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It is known that the $\beta$-conversions of the full intuitionistic propositional calculus ($\mathbf{IPC}$) translate into $\beta\eta$-conversions of the atomic polymorphic calculus ${\mathbf{F}}_{\mathbf{at}}$. Since ${\mathbf{F}}_{\mathbf{at}}$ enjoys the property of strong normalization for $\beta\eta$-conversions, an alternative proof of strong normalization for $\mathbf{IPC}$ considering $\beta$-conversions can be derived. In the present article, we improve the previous result by analysing the translation of the $\eta$-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus ${\mathbf{F}}_{\mathbf{at}}^{\wedge}$). In fact, from the strong normalization of ${\mathbf{F}}_{\mathbf{at}}^{\wedge}$ we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard ($\beta$ and $\eta$) conversions.

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... 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]. 2 For 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. ...

... 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. ...

... We define D ↓ by a sub-induction on F . they 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. ...

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.

... This idea is implicit in [8] -we will confirm this later on in this paper. Following [5,9], we are calling Russell-Prawitz translation the translation of IPC into system F based on this translations of proofs. ...

... There are several reasons to study embeddings of IPC into F at , the foremost being that F at is a predicative fragment of F. Another reason has to do with preservation of proof identities generated in IPC by commuting conversions or η-reductions: the various embeddings into F at achieve that preservation [4,5,2], while the Russell-Prawitz translation into F does not [6,10,4,5,9]. This seems an indication that other conversion principles are missing in F, besides the βη ones. ...

... There are several reasons to study embeddings of IPC into F at , the foremost being that F at is a predicative fragment of F. Another reason has to do with preservation of proof identities generated in IPC by commuting conversions or η-reductions: the various embeddings into F at achieve that preservation [4,5,2], while the Russell-Prawitz translation into F does not [6,10,4,5,9]. This seems an indication that other conversion principles are missing in F, besides the βη ones. ...

Given the recent interest in the fragment of system F where universal instantiation is restricted to atomic formulas, a fragment nowadays named system F_at, we study directly in system F new conversions whose purpose is to enforce that restriction. We show some benefits of these new atomization conversions: (1) They help achieving strict simulation of proof reduction by means of the Russell-Prawitz embedding of IPC into system F; (2) They are not stronger than a certain "dinaturality" conversion known to generate a consistent equality of proofs; (3) They provide the bridge between the Russell-Prawitz embedding and another translation, due to the authors, of IPC directly into system F_at; (4) They give means for explaining why the Russell-Prawitz translation achieves strict simulation whereas the translation into F_at does not.

... 3 Two translations of NI into NI 2 at As we recalled in the introduction, Ferreira and Ferreira [4,5,6,8], developed an alternative translation of NI-derivations into NI 2 that we propose to call the FF-translation. The FF-translation agrees with the RP-translation on how to translate formulas, but not on how to translate derivations. ...

... Remark 3.1. Ferreira and Ferreira present their result in a slightly different way, by using the inductive clauses of Definition 3.2 to give a direct proof of Proposition 3.1, and they refer to what we here called the FF-translation of D as to "the canonical translation of D in F at provided by instatiation overflow" [8]. This minor difference in presentation is however inessential for the results presented here. ...

... Moreover, as Ferreira and Ferreira observed in [8], Theorem 3.2 fails if NI is extended to include also conjunction. On the other hand, our Theorem 2.1 keeps on holding. ...

In this paper (which is a prosecution of "The naturality of natural deduction", Studia Logica 2019) we investigate the exact relationship between the Russell-Prawitz translation of intuitionistic propositional logic into intuitionistc second-order propositional logic (System F), and its variant proposed by Fernando Ferreira and Gilda Ferreira into the atomic fragment of System F (System Fat). In the previous paper we investigated the Russell-Prawitz translation via an extended equational theory for System F arising from its categorical semantics. The main result of this paper is that the Russell-Prawitz translation and Ferreira and Ferreira's translation are equivalent modulo this extended equational theory. This result highlights a close connection between our previous work and that of Ferreira and Ferreira. We argue however that the approach obtained by coupling the original Russell-Prawitz translation with our extended equational theory is more satisfactory for the study of proof identity than the one based on System Fat.

... F at expresses the connectives of IPC in a uniform way avoiding bad connectives (according to Girard [GLT89], page 74) and avoiding commuting conversions. This explains the usefulness of the system in proof theoretical studies [Fer17a,Fer17b]. For related work in the topic see [PTP22], where the authors investigate predicative translations of IPC into system F at using the equational framework presented in [TPP19] and giving an elegant semantic explanation (relying on parametricity) of the syntactic results on atomic polymorphism. ...

It is well-known that typability, type inhabitation and type inference are
undecidable in the Girard-Reynolds polymorphic system F. It has recently been
proven that type inhabitation remains undecidable even in the predicative
fragment of system F in which all universal instantiations have an atomic
witness (system Fat). In this paper we analyze typability and type inference in
Curry style variants of system Fat and show that typability is decidable and
that there is an algorithm for type inference which is capable of dealing with
non-redundancy constraints.

... Interestingly, at the level of conversions, this encoding is stronger than the usual one: it translates not only β-reductions, but also the permutative conversions and a restricted form of η-conversion for sums, into β and η-reductions of F at (see [11,14,9]). ...

Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most well-known being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.

... F at expresses the connectives of IPC in a uniform way avoiding bad connectives (according to Girard [GLT89], page 74) and avoiding commuting conversions. This explains the usefulness of the system in proof theoretical studies [Fer17a,Fer17b]. For related work in the topic see [PTP19], where the authors investigate predicative translations of IPC into system F at using the equational framework presented in [TPP19] and giving an elegant semantic explanation (relying on parametricity) of the syntactic results on atomic polymorphism. ...

It is well-known that typability, type inhabitation and type inference are undecidable in the Girard-Reynolds polymorphic system F. It has recently been proven that type inhabitation remains undecidable even in the predicative fragment of system F in which all universal instantiations have an atomic witness (system Fat). In this paper we analyze typability and type inference in system Fat and show that these two problems are decidable in the atomic polymorphic system.

Atomic polymorphism Fat is a restriction of Girard and Reynold’s system F
F(or λ2) which was first introduced in Ferreira [ 2] in the context of a philosophical commentary on predicativism. λ2 is a well-known and powerful formal tool for studying polymorphic functional programming languages and formal methods in program specification and development, but its computational power far exceeds the recursive level of interest in applications. Hence, the interest of studying subsystems of λ2
with weaker computational power. Fat is defined by restricting instantiation to atomic variables only. It turns out that the type system is still sufficiently powerful to possess embeddings of full intuitionistic propositional calculus [ 3, 4], and since the calculus has fewer connectives and strong normalizability is simple to prove [ 3], this result allows us to circumvent many of the extra computational complexities present when dealing with the proof theory of IPC. It is natural to inquire whether type inhabitation, i.e. provability in the corresponding fragment of second-order intuitionistic propositional logic, is decidable or not and in general to see whether the negative results involving the undecidability of type inhabitation, typability and type-checking for F still hold in this fragment. A further theme would be to study the result of adding type constructors, recursors or even dependent types to Fat. In this paper, we show that type inhabitation for Fat is undecidable by codifying within it an undecidable fragment of first-order intuitionistic predicate calculus, adapting and modifying the technique of Urzyczyn’s [ 1, 7] purely syntactic proof of the undecidability of type inhabitation for F.

We show that the number-theoretic functions definable in the atomic polymorphic system (${\mathbf{F}}_{\mathbf{at}}$) are exactly the extended polynomials. Two proofs of the above result are presented: one, reducing the functions’ definability problem in ${\mathbf{F}}_{\mathbf{at}}$ to definability in the simply typed lambda calculus ($\lambda ^{\rightarrow }$) and the other, directly adapting Helmut Schwichtenberg’s strategy for definability in $\lambda ^{\rightarrow }$ to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda calculus, is emphasized.

Given the recent interest in the fragment of system $\mathbf{F}$ where universal instantiation is restricted to atomic formulas, a fragment nowadays named system ${\mathbf{F}}_{\textbf{at}}$, we study directly in system $\mathbf{F}$ new conversions whose purpose is to enforce that restriction. We show some benefits of these new atomization conversions: (i) they help achieving strict simulation of proof reduction by means of the Russell–Prawitz embedding of $\textbf{IPC}$ into system $\mathbf{F}$, (ii) they are not stronger than a certain ‘dinaturality’ conversion known to generate a consistent equality of proofs, (iii) they provide the bridge between the Russell–Prawitz embedding and another translation, due to the authors, of $\textbf{IPC}$ directly into system ${\mathbf{F}}_{\textbf{at}}$ and (iv) they give means for explaining why the Russell–Prawitz translation achieves strict simulation whereas the translation into ${\mathbf{F}}_{\textbf{at}}$ does not.

We study an alternative embedding of IPC into atomic system F whose translation of proofs is based, not on instantiation overflow, but instead on the admissibility of the elimination rules for disjunction and absurdity (where these connectives are defined according to the Russell–Prawitz translation). As compared to the embedding based on instantiation overflow, the alternative embedding works equally well at the levels of provability and preservation of proof identity, but it produces shorter derivations and shorter simulations of reduction sequences. Lambda-terms are employed in the technical development so that the algorithmic content is made explicit, both for the alternative and the original embeddings. The investigation of preservation of proof-reduction steps by the alternative embedding enables the analysis of generation of “administrative” redexes. These are the key, on the one hand, to understand the difference between the two embeddings; on the other hand, to understand whether the final word on the embedding of IPC into atomic system F has been said.

We present a purely proof-theoretic proof of the existence property for the full intuitionistic first-order predicate calculus, via natural deduction, in which commuting conversions are not needed. Such proof illustrates the potential of an atomic polymorphic system with only three generators of formulas – conditional and first and second-order universal quantifiers – as a tool for proof-theoretical studies.

We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (\(\mathbf {IPC}\)), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of \(\mathbf {IPC}\) into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of \(\mathbf {IPC}\) (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.

We give an elementary proof (in the sense that it is formalizable in Peano arithmetic) of the strong normalization of the atomic polymorphic calculus Fat (a predicative restriction of Girard's system F).

We show how to interpret intuitionistic propositional logic in a predicative second-order intuitionistic propositional system having only the conditional and the universal second-order quantifier. We comment on this fact. We argue that it supports the legitimacy of using classical logic in a predicative setting, even though the philosophical cast of predicativism is non-realistic. We also note that the absence of disjuction and existential quantifications allows one to have a process of normalization of proofs that avoids the use of “commuting conversions.”

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that 'one tends to think that natural deduction should be modifled to correct such atrocities.' We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions.

The intuitionistic fragment of the call-by-name version of Curien and
Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly
normalising by means of an embedding into the simply-typed lambda-calculus. Our
embedding is a continuation-and-garbage-passing style translation, the
inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's
\lambda\_mu-calculus. The embedding strictly simulates reductions while usual
continuation-passing-style transformations erase permutative reduction steps.
For our intuitionistic sequent calculus, we even only need "units of garbage"
to be passed. We apply the same method to other calculi, namely successive
extensions of the simply-typed λ-calculus leading to our intuitionistic
system, and already for the simplest extension we consider (λ-calculus
with generalised application), this yields the first proof of strong
normalisation through a reduction-preserving embedding. The results obtained
extend to second and higher-order calculi.

The well-known embedding of full intuitionistic propositional calculus into the atomic polymorphic system Fat is possible due to the intriguing phenomenon of instantiation overflow. Instantiation overflow ensures that (in Fat) we can instantiate certain universal formulas by any formula of the system, not necessarily atomic. Until now only three types in Fat were identified with such property: the types that result from the Prawitz translation of the propositional connectives (⊥, ∧, ∨) into Fat (or Girard's system F). Are there other types in Fat with instantiation overflow? In this paper we show that the answer is yes and we isolate a class of formulas with such property.

It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system F
at, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the intuitionistic propositional logic in which commuting conversions are not needed.

It has been known for six years that the restriction of Girard's
polymorphic system $\text{\bfseries\upshape F}$ to atomic universal instantiations
interprets the full fragment of the intuitionistic propositional calculus.
We firstly observe that Tait's method of “convertibility” applies
quite naturally to the proof of strong normalization of the restricted
Girard system. We then show that each $\beta$-reduction step of the full
intuitionistic propositional calculus translates into one or more
$\beta\eta$-reduction steps in the restricted Girard system. As a consequence,
we obtain a novel and perspicuous proof of the strong normalization property
for the full intuitionistic propositional calculus. It is noticed that this
novel proof bestows a crucial role to $\eta$-conversions.

This paper proposes a new proof method for strong normalization of classical natural deduction and calculi with control operators. For this purpose, we introduce a new CPS-translation, continuation and garbage passing style (CGPS ) translation. We show that this CGPS-translation method gives simple proofs of strong normalization of λμ→∧∨⊥, which is introduced in [P. de Groote, Strong normalization of classical natural deduction with disjunction, in: S. Abramsky (Ed.), Typed Lambda Calculi and Applications, 5th International Conference, TLCA 2001, in: Lecture Notes in Comput. Sci., vol. 2044, Springer, Berlin, 2001, pp. 182–196] by de Groote and corresponds to the classical natural deduction with disjunctions and permutative conversions.

Natural Deduction. Almkvist & Wiksell, 1965. Reprinted, with a new preface

- D Prawitz

The faithfulness of atomic polymorphism

- F Ferreira
- G Ferreira