⇑
h⇔ ⟦ Q ⟧
⃗xD= ∅ The following is an immediate lemma:
L emma 2.11. If D is sensible for Λ
τ(D)then:
(⃗αβ, γ) ∈ ⟦ M ⟧
⃗xy⇔ (⃗α, γ) ∈ ⟦ M[¯ ϵ
β/y] ⟧
⃗x, (⃗α, γ) ∈ ⟦ M ⟧
⃗x⇔ ⃗α ∈ ⟦ τ
γ(M) ⟧
⃗x. T heorem 2.12 (D efinability ). If D is sensible for Λ
τ(D)then:
(⃗α, β) ∈ ⟦ M ⟧
⃗x⇔ τ
β(M[(¯ ϵ
αi/x
i)
i≤n])⇓
h.
P roof . If (⃗α, β) ∈ ⟦ M ⟧
⃗xthen ⟦ τ
β(M[(¯ ϵ
αi/x
i)
i≤n]) ⟧ is not empty by Lemma 2.11, thus it converges by sensibility.
Conversely, if τ
β(M[(¯ ϵ
αi/x
i)
i≤n]) ⇓
h, since it has no variable, its interpretation is either empty or { () } , it has to be the
second by sensibility, which means (⃗α, β) ∈ ⟦ M ⟧
⃗x(by Lemma 2.11).
T heorem 2.13 ( full abstraction ). For any DEFiM D, if D is sensible for Λ
τ(D), then D is inequationally fully abstract for the observational preorder of Λ
τ(D):
⟦ M ⟧ ⊆ ⟦ N ⟧ ⇔ ∀ C ∈ T
(|·|τ(D)),C( | M | )⇓
h⇒ C( | N | )⇓
h.
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• P roof . Let ⟦ M ⟧ ⊆ ⟦ N ⟧ and C( |M| )⇓
h. Then by sensibility we have that ⟦ C( |M| ) ⟧ is non-empty. Moreover, by Proposition 2 we have that ⟦ C( |M| ) ⟧ ⊆ ⟦ C( |N| ) ⟧ . Thus ⟦ C( |N| ) ⟧ is non-empty and by sensibility, C( |N| )⇓
h.
change in the thesis!!!
change in the
thesis!!! Conversely, suppose that for all context C ∈ T
(τ(D)|·|), C( |M| )⇓
h⇒ C( |N| )⇓
hand let (⃗α, β) ∈ ⟦ M ⟧
⃗x:
Then by Theorem 2.12, τ
β(M[( ϵ ¯
αi/x
i)
i≤n])⇓
hwhere n is the length of ⃗α. Thus, after stating the context C = τ
β((λx
1...x
n.( |.| )) ϵ ¯
α1· · · ϵ ¯
αn), we have C( |M| ) →
nhτ
β(M[(¯ ϵ
αi/x
i)
i≤n])⇓
hwhich implies that C( |N| )⇓
h. However, there is no choice for the n first head reductions of C( |N| ), those are forced to be C( |N| ) →
nhτ
β(N[(¯ ϵ
αi/x
i)
i≤n]) so that this term is also head-converging. Then by applying the reverse implication of Theorem 2.12 we conclude (⃗α, β) ∈ ⟦ N ⟧
⃗x. 3 COLLAPSING SENSIBILITY AND APPROXIMABILITY FOR TESTS
Once we have said that sensibility and full abstraction are equivalent properties for test, it should not surprise the reader to learn that approximability is also equivalent to those properties. Indeed, approximability usually corresponds to the adequation of the B¨ohm-tree’s equality, which is a property between sensibility and full abstraction. However, the situation is a bit mere subtle: if the properties of sensibility and full abstraction for Λ
τ(D)strongly refer to tests mechanisms, the property of approximability is defined independently from tests. This really means that D-tests will behave well exactly whenever D is approximable.
First we extend the languages of approximants with tests (or rather the language of tests with approximants):
T heorem 3.1. Each of the above properties are still true when adding to the calculus with D-test the term ⊥ and the rules:
λx.⊥ → ⊥ ⊥ M → ⊥ τ
α(⊥) → 0.
P roof . The term ⊥ behave similarly to ¯ τ
α(0) (for any α). The only di ff erence is that the rule ¯ τ + is delayed. But this rule is cosmetic, one can work with or without it since it distributes with every other rules.
We can now use the approximants of Definition 1.9 together with tests:
L emma 3.2. For any DEFiM D, any sequence ⃗α ∈ D
⃗x, any β ∈ D −{ ω } and any M ∈ Λ (with free variables ⃗ x), the following are equivalent:
∙ the test τ
β(M[¯ ϵ
α/⃗ x]) is may-head converging without β-reduction,
∙ the test with approximants τ
β(a p(M)[¯ ϵ
α/⃗ x]) is may-head converging,
∙ (⃗α, β) ∈ ⟦ a p(M) ⟧
⃗x.
P roof . Considering that ⊥ is a notation for ¯ τ
α(0), the second and third points are equivanent by Theorem 2.12.
The equivalence between the two first points is obtained by induction on a p(M):
∙ Immediate when a p(M) = M = x
i.
∙ When a p(M) = λy.a p(N) for M = λy.N, we can use the induction hypothesis on N.
∙ When a p(M) = ⊥, this means that τ
β(M[ ϵ ¯
α/⃗ x]) →
*τ
′β((λy.M
′) M
1· · · M
n) cannot converges without performing a β-reduction.
∙ Otherwise, a p(M) = x
ia p(N
1) · · · a p(N
n) with M = x
iN
1· · · N
nthus the terms τ
β(M[¯ ϵ
α/⃗ x]) and τ
β(a p(M)[¯ ϵ
α/⃗ x]) can performe the same sequence of ¯ τ-reductions followed by a τ¯ τ-reduction which results in a sum and product combination of tests behaving the same way by induction hypothesis.
This clearly shows that taking the approximants is an operation that distribute with the semantics. This is su ffi cient to get the approximation theorem whenever the extension with tests is sensible.
T heorem 3.3. Any extensional filter model D, is approximable if and only if it is sensible for D-tests.
P roof . Both implications are considered separatelly.
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∙ If D is sensible for Λ
τ(D)then it is approximable:
Let ⃗α ∈ D
⃗x, β ∈ D−⊥ and M ∈ Λ .
– If (⃗α, β) ∈ ⟦ a p(N) ⟧
⃗Dxfor some M →
*N, then τ
α(N[¯ ϵ
α/⃗ x])⇓
hhby Lemma 3.2, thus τ
α(M[ ϵ ¯
α/⃗ x])⇓
hhand (⃗α, β) ∈ ⟦ a p(M) ⟧
⃗xD.
– If (⃗α, β) ∈ ⟦ a p(M) ⟧
⃗Dx, then τ
α(M[¯ ϵ
α/⃗ x]) ⇓
hh. By Theorem 2.5, M →
*βN with τ
α(N[ ϵ ¯
α/⃗ x]) that may-head converges without β-reduction. Thus, (⃗α, β) ∈ ⟦ a p(N) ⟧
⃗xby Lemma 3.2.
∙ If D is approximable then it is sensible for Λ
τ(D): Let ⃗α ∈ D
⃗x, β ∈ D−⊥ and M ∈ Λ .
– If τ
α(M[¯ ϵ
α/⃗ x])⇓
hh, then by Theorem 2.5, M →
*βN with τ
α(N[¯ ϵ
α/⃗ x]) that may-head converges without β-reduction. Thus, by Lemma 3.2, (⃗α, β) ∈ ⟦ a p(N) ⟧
⃗x, which is included in ⟦ M ⟧
⃗xby approximability.
– If (⃗α, β) ∈ ⟦ a p(M) ⟧
⃗xD, then there is M →
*N such that (⃗α, β) ∈ ⟦ a p(N) ⟧
⃗x. By Lemma 3.2 we have τ
β(M[ ϵ ¯
α/⃗ x]) that is may-head converging so that τ
α(M[ ϵ ¯
α/⃗ x])⇓
hh.
4 SUFFICIENT CONDITION FOR THE SENSIBILITY OF TESTS
So far we could not find a generic proof of the approximation property in the literature for standard filter model.
2Hence, we give a su ffi cient condition (Def 4.11) for a filter model D to be approximable (Th. 4.16). Indeed, we use this condition in Example 4.20 for stating the approximation property of the models of Example 1.4 (save for P
∞) and Example ??.
Here, we make a strong use of the equivalence between approximability and sensibility with tests (Th. 3.3) proven in the previous chapter. Indeed, if approximability is also proved using Tait reducibility methods [22], the process is not as well understood as in the proofs of sensibility. By directly relying on the connection with tests, we can get the more refined analysis of the theorem of approximation that we have ever find.
After our detailed analysis, we describe a su ffi cient, but not necessary, condition for the approximability.
Generalizing the study of sensible models carried out by Berline [3] and her students (Kerth [16] in particular). In fact, we include (by far) all filter models proven sensible in the literature!
4.1 Realizers
In this section, we are adapting Tait proof of reducibility to the filter models and the calculi with tests. The adaptation for filter models (or restrictions) have already been extensively studied; for example, see Berline’s survey [3]. However, the adaptation to tests is new and quite interesting. Indeed, we will see that the set of realizers we are looking into is much more coarser and refined, making the search more readable.
The first step is to defined what is a correct realizer:
3Definition 4.1. A saturated set is a set of term G ⊆ Λ
τ(D)that is close by backward reduction.
Given two saturated sets G, H, we denotes by G ↦→ H the saturated set of terms M such that (M N) ∈ H whenever N ∈ G.
2Save Chapter 17.3 of the book of Barendregt, Dekkers ans Statman [1] where this proof is done in parallel for several models of different classes.
3Notice that this notion of Realizers is not exactly what Tait call a realizer, but more like aD-indexed set of those. However, since we will look into a set of suchD-indexed sets of Realizers; we changed the level of abstraction...
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• Definition 4.2. A realizer of D in Λ is a function R from D to saturated subsets of Λ
τ(D)such that for all α, β ∈ D, we have
R(α ∧ β) = R(α) ∩ R(β) R(α→β) = R(α) ↦→ R(β).
Given any D-indexed sequence S of saturated sets, a realizer R of D in Λ is a S -realizer if for all α, R(α) ∈ R
α. This definition trivially is extended for a partial DEFiM J in place of D.
Intuitively, a S -realizer is a proof that a certain property represented by S is true for every typable terms. This
“certain property represented by S ” is basically the commune property of elements of S . In our case, we are looking for sensibility, this gives us following, the quite complex but also quite refined, set:
Definition 4.3. We denote, for all α ∈ D−ω:
∙ 𝒩
α+: = {M ∈ Λ
τ(D)| ∀β ≥
Dα, τ
β(M)⇓
h},
∙ 𝒩
α−: = {( ∑︀
i
ϵ ¯
βi+ L | ⋀︀
i
β
i≤
Dα, L ∈ Λ
τ(D)},
∙ S
αis the set of τ(D)-saturated subsets of 𝒩
α+that contains 𝒩
α−,
∙ S
ω= Λ
τ(D)∙ S
τ(D)D= (S
ατ(D))
α∈Dis the set of D-indexed collections of elements of S
ατ(D). The definition is extended for partial models.
L emma 4.4. Let R be a S
Dτ(D)-realizer in D.
if (⃗α, β) ∈ ⟦ M ⟧
⃗xand (∀i, L
i∈ R(α
i)) then M[ ⃗ L/⃗ x] ∈ R(β) if ⃗α ∈ ⟦ Q ⟧
⃗xand (∀i, L
i∈ R(α
i)) then Q[ ⃗ L/⃗ x] →
*ϵ P roof . By induction on M and Q.:
∙ M = x
i: then α
i≤
Dβ. Thus if L
i∈ R(α
i) ⊆ R(β), we have M[ ⃗ L/⃗ x] = L
i∈ R(β).
∙ M = N
1N
2: there exists (γ
j, β
j)
j≤nsuch that β = ⋀︀
j
β
j, (⃗α, ⋀︀
j
γ
j→β
j) ∈ ⟦ N
1⟧
⃗xand (⃗ a; ⋀︀
j
γ
j) ∈ ⟦ N
2⟧
⃗x. Thus, by induction hypothesis, if for all i, L
i∈ R(α
i), N
1[ ⃗ L/⃗ x] ∈ ( ⋂︀
j
(R(γ
j) ↦→ R(β
j))) and N
2[ ⃗ L/⃗ x] ∈ ⋂︀
j
R(γ
j).
We conclude by (N
1N
2)[ ⃗ L/⃗ x] ∈ ⋃︀
j
R(β
i).
∙ M = λy.N : then β = ⋀︀
j
γ
j→β
jand ((⃗α, ⋀︀
i
γ
i); ⋀︀
i
β
i) ∈ ⟦ N ⟧
⃗xy. We want to show that whenever ∀i ≤ |⃗ x|, L
i∈ R(α
i) and j ≤ n, we have λy.N[ ⃗ L/⃗ x] ∈ R(γ
j) ↦→ R(β
j). But if L ∈ R( ⋀︀
i
γ
i) for all i, the induction hypothesis give us that for any j, N[ ⃗ L/⃗ x][L/y] ∈ R(β
j).
∙ M = Σ
j∈Jτ ¯
γj(Q
j) : there is J
′⊆ J such that β ⪯ ⋀︀
j∈J′
γ
jand ⃗α ∈ ⋂︀
j∈J′
⟦ Q
j⟧
⃗x.
By induction hypothesis, when given L
i∈ R(α
i) for each i ≤ |⃗ x|, we get Q
j[ ⃗ L/⃗ x] →
*ϵ for any j ∈ J
′. Thus, for all j ∈ J
′, M[ ⃗ L/⃗ x] →
*M
′+ ϵ ¯
γj∈ 𝒩
γ−j
⊆ R(γ
j), so that M[ ⃗ L/⃗ x] ∈ R( ⋀︀
j∈J′
