The Naturality of Natural Deduction (II): On Atomic Polymorphism and Generalized Propositional Connectives

In a previous paper (of which this is a prosecution) we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models of polymorphic type theory. A different approach to extract proof-theoretic properties of natural deduction derivations was proposed in a recent series of papers on the basis of an embedding of intuitionistic propositional logic into a predicative fragment of System F, called atomic System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification in terms of parametricity.


Introduction
Russell was the first to observe that propositional connectives like disjunction and conjunction can be defined using only implication and propositional quantification.Later Prawitz, in his monograph on natural deduction, showed how the natural deduction system for intuitionistic propositional logic (henceforth NI) can be embedded into the implicational fragment of second-order propositional intuitionistic logic (also known as System F, and here referred to as NI 2 ).We will refer to this embedding as the Russell-Prawitz translation (shortly RP-translation).
Taking inspiration from this embedding, some recent work in the field of proof-theoretic semantics (see for instance [18] and [9]) has suggested the use of NI2 as a convenient meta-language to investigate the proof theory of propositional connectives.However, this way of looking at NI 2 faces two kinds of difficulties.The equivalence-preservation problem In proof-theoretic semantics, one is not only concerned with provability-i.e. with whether there is a derivation of a given formula in a certain system-but also with identity of proofsi.e. with whether two distinct derivations of the same formula can be viewed as different syntactic representations of the same proof (understood as an abstract object).
A common way to characterize identity of proofs is by declaring two derivations equivalent when they converge, under the usual conversions used for normalization, to the same normal derivation.Equivalent derivations are then taken to represent the same proof.This intuition can be made precise using the categorical semantics for natural deduction systems.For instance NI can be interpreted in any bi-cartesian closed category, with equivalent derivations being mapped onto the same morphism.
When not only provability but also identity of proofs is considered, the RP-translation appears as not entirely satisfactory.In fact, equivalent derivations in NI need not translate into equivalent derivations in NI 2 .Although the translation works for the equivalence induced by β-conversions only, it fails for the one induced by η-conversions and permutations, here referred to as γ-conversions.In categorical terms, the RP-translation of, say, a disjunction, is not interpreted as a co-product in every categorical model of NI 2 , but only as a "weak" variant of it.We will refer to this fact as the equivalence-preservation problem of the RP-translation.
In a previous paper, of which the present one is a follow-up, we explored a solution to the equivalence-preservation problem based on the fact that the RP-translation of conjunctions and disjunctions does yield categorical products/co-products in the class of parametric models of NI 2 [1,14,25].With the goal of making this result accessible to the proof theory community at large, in [33] we provided a purely syntactic reconstruction of it: we introduced an equational theory extending the one arising from the usual βand η-conversions for NI 2 -derivations using a new class of conversions-that we called ε-conversions-expressing a naturality condition for NI 2 -derivations which is satisfied by all parametric models of NI 2 .We then showed that the RP-translation does preserve the full equivalence of NI-derivations as soon as NI 2 -derivations are considered under this stronger equivalence. 1mpredicative versus predicative translations A second difficulty is of a foundational nature and stems from the fact that the RP-translation and, more generally, the second-order encoding of inductive types (e.g. the types of natural numbers and well-founded trees) inside NI2 are impredicative.In fact, the embedding of NI into NI 2 requires the full power of second-order quantification: in the elimination rule for the second-order quantifier ∀E

∀X.A ∀E A[[B/X]]
no restriction can be imposed on the choice of the formula B (called the witness of the rule application).
A solution to this problem can be found in a recent series of papers by Fernando Ferreira and Gilda Ferreira, who proposed a variant of the RPtranslation (to which we will refer to as FF-translation) which encodes NI in atomic System F (here referred to as NI 2 at ), a weak predicative fragment of NI 2 in which the witnesses of ∀E are required to be atomic formulas.A further refinement of the FF-translation was later proposed by José Espírito Santo and Gilda Ferreira in [5] (we will refer to it as the ESF-translation).
Besides being predicative, the FF-and ESF-translations have another significant advantage over the RP-translation: they do preserve the equivalence arising not only from β-conversions, but also from ηand γ-conversions for disjunction and ⊥ [5,8,11]. 2For these reasons the predicative translations were advocated in [9] as evidence in favor of taking NI 2 at as a convenient meta-language to investigate propositional connectives.

From impredicative to atomic polymorphism through ε-conversions
In this paper we exploit the equational framework we developed in the first paper of this series to investigate the predicative translations of intuitionistic propositional logic into NI 2 at .In particular, we show that the usual, impredicative, translation can be related to the predicative ones using the ε-conversion, thus providing a semantic explanation ultimately relying on parametricity for some syntactic results based on atomic polymorphism.
Our first observation is that the results of Ferreira and co-authors do not hold only for ∨ and ⊥, but for the class of connectives that are definable in NI by arbitrarily composing ∧, ∨, and ⊥ (to describe such connectives we borrow another concept from the toolbox of category theory, that of a finite polynomial functor [13]).
By extending the RP-translation to a natural deduction system for this class of propositional connectives (called NI p ), we are led to consider another fragment of NI 2 , that we call the Russell-Prawitz fragment (NI 2 RP ).Unlike the atomic fragment NI 2  at , the fragment NI 2 RP is impredicative, since no restriction is imposed on the witnesses of the applications of ∀E.
Nonetheless, we show that every derivation in NI 2  RP can be "atomized", i.e. it can be mapped onto a derivation in NI 2  at with the same conclusion and the same assumptions, by applying instances of η-expansion and the εconversion.By composing the RP-translation from NI p to NI 2  RP with the atomization from NI 2  RP to NI 2 at one thereby obtains another predicative translation from NI p to NI 2  at (we call it the ε-translation) which only differs from the FF-and ESF-translation by some β-reduction steps.
An immediate consequence of this fact is that the RP-translation and its three predicative variants are all equivalent modulo β-, ηand ε-conversions.Thus, under the notion of identity of proofs induced by ε-conversions, all translations of a given propositional derivation are different syntactic descriptions of the same second-order proof (that is, all these translations interpret an NI p -derivation as the same morphism in all parametric models of NI 2 ).

Limitations of the predicative translations
The comparison of the impredicative and predicative translations leads us to highlight two limitations of the approach based on atomic polymorphism: first, the predicative translations do not preserve the full η-rule needed to interpret disjunction as a categorical co-product, and thus fail to provide a full solution to the equivalence-preservation problem.Moreover, we show that once ∀E is restricted to atomic witnesses it is not possible to prove the logical equivalence between a propositional formula and its second-order translation (with the terminology of [22], this means that connectives are not strongly definable, but only weakly definable in NI 2 at ).

Goals and plan of the paper
One of the motivations for the previous and present papers is that of making some ideas underlying the categorical semantics of System F accessible to the proof-theoretic community at large, and to show that these ideas can be fruitfully connected with strands of research arisen within more philosophically-oriented areas of proof theory.This was the reason for reformulating in the first paper categorical notions such as functors and natural transformations in the language of natural deduction, at the expenses of typographic conciseness.
To keep the presentation compact and readable for the largest audience, we chose to present the main results of the paper using the natural deduction notation and restricting the attention only to the case of a particular ternary connective •(A, B, C).Full proofs for the whole class of connectives we consider are postponed to a (large) technical appendix written using the drastically more economical λ-calculus notation.
In Section 2, we introduce the natural deduction calculus NI p for the class of propositional connectives we intend to investigate and a fragment of NI 2 , that we call the Russell-Prawitz fragment (noted NI 2 RP ).Both NI p and NI 2  RP are inspired by the notion of finite polynomial functor from category theory, and we introduce a generalization of the usual RP-translation as a derivability-preserving embedding between these two systems.In Section 3 we recall the framework introduced in our previous paper to describe functors and natural transformations within natural deduction, based on the ε-conversion.In Section 4 we generalize the FF-and the ESF-translations to NI p and we investigate their relationship to the RP-translation.To do this, we first show how the FF-translation can be analyzed as the composition of the RP-translation and of an embedding from the fragment NI 2 RP into NI 2 at that we call FF-atomization, and then defining an alternative embedding from NI 2 RP into NI 2 at using the ε-conversions, the ε-atomization.In Section 5 we discuss some limitations of the predicative translations for the study of identity of proofs of propositional connectives .In Section 6 we briefly summarize the results of the paper, we draw some connections with related work, and we suggest further directions of investigation.Finally, the rich appendix provides full proofs (in λ-calculus notation) of the results discussed or simply sketched in the main text.

Polynomial Connectives and Their RP-Translation
In this section we introduce a formal framework for natural deduction which extends the one from [33] to a more general class of propositional connectives.

Polynomial Connectives
As suggested in the previous paper (cf.[33] Section 4.3), the results we are concerned with are not limited to the standard intuitionistic connectives, but scale smoothly to a wider class of connectives investigated in proof-theoretic semantics (see e.g.[24,26]).These are the connectives that are obtained by freely composing ∨, ∧, and ⊥, such as the ternary connective whose introduction and elimination rules are as follows: In general, each such connective is definable in NI as n p=1 m p q=1 A pq for some choice of natural numbers n, m p s and formulas A pq s.We take here the viewpoint of treating such connectives as logical primitives.
Borrowing ideas from the theory of finite polynomial functors [13], each such connective can be described as determined by three finite lists I, J , K (to be thought of as lists of indices) and two functions f : J → I and g : J → K that we depict in a diagram as follows:3 Any such diagram determines what we will call a polynomial connective to be indicated with † (f,g) , or simply † when f, g are clear from the context, in the following way: • The length |I| of I measures the arity of †, so that when † is applied to an I-indexed list of formulas A i i∈I 4 one obtains a new formula † A i i∈I .
• The length |K| of K is the number of distinct introduction rules of †.
• Any element k in K determines a sublist of J , namely the list of all j ∈ J such that g(j) = k, that we indicate with g −1 (k) and whose length |g −1 (k)| is the number of premises of the k-th introduction rule of †. 5Using the functions f and g we can describe the introduction and elimination rules for † as follows.Given an I-indexed list of formulas A i i∈I , the k-th introduction rule †I k for †, allows us to infer † A i i∈I from the list of premises A f (j) j∈g −1 (k) .Given † A i i∈I and a K-indexed list of derivations of an arbitrary formula C from (respectively) the premises of the k-th introduction rule A f (j) j∈g −1 (k) , we can infer C thereby discharging in the k-th derivation of C the assumptions A f (j) j∈g −1 (k) .We depict the rules as follows:

C
k∈K †E C Remark 2.1.Let 0 be the empty list, and 1, 2, 3, 4, 5 be the lists 1 , 1, 2 , 1, 2, 3 , 1, 2, 3, 4 , 1, 2, 3, 4, 5 respectively.The connective given by 3 where I A indicates the identity function on the list A and g : {1, 2 → 1; 3 → 2}.Similarly, the usual connectives ∨, ∧, , ⊥ are obtained through the configurations 2 respectively (where ∅ indicates the empty function and 1 the constant function with value 1.).We observe that some arguments of a connective may play a "dummy" role or may be used more than once, such as in the connective (A where f : {1 → 2; 2, 4 → 3; 3 → 4} and g : {1, 2 → 1; 3, 4 → 2}.Observe that, informally speaking, for every connective † determined by the , the elements of J code the positions of the arguments of † in the formula of the language of NI that define †.Remark 2.2.Treating ∨ and ⊥ as polynomial connectives (see Remark 2.1) yields their usual introduction and elimination rules.This is not the case for ∧ (and ).When treated as a polynomial connective, conjunction has a unique elimination rule ∧E p instead of the usual ∧E 1 and ∧E 2 (see also Remark 2.4 below): 3. We will adopt the convention of using i for indices in I, k for indices in K and j for indices in g −1 (k).Hence, to enhance readability, we will often omit the indication of the index set, so that for instance A i i∈I will be abbreviated as A i .
Given a set of propositional variables V, indicated as X, Y , Z, . . ., the formulas of the language L 2p will be constructed using implication, the universal quantifier and the family of polynomial connectives † (f,g) .Besides L 2p , we will mainly be concerned with two restrictions thereof, L p and L 2 : We moreover indicate with L ∨ (respectively L • ) the restriction of L p in which ∨ (resp.•) is the only connective besides ⊃, and similarly for L 2∨ and L 2• .The natural deduction system NI 2p over the language L 2p is obtained by adding to the rules of the standard system NI 2 for L 2 (recalled in Definition 1.1 and Table 1 on pages 197-198 of [33]), all introduction and elimination rules for the connectives † (f,g) .The system NI 2p and its equational theory are described in detail in λ-notation in Appendices A and D respectively.Finally, by restricting NI 2p to the languages L p , L where ∀ Y i (for 1 i n+1) indicates a (possibly empty) list of consecutive quantifications.Moreover, we let at(A) = X indicate the rightmost atom of A. Finally, we will often abbreviate we assume right-associativity for implication).

Polynomial Formulas and the RP-Fragment of NI 2
The RP-translation of standard connectives ∧, ∨, ⊥, is recalled in Table 1.
Although the connective • can be translated by composing the translations of ∧ and ∨, a natural and more economical way to encode • in NI 2 is given The universal formula above shares a common structure with those in Table 1: all such formulas are of the form ∀X.A 1 ⊃ • • • ⊃ A n ⊃ X, where each A i has a unique occurrence of X in rightmost position.This suggests the following definition: Remark 2.6.We introduce the following compact notation for NI 2 -formulas.Given a finite list A = a 1 , . . ., a k , an A-indexed list of formulas A a 1 , . . ., A a k and a formula B, we let When A is clear from the context the index a ∈ A will be omitted so we simply write A a ⊃ B.
A universal polynomial formula A can thus be written as ∀X.B b b∈B ⊃ X, where B is some list and the B b s are sp-X.Moreover, any sp-X formula B can in turn be written as A a a∈A ⊃ X, for some list A such that each A a has no occurrence of X.Hence, a universal polynomial formula can be written as ∀X.A a a∈A b ⊃ X b∈B ⊃ X for some list B and family of lists A b indexed by B. Since a diagram I f ← J g → K describes a family of lists indexed by the elements of a list (see footnote 5 above), it can be used to associate to each I-indexed family of formulas A i i∈I the universal polynomial formula ∀X.A * f (j) j∈g −1 (k) ⊃ X k∈K ⊃ X, which in turn we propose to take as the RP-translation of the propositional formula † f,g A i i∈I .
The Russell-Prawitz translation can thus be generalized to the whole of L 2p by translating the formulas of L 2p whose outermost connective is polynomial with a universal polynomial formula of L 2 : Definition 2.7.(RP-Translation of Formulas) We define a translation * from formulas of L 2p to formulas of L 2 as follows: for any diagram For readability, we will abbreviate (A ⊃ X) ⊃ ((B ⊃ X) ⊃ X) as A B (and thus (A ∨ B) * as ∀X.(A * B * )), and similarly The RP-translation scales well from formulas to derivations, yielding an embedding * : (For a proof for the whole NI 2p , see Appendix C).If we restrict the language of NI 2 by requiring every universal formula to be universal polynomial, then we obtain a fragment of NI 2 that we will call the Russell-Prawitz fragment.Remark 2.11.The RP-fragment of L 2 can be equivalently defined inductively as follows: Observe that not only the translation of a formula † f,g A i is a universal polynomial formula, but for every universal polynomial formula A there is at least (but in general more than) one configuration . ., n , f to be the identity function and g = π 2  1 : i, j → i.Given this it is easily seen by induction that for any A ∈ L 2  RP there is at least (but in general more than) one B ∈ L p such that B * = A. Note however that not every NI 2 RP -derivation is the image of some NI p -derivation under the RP-translation.
Remark 2.15.In [33] we alluded to the fact that the translation of polynomial connectives can be described through a class of formulas called nested sp-X.There we used "sp-X" as short for "strictly positive" rather than "strongly positive", where strictly positive formulas are a slight generalization of strongly positive formulas in which X may not occur at all, and nested sp-X formulas roughly stand to universal polynomial as strictly positive formulas stand to strongly positive formulas.
The class of polynomial formulas we consider here is thus a proper subset of the class of nested sp-X formulas and it is already sufficient for the goal of RP-translating polynomial connectives.However, most of the results we prove for NI 2  RP can be extended to a similar (and slightly larger) system defined using nested sp-X (more generally these systems are fragments of a more general system Λ2 κ≤2 studied in [20]).

The ε-Conversions
In this section we introduce some notational conventions and shortly recall some notions and results from the previous paper.These are based on the introduction of a notation for functors in natural deduction and of a class of conversions, called the ε-conversions, that express a naturality condition for natural deduction derivations.

Weak Expansion
Definition 3.1.Given a formula A ∈ L 2 (whose structure can be described as in ( * ), see end of Section 2.1 above), the weak expansion of A is the derivation that one obtains by repeatedly applying η-expansions to the derivation consisting only of the assumption of A until the minimal formula of the main branch is atomic, i.e.: indicate (possibly empty) sequences of applications of ∀E and ∀I.

We will indicate by El m
A the "first half" of the derivation above, consisting of a chain of elimination rules depending on indices m = m 1 , . . ., m n for the undischarged assumptions

Similarly, we will indicate by
In( m) the "second half" of the derivation above, consisting of a chain of introduction rules discharging the assumptions with indices m (a more rigorous description of this construction is provided in Appendix E through the notion of expansion pair ).
Remark 3.2.The weak expansion of A might differ from the expanded normal form (also known as η-long normal form) of the derivation consisting only of the assumption of A, since in the latter not only the minimal formula in the main branch, but the minimal formulas of all branches are atomic, see [23,§II.3.2.2.].

C-Expansion
Δ defined by induction on n as follows: and D is sp-X.We define: Remark 3.4.In ( [33], sec.3.2) we defined the notion of C-expansion for a broader class of formulas called pn-X formulas.The restriction of the definition to the class of sp-X formulas allows a straightforward reformulation of the definition using the notion of weak expansion that we give in Appendix E.

The ε-Conversions
As recalled in the introduction, it is common to characterize identity of proofs using an equivalence induced by (the symmetric closure of) some reduction relation over derivations.The equivalence βη generated by the βand η-conversions for NI 2 is standard (and recalled in Appendix D).Propositional connectives like ∨ require, in addition to βand η-conversion rules, the so-called permutative conversions (which allow to permute an elimination rule upwards a ∨-elimination rule), that we call here γ-conversions, generating an equivalence relation βηγ .Analogous conversions can be defined for all polynomial connectives, see Appendix D.
Remark 3.8.The equivalence relation βηγ can be further extended in two equivalent ways: either by replacing γ by a stronger permutation γ + which allows the permutation of an arbitrary derivation upwards across an application of †E, or by replacing η by a stronger η + which expresses in categorical terms the universality of the connective (η + in fact subsumes both γ and γ + ), see Appendix D.
The equivalence-preservation problem of the RP-translation can be formulated as the failure of the implication below (see [33]): In [33] (see Section 4.1) we showed that the implication above does hold when the equivalence we consider for NI 2 is the one induced by adding to βand η-conversions a new class of conversions, called ε-conversions (for the case of • these are shown in Table 2, for arbitrary universal polynomial formulas see Appendix F).
Semantically, the ε-conversions express a naturality condition for NI 2derivations.Thus, the results of our previous paper were a syntactic reformulation of some well-known properties which hold in parametric models of NI 2 (see [1] and [14]).2.

Predicative Translations via Atomization
In this section we show that the RP-translation can be related to the predicative translations by embedding the NI 2 RP fragment of NI 2 into the atomic fragment NI 2 at .

The FF-and ESF-Translations
As we recalled in the introduction, Ferreira and Ferreira [7][8][9]11], proposed an alternative translation of NI-derivations into NI 2 that we call the FFtranslation.The FF-translation agrees with the RP-translation on how to translate formulas, but not on how to translate derivations.In particular, the FF-translation does not use the full power of the ∀E rule of NI 2 , but rather it maps derivations in NI into derivations in NI 2 at .The FF-translation exploits the property of instantiation overflow :

∀X.A NI 2 at A[[B/X]]
While Ferreira and Ferreira only address standard intuitionistic connectives, it is easily seen that the instantiation overflow property holds for all universal polynomial formulas ∀X.A ∈ L 2 RP (and actually for many more, see [19]).The results of Ferreira and Ferreira thus scale to the whole system NI p .For simplicity we will focus here on the connective • and on the fragments NI • and NI 2 • of NI p and NI 2 RP respectively.However, all definitions and results here presented scale to all polynomial connectives, as shown in appendix.
By reformulating Ferreira and Ferreira's insight, we define an embedding of NI 2 • into NI 2 at that we call FF-atomization, and using it we define the FF-translation from NI • into NI 2 at as the composition of the RP-translation and FF-atomization.• -derivation, the FF-atomization of D, which we indicate as D ↓ , is the NI 2 at -derivation defined by induction on D as follows.We only consider the case in which the last rule D is ∀E with a non-atomic witness, since all other rules are translated in a trivial way.In this case observe that We define D ↓ by a sub-induction on F .
These two translation not only allow one to translate NI • into a very weak predicative fragment of NI 2 , but they also preserve slightly more equations than the RP-translation.In particular, the equivalences induced by γand η-conversion are preserved without appealing to additional conversions for NI 2 beside βand η-conversions.In fact, both translations map not only β-, but also η-reduction steps in NI onto chains of βη-reduction steps in NI 2 and γ-reduction steps in NI onto chains of βη-reduction and expansion steps in More precisely: Proof.For the FF-translation * ↓, point 1 and 3 of the proposition were proven for the whole of NI by Ferreira and Ferreira [9] and [8] respectively, and point 2 by Ferreira [11] (although in this case by adding a primitive conjunction to NI 2 , see below Section 5.1), and we claim that, at the cost of tedious computations, those proofs can be scaled to the whole of NI p .In Appendix I we give a proof of the proposition for the ESF-translation scaled to the whole of NI p , that generalizes the analogous result for NI of Espírito Santo and Ferreira in [5].

The ε-Translation
We now show that the ε-rule can be used to clarify the relationship between the RP-translation and the FF-and ESF-translations.
where F is not atomic and Z is the variable in rightmost position in F , using the notation introduced in Sections 3.1-3.2(see in particular Remark 3.7), we define : The name "ε-atomization" is justified by the fact that, as can be easily verified in the case of •, the ε-atomization D ↓ ε of an NI 2 • -derivation D is the result of applying η-expansions and ε-conversions to D: Proof.See Appendix J.
The relationship between the three predicative translations is very close.In fact, they yield β-equivalent derivations, with the ε-translation lying between the FF-translation and the ESF-translation, in the sense below: Proof.See Appendix K.
By putting together Proposition 2 and 4 we can deduce that also the ε-translation preserves the equivalence induced by permutative conversions and η-conversions: 2 .Remark 4.7.The statement of Corollary 4.6 cannot be expressed in terms of reduction, but only in terms of equivalence, due to the fact that neither The relationship between the four different translations is illustrated in Table 3.
Summing up, all three predicative translations are equivalent, modulo βηε-equivalence to the RP-translation and thus, semantically, they all interpret an NI • -derivation using the same second-order proof (provided one understands second-order proofs as satisfying the naturality conditions expressed by ε-conversions).
Moreover, our results offer a new insight for the fact that the predicative translations do preserve the ηand γ-equivalences of NI • resorting only to the βη-equivalences of NI 2 : the predicative translations encode those bits of ε-equivalences (or, semantically speaking, of naturality) needed to translate the ηand γ-equivalences of NI • into NI 2 .

Limitations of the Predicative Translations
By comparing Propositions 1 and 2 of the previous sections, it might be tempting to say that the approach based on atomic polymorphism provides Table 3. Relationship among the RP-, FF-, ESF-and ε-translations a fully syntactic alternative to categorical semantics and related techniques for the study of identity of proofs for propositional connectives.
In this section we argue that this is not entirely the case, by stressing two important limitations of the predicative translations.First, they do not preserve the whole equational theory of propositional connectives, and in particular the predicative translations of ∧, ∨ do not yield products and co-products in NI 2 at .Hence, strictu sensu the predicative translations do not fully solve the equivalence-preservation problem.Second, the ∀E rule restricted to atomic witnesses is too weak to prove the logical equivalence between a connective and its second-order translation.In Prawitz' terminology ( [22]), we show that the connective ∨ is not strongly definable in NI 2 at but only weakly definable.

Predicative Translations and Generalized Permutations
As is well known, in order to obtain a perfect match between the syntax of NI and the free bi-cartesian closed category, it is necessary to consider the following stronger permutative conversions in which every chunk of derivation (and not just those consisting of applications of elimination rules) can be permuted-up across an application of disjunction elimination:6,7 Point 3 of Proposition 1 can be strengthened by replacing γ with γ + , so that for any instance of γ + , the RP-translation of its left-hand side βεreduces to the RP-translation of the right-hand side (see Appendix G).
However, Point 3 of Proposition 2 ceases to hold as soon as one replaces γ with γ + .That is, although the predicative translations preserve the equivalences induced by ηand γ-conversions, they do not preserve all equivalences needed to interpret NI as a bi-cartesian closed category.
To see this, it is enough to consider the instance of γ + in which A, B, C and D are atoms, D (respectively D 1 , D 2 ) consists only of the assumptions of A∨B (resp.C), and . As the reader can easily check, the predicative translations of the left and right-hand side of this instance of γ + are not βη-equivalent, just like their RP-translations.
That the predicative translations fail to preserve the equivalence induced by the stronger permutations provides an explanation for another puzzling aspect of the approach of Ferreira and co-authors.As observed in [11], Point 2 of Proposition 2 fails if NI ∨ is extended to include the standard rules of conjunction.In particular, the predicative translations fail to preserve all instances of the η-equation for the standard conjunction rules.Nonetheless, when the two standard elimination rules ∧E 1 and ∧E 2 are replaced by the general elimination rule ∧E p as in NI p (see above Remarks 2.2 and 2.4), Proposition 2 applies to the equivalence relation induced by the β-, ηand γconversions for "generalized" conjunction (see Appendix D for a formulation of these conversions in λ-calculus notation and Appendix I for a proof of Proposition 2 scaled to NI p ).
The standard elimination rules can be defined using the general elimination rule as follows: and given these definitions, the η-equations for the standard conjunction rules become the following: The different behavior of the standard and the general elimination rules is explained by the fact that, as in the case of disjunction, the predicative translations of two γ + -equivalent NI p -derivations need not be βη-equivalent NI 2 at -derivations, and that the γ + -conversion for conjunction: is essential to define the η-conversion for the standard elimination rules using the η-conversion for the general elimination rule (see Remark D.3 in Appendix D for a proof).

at
A second disadvantage of the predicative translations arises when one considers the extension NI 2∨ of NI 2 with a primitive disjunction (for an early semantic investigation of this system see [12,30]), in which the following holds: Proposition 5.For all A ∈ L 2∨ : That is, as soon as one extends NI 2 with a primitive disjunction , one can show in the extended system NI 2∨ that every formula is interderivable with its RP-translation. 9 In contrast to what happens in NI 2∨ , a propositional formula A and its RP-translation A * may fail to be interderivable in the extension of NI 2 at 8 The same remarks of footnote 7 apply. 9Actually, A and its RP-translation are not just interderivable, but can be shown to be isomorphic modulo βηγ + ε (see for details [20]).
with a primitive disjunction.By inspecting one direction of the proof of Proposition 5 it is clear that A NI 2∨ at A * .However, the inspection of the other direction clearly suggests that, at least in some cases, in order to establish that A * NI 2∨ at A it is essential to apply ∀E with a non-atomic witness.In particular, Proposition 6.For any two distinct propositional variables Y, Z, (Y ∨ Z) at (see Proposition 22 in Appendix L).Notice that A * = A and (∃X.A ⊃ X) * = ∃X.A ⊃ X.On the other hand, one can show that NI 2 at (A ∨ (∃X.A ⊃ X)) * as follows:

Another way of highlighting the difference between (A∨B) * and (A∨B) in NI 2∨
at is by observing that, whereas the disjunction elimination rule warrants that at iff for all A, B ∈ L ∨ there is an 10 Then, not only Proposition 6 shows that disjunction is not strongly defined by its RP-translation, but there is in fact no other formula strongly defining it in NI 2 at : Proposition 8. ∨ is not strongly definable in NI 2 at .
Proof.See Appendix L

Summary of the Results
In this paper we have shown how the category-theory-inspired ε-conversions introduced in our previous paper can be used to clarify the relationship between the alternative translations proposed by Ferreira, Ferreira and Espírito Santo and the original Russell-Prawitz translation, and to provide semantic insights on the proof-theoretic properties of the former translations.
Our approach consisted in focusing on an atomizing translation from a suitable fragment of NI 2 (the fragment that we called NI 2  RP , in which universal formulas correspond to the translation of connectives) into NI 2 at , that we defined using the ε-conversions.This made it possible to show that the predicative translations produce derivations that are equivalent to the RP-translation modulo the εconversions (hence, in semantic terms, modulo parametricity), and that their proof-theoretic properties (the preservation of the equivalence induced by ηand permutative conversions) result from the fact that ε-conversions express a naturality condition for the proofs denoted by NI 2 -derivations.
We thus used the connections of ε-conversions with category theory to provide an explanation of how and why the predicative translations work, by embedding the syntactic results about them into a wider picture.
Moreover, we showed that the predicative translations do not offer a full solution to the equivalence preservation problem, due to the fact that the predicative translations of the right-hand and left-hand side of generalized permutations in NI are NI 2  at -derivations which in general may fail to be βηequivalent.This suggests that the system NI 2 at coupled with βη-equivalence is not the best setting to study the notion of identity of proof resulting from the full equational theory on NI-derivations.We provided further hints for this conclusion by showing that disjunction is only weakly and not strongly definable in NI 2 at .

Related and Further Work
The βηγ + -equational theory has been recently shown to be the maximum consistent equational theory in NI [28].As shown by the first and second author in [20], the βηε-equational theory in the fragment NI 2 RP is equivalent (in the sense of category theory) to the βηγ + -equational theory of NI.From this they inferred both the maximality of the βηε-equational theory in NI at , see [21] for an overview of these results).Although a generalization of our ε-conversions (expressing a dinaturality condition, rather than just naturality) can be formulated for any formula ∀X.A of L 2 , the tight connection between the ε-conversions and the phenomenon of instantiation overflow underlying the present paper seems to be limited to the class of universal polynomial formulas.In fact, in [33] the ε-conversions were defined for a broader class of formulas, that of nested p-X formulas, all enjoying the instantiation overflow property.However, it does not seem possible to define a uniform atomization procedure the style of our ε-atomization for the NI 2 -derivations in which the premises of all applications of ∀E are nested p-X (rather than universal polynomial) formulas.For a characterization of the class of L 2 -formulas enjoying instantiation overflow, see [3] and [19].
As to further directions of investigations, we observe that, in the extension of System F with primitive propositional connectives, not only every proposition and its Russell-Prawitz translation are interderivable, but they are actually isomorphic (modulo the equivalence relation induced by β-, η-, γ + -, and ε-equations).From a categorical perspective, such isomorphisms belong to a more general class of isomorphisms induced by a proof-theoretic formulation of the Yoneda lemma.This and further categorical aspects underlying the present work have been investigated by the authors in [20].
As observed, the ε-conversions allow one to translate not only the γbut also γ + -permutations.As a consequence, the study of the reduction (as opposed to equivalence) induced by ε-conversions can be expected to be rather involved.In particular, the disadvantages of the fully extensional reduction theory for disjunction, such as non-confluence and non-termination (see e.g.[17] for a discussion) would be imported inside NI 2 .
As in the case of disjunction, these problems do not exclude the possibility of considering restricted forms of ε-conversion (essentially, those in which only elimination rules are permuted across an application of ∀E) corresponding to the restricted permutations of NI, and it is not implausible to thereby obtain a well-behaving reduction theory.
In a recent paper [6], Espírito Santo and Ferreira developed an analogous approach by devising further conversions in NI 2 corresponding to those steps of ε-conversions and of η-expansions needed to atomize applications of ∀E in the RP-translation of NI-derivations.Using the reduction relation induced by the β-, ηand the newly introduced conversions in place of ε-conversions they could obtain a result analogous to our Proposition 1 but establishing the preservation of reduction rather than just equivalence also in the case of η-conversions.
Acknowledgements.Many thanks to the referees for the many suggestions that substantially contributed to improve the paper.Many thanks also to Gilda Ferreira, Fernando Ferreira and José Espírito Santo for several discussions on atomic polymorphism.
Funding Open Access funding enabled and organized by Projekt DEAL.Mattia Petrolo gratefully acknowledges the support of the São Paulo if t a is an A-indexed list of terms, for any u, we let u t a a∈A (often abbreviated as u t a ) indicate the term ut a 1 . . .t a k .
As usual, by a typing context we indicate a finite set of type declarations x : A where all declared variables are distinct.We indicate typing contexts with Γ, Δ, Σ.
The typing rules for NI 2p are the following: We write Γ NI 2p t : A iff there is a derivation of Γ t : A using the above rules.Similarly for Γ NI 2 t : A and all other systems introduced in Sections 2.1 and 2.2.

B. Contexts
Let A be a finite (possibly empty) list.By an A-multicontext T (henceforth simply referred to as context, or term context to avoid confusion with typing contexts) we indicate a term containing exactly one occurrence of |A| distinct special variables noted as [ ] a (and referred to as "holes"), for each a ∈ A.
Given an A-multicontext T, we let T[t] (resp.T[U]) indicate usual-i.e.non variable-capturing-substitution (short n.v.c.-substitution) of the term t (resp.context U) for each [ ] a in T, and T{t} (respectively T{U}) indicate variable-capturing substitution (short v.c.-substitution) of the term t (resp.context U) for [ ] a in T.
We will use the following special families of contexts: • the principal contexts (indicated as C, D, . . . ) are defined by the grammar below: • the introduction contexts are defined by dropping the cases Cu, CB and δ † (E y j .sk ) from the grammar above; • the elimination contexts are defined by dropping the cases λx.C, ΛY.C and | from the grammar above; • the atomic elimination contexts are defined by restricting B to be a type variable in the grammar for elimination contexts.
Observe that all principal contexts are contexts in standard sense (i.e. they all contain exactly one occurrence of the hole).Note that if C, D are principal contexts, then C{D} and C[D] are principal contexts as well.
It is easily checked that if E is an elimination context, then for all terms t, E{t} cannot capture variables of t, whence E{t} = E[t].
We let T : A Γ B be a shorthand for Γ, x : A T[x] : B (where x is a fresh term variable).Note that if T : A Γ B and Γ t : A, Γ T[t] : B holds, but Γ T{t} : B in general does not hold.At crucial points below we will however rely on an analogous (but weaker) property for v.c.-substitution (namely Fact B.2 below), which requires a more fine-grained control over v.c.-substitution in order to be formulated and proven.The remaining part of this appendix is devoted to making this formally precise.
Given an A-multicontext T and an A-indexed finite list of terms u a a∈A , we let T{ u a a∈A } be the term obtained by simultaneous variable-capturing substitution of [ ] a by u a .(Note that T{t} is just an abbreviation of T{ u a a∈A } when u a = t for all a ∈ A.) We now introduce a refined typing for contexts, where a refined typing T : A Γ Δ a a∈A B indicates, informally speaking, that in T{ u a a∈A } the variables declared in Δ a are bound in the term u a inserted in the a-indexed hole in T. Refined typing judgments are defined inductively by appropriate rules.Being rather obvious, we omit some of them.In the refined typing rules, when A and B are lists we indicate with A * B the concatenation of A and B.
Remark B.1.The refined typing judgements could be generalized to the following: T : A a a∈A Γ Δ a a∈A B, i.e. the holes could have different types, and an opportune generalization of Fact B.2 below would hold for this generalized refined typing.As we make no use of the more general case, to avoid additional notational overhead we restrict our presentation to the relevant special case.
Observe that if T : A Γ Δ a a∈A B holds, then Γ ∪ Δ a is a well-defined context, for any a ∈ A.
The following fact can be easily established by induction on derivation of T : A Γ Δ a a∈A B (for some explanatory remarks, see below): Fact B.2.If T : A Γ Δ a a∈A B and u a a∈A is an A-indexed family of terms such that Γ ∪ Δ a u a : A for all a ∈ A, then Γ T{ u a a∈A } : B. Remark B.3.We observe that: • the conditions for applying the induction hypothesis in the case of rules that discharge hypothesis are guaranteed by the fact that the variable(s) to be bound swaps between Γ and the Δs in passing from the premise to the conclusion of the refined typing rules; • it is crucial that the refined typings keep apart the variables captured by the distinct occurrences of the hole: Had one merged all the Δ a s in each refined typing into a single Δ, the inductive steps for rules with more than one premise would not go through, since the fact that e.g., in the case of • in the refined typing judgments there is no need of keeping track also of bound propositional variables, as they play no role in the proof; • in the case in which the same term t is inserted in all holes of a context, the premise of the lemma is to be understood as meaning that F V (t) ⊆ a∈A (F V (Γ) ∪ F V (Δ a )) (where F V (t) indicates the set of free variables of t).

C. The RP-Translation
Definition C.1.(RP-translation) Given the RP-translation of formulas (Definition 2.7 in Section 2.2), for every term u such that Γ NI 2p u : A we define a term u * as follows:

D. The Standard Equivalence on Derivations
The rules of equivalence are the following: together with reflexivity, transitivity, symmetry and congruence rules.
Remark D.1.As in the case of disjunction (see, e.g.[29] and [17]) the "generalized" rule η + for †-connectives can be decomposed into a simple form of η-rule and a "generalized" permutation rule: The rule †γ expressing the standard permuting conversions used in establishing the subformula of normal derivations in NI is the special case of †γ + in which the context U consists in a single elimination rule (with the hole [ ] corresponding to the major premise).
We where the three applications of †γ + can be described as:

E. Weak Expansion and A-Expansion
Definition E.1.For any A ∈ L 2 we define a family of atomic elimination contexts Elim(A) and a family of introduction contexts Intro(A) by induction as follows: We list some useful and easily established facts about introduction and elimination contexts (recalling that for all elimination contexts E and terms t, E{t} = E[t]): Fact E.4.If (E 1 , I 1 ) and (E 2 , I 2 ) are two expansion pairs for A, then for any U : at(A) Γ at(A) and term Γ t : A, if no variable free in U and t is bound in either I 1 or I 2 , then In the following we will suppose fixed for any formula A ∈ L 2 an expansion pair (El A , In A ).

Definition E.5. (Weak expansion and A-expansion) For all A ∈ L 2 , the weak expansion of A is the context In
Observe that by Fact E.4, the context A X U does not depend on the chosen expansion pair for A. Moreover, it is easily checked that for all sp-X formulas Remark E.6.In [33] we defined the notion of A-expansion in a more direct manner for the broader class of strictly positive formulas (and not, as done here, strongly positive, see Remark 2.15).The interested reader can easily check that the two definitions coincide in the case of strongly positive formulas.

F. The ε-Equation
We write Γ t ε u : A iff there is a derivation of Γ t u using the rules of congruence, symmetry, transitivity, reflexivity and any instance of the rule below: For the use of e.g.ηε and ε we adopt the same conventions introduced in Appendix D.
Remark F.1.As for the rules †η + and †γ + , we formulated the rule ε using n.v.c.-substitution of a term for the hole of U. Similarly to the other cases (see Remark D.2), a variant of ε with v.c.-substitution in place of n.v.c.substitution can be derived from the rule ε along with βand η-conversions and symmetry.

G. Proof of Proposition 1
We will establish following strengthening of the generalization of Proposition 1 to NI 2p : We need the following two lemmas: Proof.The lemma is easily established by induction on U.

Moreover, if no free variable of El
where C = ΛX.λf k k∈K .[] and the variables X, f k are chosen so as not to occur free in the contexts Proof.By induction on the typing derivation D of Γ NI 2p u γ + v : A. If D ends with an application of either reflexivity, transitivity or of one of the congruence rules, it is enough to apply the induction hypothesis to the derivations of the premises, and then applying the reflexivity, transitivity or the congruence rules.
If D ends with an application of †γ + , then for some context U : Proof of Proposition 9.It suffices to check rule by rule that points 1-3 of the proposition hold.For point 1., the only critical case is †β and its verification is a straightforward generalization of the preservation of the β-reduction for disjunction by the RP-translation [15].For points 2. and 3., the critical cases are †η and †γ + which follow from Proposition 10 and Proposition 11 respectively.

H. The Three Translations into NI
To prove the proposition we will need the following lemma, whose formulation relies on the refined typing for contexts introduced in Appendix B.

Lemma I.1. Let
• A, B be formulas of NI 2 ; • u a a∈A be a finite list of terms such that Γ, Δ a NI 2 at u a : A; Δ a a∈A X be an A-multicontext in NI 2 such that X does not occur free neither in Γ nor in the Δ a .

Moreover, suppose that T binds no term or type variable of El A and that
In A binds no term or variable of T, C and of u a for all a ∈ A. Then where T Y is shorthand for T[[Y /X]].
Proof.We argue by induction on the principal context C: If D ends with an application of either reflexivity, transitivity or of one of the congruence rules, it is enough to apply the induction hypothesis to the derivations of the premises, and then applying the reflexivity, transitivity or the congruence rules.
If D ends with an application of †γ, u = E[δ † (t, y j .sk )] and v = δ † (t, y j .E[s k ] ) for some elimination context E : B Γ A (hence Γ NI p δ † (t, y j .sk ) :B and Γ NI p t : † A i for some †).We show how to construct a derivation D of Γ * NI 2 at u β v : A * .Let Z = at(A * ) and Z = at(B * ).By Remark H.7 we know that E is a principal context and that u = (E[δ † (t, y j .sk )]) = (E [(δ † (t, y j .sk )) ]) = (E {(δ † (t, y j .sk )) }).Moreover, consider the K-multicontext T : X Γ Δ k k∈K X given by T = (t X) λ y j j∈g −1 (k) .[] k k∈K , where Δ k = y j : A f (j) j∈g −1 (k) and X is a fresh variable.The reader can check that all conditions of Lemma I. 1 From these facts we deduce in turn: We now deduce that Z, since the only γ ≥ ⊥ such that γ M[X →a] Y ⊃ X and for all γ ≥ γ, γ M[X →a] Z ⊃ X is α and α M[X →a] X; • if a = {β}, then ⊥ M[X →a] Y Z, since the only γ ≥ ⊥ such that γ M[X →a] Y ⊃ X and for all γ ≥ γ, γ M[X →a] Z ⊃ X is β and β M[X →a] X.
We deduce then (using Fact L.1) that ⊥ M ∀X.Y Z, and since ⊥ M Y ∨ Z, we conclude that (∀X.Y Z) Y ∨ Z.

Definition 2 .
10. (Russell-Prawitz fragment) We let L 2 RP be the subset of L 2 in which all universal formulas are universal polynomial, and we let NI 2 RP , called the Russell-Prawitz fragment of NI 2 , be the fragment of NI 2 obtained by restricting to L 2 RP -formulas.The system NI 2 • is the subsystem of NI 2 RP in which universal formulas ∀X.A are all of the form ∀X.•(A 1 , A 2 , A 3 ).

Remark 3 . 5 .
In the functorial semantics of NI2 , what we call the Cexpansion of a derivation D is just the result of applying the functor interpreting C to the morphism interpreting D. As any L 2 -formula can be interpreted as a functor, the notion of C-expansion can be extended to any C ∈ L 2 .Remark 3.6.Whenever X is clear from the context, with the notation for substitution introduced in ([33], sec.3.2), we write C[[B/X]] and C[[A/X]] as C B and C A , and leaving the main assumption A implicit we indicate the C-expansion of D as C D .Remark 3.7.In the present paper, we will only be concerned with the Cexpansion of derivations of the form El m A .Observe that, by the definition of C-expansion, the undischarged assumptions of C El m A besides C A are the same as the undischarged assumptions of El m A besides A, namely (see Definition 3.1 above)

Table 2 .
ε-conversion Remark 3.9.The proof of Proposition 1 scales straightforwardly to the whole of NI p (see Appendix G), and it actually holds if one replaces η (resp.γ) with the more general η + (resp.γ + ) (see Remark 3.8 above, Remark D.1 in Appendix D and Section 5.1 below).The class of ε-conversions needed to establish the analog of Proposition 1 for NI • are depicted in Table

Definition 4 . 1 .
A formula ∀X.A ∈ L 2 enjoys the instantiation overflow property iff for all formulas B ∈ L2 Definition 4.2.(FF-atomization, FF-translation) If D is an NI 2 the clause is analogous to the previous one (for a fully detailed definition, see Definition H.1 in Appendix H) If D is an NI • -derivation, we call the NI 2 at -derivation D * ↓ the FF-translation of D. Remark 4.3.Ferreira and Ferreira present their result in a different way, by using the inductive clauses of Definition 4.2 to give a direct proof of instantiation overflow for the universal formulas of the form (A ∨ B) * , andthey refer to what we here called the FF-translation of D as to "the canonical translation of D in F at provided by instantiation overflow"[11].This difference in presentation will allow a more straightforward formulation of our results.Whereas the FF-translation is defined by combining the RP-translation from NI • into NI2  RP with the FF-atomization embedding NI 2 RP into NI 2 at , more recently Espírito Santo and Ferreira[5] introduced an alternative translation by "directly" defining an embedding from NI into NI 2 at .We will refer to this translation as the ESF-translation.Using the notation we introduced, the definition of can be adapted to NI • (the definition actually scales to the whole of NI p , see Def.H.5 in Appendix H), and the crucial case of the definition, that of an NI • -derivation ending with an application of •E runs as follows (assuming at(C * ) = Z): By combining Proposition 3 with Proposition 4 we deduce that the FFand ESF-translations are βηε-equivalent to the RP-translation: Corollary 4.8.For all NI • -derivations D, D * βη D * ↓ and D * βη D .
free in E and E ∈ Elim(C)} and Intro(A) = {λx.I | I ∈ Intro(C)and x does not occur in I} • If A = ∀Z.C, then Elim(A) = {E[[ ]X] | X not free in E and E ∈ Elim(C)} and Intro(A) = {ΛX.I | I ∈ Intro(C) and X does not occur in I} An expansion pair for A is any pair of contexts (El, In) such that El ∈ Elim(A), In ∈ Intro(A) and In{El} η [ ].

Fact E. 2 .
If Γ NI 2 t : A then for all expansion pairs (El, In) for A, Γ t η In{El[t]} : A Fact E.3.For all expansion pairs (El, In) for A ⊃ B there exists x ∈ T V such that El = E[[ ]x] and In = λx.I, where (E, I) is an expansion pair for B. Similarly, for all expansion pairs (El, In) for ∀X.A exists X ∈ V such that El = E[[ ]X] and In = ΛX.I, where (E, I) is an expansion pair for A.
A a a∈A ⊃X is bound in U, then Γ ( Aa a∈A ⊃ X) X U {λ xa a∈A .t}β λ xa a∈A .U{t} : Aa a∈A ⊃ C Proof.The lemma is easily established by induction on the length of the list A. Proposition 10.For all u, v if Γ NI 2p u η v : A then Γ * NI 2 u * βηε v * : A * .Proof.We can argue by induction on the derivation D of Γ NI 2p u η v : A. The only non-trivial case is when D ends with an application of †η.So suppose

2 atC
the claim is immediate;• if C = C v, then C : A Γ NI ⊃ B and Γ NI 2at v : C, and noticing that at(B) = at(C ⊃ B) we then haveC In A T at(A) { El A [u a ] a∈A } = C In A T at(A) { El A [u a ] a∈A } v I.H. β In C⊃B T at(B) { El C⊃B [C [u a ]] a∈A } v = λx.In B T at(B) { El B [C [u a ]x] a∈A } v β In B T at(B) { El B [C [u a ]v] a∈A } = In B T at(B) { El B [C[u a ]] a∈A } • if C = C W , then C : A Γ ∀Y.B , B = B [[W/Y ]]; by noticing that (at(∀Y.B ))[[W/Y ]] = at(B) we then have C In A T at(A) { El A [u a ] a∈A } = C In A T at(A) { El A [u a ] a∈A } W I.H. β In ∀Y.B T at(∀Y.B ) { El ∀Y.B [C [u a ]] a∈A } W = ΛY.In B T at(∀Y.B ) { El B [C [u a ]Y ] a∈A } W β In B T at(B) { El B [C [u a ]W ] a∈A } = In B T at(B) { El B [C[u a ]] a∈A } • if C = λw.C , then B = B 1 ⊃ B 2 and C : A Γ,w:B 1 B 2 ,so by noticing that at(B) = at(B 2 ) we haveIn B T at(B) { El B [C[u a ]] a∈A } = λw .In B 2 T at(B) { El B 2 [(λw.C [u a ])w ] a∈A } β λw .In B 2 T at(B) { El B 2 [C [[w /w]][u a ]] a∈A } I.H. β λw .C [[w /w]] In A T at(A) { El A [u a ] a∈A } = C In A T at(A) { El A [u a ] a∈A }• if C = ΛW.C , then B = ∀W.B and C : A Γ B , so by noticing that at(B) = at(B ) we haveIn B T at(B) { El B [C[u a ]] a∈A } = ΛW .In B T at(B) { El B [(ΛW.C [u a ])W ] a∈A } β ΛW .In B T at(B) { El B [C [[W /W ]][u a ]] a∈A } I.H. β ΛW .C [[W /W ]] In A T at(A) { El A [u a ] a∈A } = C In A T at(A) { El A [u a ] a∈A } Proposition 13.For all u, v, if Γ NI p u γ v : A, then Γ * NI 2 at u β v : A * .Proof.By induction on the typing derivation D of Γ NI p u γ v : A.

Table 1 .
RP-Translation of Standard Connectives the diagram determining a polynomial connective.A quantified formula of L 2 RP may therefore have quantified formulas as proper subformulas, provided they are in turn universal polynomial formulas.Remark 2.12.It is clear that the restriction of the RP-translation to L p is into L 2 RP , and that if D is an NI • -derivation then D * is an NI 2 • -derivation.Remark 2.13.It is easy to check that the equational theory of NI 2 RP is well-defined, since L 2 RP is closed under substitution.Remark 2.14.The system NI 2 RP is the smallest fragment of NI 2 closed under ⊃ and containing the RP-translation of all polynomial connectives.
To see how, we define an alternative atomization procedure yielding yet another translation of NI • into NI 2 at (also this atomization scales to the whole of NI p , see Definition H.2 in Appendix H): Definition 4.4.(ε-atomization, ε-translation) The definition differs from Definition 4.2 in the following respect: In case An immediate consequence of Proposition 6 concerns the faithfulness of the RP-translation.We recall that a translation (•) fromN to N' is faithful iff for all Γ, A, if Γ N' A then Γ N A. Although the FFtranslation is a faithful translation of NI ∨ into NI 2 at ([10]), it is not a faithful translation of NI 2∨ at into NI 2 at .In fact, keeping in mind that the FF-and the RP-translations of formulas coincide, if we let Γ = (Y ∨ Z) Remark 5.2.In addition to the failure of faithfulness, also the disjunction property fails for the translation of disjunction in NI 2 at in the following sense: there exist formulas A, B of NI 2∨ at such that NI 2 * * and A = Y ∨ Z we have that ((Y ∨ Z) * ) * = (Y ∨ Z) * and hence obviously Γ * NI 2 at A * , but Γ NI 2∨ at A does not hold by the above proposition.at (A ∨ B) * holds but neither of NI 2 at A * and NI 2 at B * holds.The following example establishes this fact.For readability, we write ∃Z.C as shorthand for ∀W.(∀Z.C ⊃ W ) ⊃ W (with W a fresh propositional variable) and the same does not hold if one replaces A ∨ B with ∀X.(A B): As a consequence, in contrast to what happens in NI 2 at , the instantiation overflow property fails for ∀X.(AB) in NI 2∨ at .Given Proposition 6, the analog of Proposition 5 cannot hold for NI 2∨ at .One may therefore wonder in which sense, if at all, can one say that disjunction is definable in NI 2 at .Following Prawitz ([22], p. 58), we can distinguish between strong and weak definability of a connective.In Prawitz' terminology, disjunction is weakly definable in NI 2 at if and only if there is a faithful translation of NI ∨ into NI 2 at , which is the case, since the FF-translation is faithful (see Remark 5.1 above); on the other hand, disjunction is strongly definable in NI 2

2
RPand some decidability results for the fragment NI 2 RP (decidability of type inhabitation, i.e. of provability, and of contextual equivalence).In contrast to what happens in NI 2 RP , provability and contextual equivalence are both undecidable in NI 2 at (although type checking and typability are decidable in NI 2 yj .sk )) * = t * A * λ yj .s* Fact C.2.If Γ NI 2p u : A, then Γ * NI 2 u Remark C.3.The clauses for ιand δ-terms the standard translation of disjunction and conjunction constructors in System F (where ι ∧ (t 1 , t 2 ) is prefix notation for the standard pairing).= ΛX.λx 1 x 2 .xu t * (u = 1, 2) (δ ∨ (t, y.s 1 , y.s 2 )) ∧ (t, y 1 y 2 .s))* = t * A * λy 1 y 2 .s* Remark C.4.It can be checked that if C : A Γ B is a context in the language of NI 2p , then C * : A * Γ * B * is a context in the language of NI 2 .Moreover, if E is an elimination context then E * is also an elimination context.

2 at
Definition H.1.(Atomization) For every term u such that Γ NI 2 RP u : A we define a term u ↓ as follows (observe that when u = tB then Γ NI 2 RP t : ∀W.F and since F is polynomial in W , by what we observed in Subsection 2.2, F can be written as A * f (j) ⊃ W ⊃ W , with the A i having no occurrence of W ): do hold with E in place of C, B * , A * in place of A, B and the s k in place of the u a .Then we have: uDef.H.5= E In B * (t Z ) λ y j .El B * [s k ] In A * (t Z) λ y j .El A * E [s k ] tB) ↓ Definition H.1+I.H β λ v k λu.((tD) ↓ ε ) λ x j .vkxj u Definition H.2 = λ v k λu.(λ y k .In D (t ↓ ε Z) ( A * f (j) ⊃ Z) Z El D [y k ] ) λ x j .vkxj u β λ v k λu.In D (t ↓ ε Z) ( A * f (j) ⊃ Z) Z El D [λ x j .vkxj u] Definition.E.5 = λ v k λu.In D (t ↓ ε Z) λ w j .El D [(λ x j .vkxj u) w j ] k λu.In D (t ↓ ε Z) λ w j .El D [v k w j u] Fact E.4 = λ v k .In C⊃D (t ↓ ε Z) λ w j .El C⊃D [v k w j ] Definition.E.5 = λ v k .In C⊃D (t ↓ ε Z) ( A * f (j) ⊃ Z) Z El C⊃D [v k ]One can argue in a similar way if B = ∀U.D (just replace C ⊃ D with ∀U.D and u with U ). Proposition 19.For all u such that Γ NI p u : A: Γ * Proof.By induction on the typing derivation D of Γ NI p u : A. If D ends with an application of either Ax, ⊃I, ⊃E or †I k , (i.e.u is either a variable or of the form λx.t, ts or ι k † t j ), then it is enough to use reflexivity or to apply the induction hypothesis to the immediate sub-derivations of D and use the congruence rules.Otherwise u = δ † (t, y j .sk ) and hence (by Definition C.1) u * = (t * A * ) λ y j .s*k , where the same remarks as in Definition H.1 apply to t * , and we informally show how to construct a derivation D of Γ NI 2 at u * ↓ ε β u : A. If A is atomic, i.e.A = Y we have that:(δ † (t, y j .sk )) * ↓ ε Definition C.1 = v Proposition 14.For all u, v if Γ NI p u η v : A then Γ * NI 2 at u βη v : A * .(βλv * .=(( t * Y ) λ y j .s*k ) ↓ ε Definition H.2 = ( t * ↓ ε Y ) λ y j .s*↓ε k I.H. β (t Y ) λ y j .skDefinitionE.1 = In Y (t Y ) λ y j .El Y [s k ] Definition H.5 = uOtherwise, assuming Z = at(A * ) we have that:(δ † (t, yj .sk )) * ↓ ε Definition C.1 = ((t * A * ) λ yj .s* k ) ↓ ε Definition H.2 = = λ z k .InA * (t * ↓ ε Z) A * f (j) ⊃ Z Z ElA * [z k ] λ yj .s* ↓ ε Z ElA * [λ yj .s* ↓ ε k ] = = InA * (t * ↓ ε Z) λ vj .ElA * [(λ yj .s* ↓ ε k ) vj ]