η-conversions of IPC implemented in atomic F

It is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus Fat. Since Fat enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present paper we improve the previous result by analyzing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus Fat). In fact, from the strong normalization of Fat we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions.


Introduction
The atomic polymorphic calculus F at [3,7] 1 is the restriction of Jean-Yves Girard's system F [9] to atomic universal instantiations.The restriction occurs only in the derivations (terms) allowed, not in the formulas (types) permitted.The formulas in F at (as in system F) are defined as the smallest class of expressions that includes the atomic formulas (propositional constants and second-order variables) and is closed under implication and second-order universal quantification.In the natural deduction calculus proofs in F at are built using the following introduction rules: where, in the second rule, X does not occur free in any undischarged hypothesis; and the following elimination rules: with C an atomic formula, free for X in A. It is the restriction to atomic instantiations in the latter rule that distinguishes F at from F. The (impredicative) system F allows, in the ∀E rule, the instantiation of X by any (not necessarily atomic) formula of the system.
The introduction of F at may be a possible answer to Girard's dissatisfaction with the natural deduction rules for ⊥ and ∨.From page 80 of [9]: "One tends to think that natural deduction should be modified to correct such atrocities [referring to the commuting conversions needed to deal with the bad connectives ⊥, ∨ and ∃]: if a connector has such bad rules, one ignores it (a very common attitude) or one tries to change the very spirit of natural deduction in order to be able to integrate it harmoniously with the others.It does not seem that the (⊥, ∨, ∃) fragment of the calculus is etched on tablets of stone." F at is an alternative to full intuitionistic propositional calculus (IPC) in the sense that IPC can be translated into F at via a sound [3,8] and faithful [6,5] embedding.Thus, any deduction in IPC can be performed into F at -a predicative system with no bad connectives, with no commuting conversions, with a simple strong normalization proof [7,4] and whose normal proofs enjoy the subformula property [3].
The embedding of IPC into F at relies on the well-known Russell-Prawitz's definition of the connectives ⊥, ∨ and ∧ in terms of → and ∀ and on instantiation overflow.To make this paper reasonably self-contained, in the next section we remember these notions.
Since ∧ is not a bad2 connective we can take it as primitive in F at .So, till the end of the present paper we will work with an atomic polymorphic calculus, we denote by F ∧ at , which has the primitive connectives ∧ (for conjunction), → (for implication) and ∀ (for second-order universal quantification) and, in addition to the introduction and elimination rules previously presented for F at , F ∧ at has also the following rules for ∧: The standard conversions of F ∧ at are the following β-conversions: where C is an atomic formula, free for X in A, and the following η-conversions: The formulas on the left hand-side of the conversion are called redexes and on the right hand-side contractums.
Note that since F ∧ at has no bad connectives there is no need for commuting conversions in the system.The reason why we choose to work within F ∧ at instead of F at will become clear in the last two sections of the paper.
It is an easy exercise to adapt the proof of strong normalization presented in [7] (for F at ) to F ∧ at .We sketch such a proof in the next section.It was shown in [7] that the β-conversions of IPC could be implemented in F at through βη-conversions and, as an application of strong normalization for F at considering βη-conversions, the strong normalization for full IPC considering β-conversions was derived.What about the η-conversions?
In the present paper we show that the η-conversions of IPC can be implemented in F ∧ at via βη-conversions.As a consequence we are able to improve the previous results: from the strong normalization of F ∧ at considering βη-conversions we can derive the strong normalization of full IPC considering all the standard (β and η) conversions.
The paper is organized as follows: In Section 2 we present the embedding of full IPC into F ∧ at and the proof of strong normalization for the latter calculus considering βη-conversions.In Section 3 we study the translation of the η-conversions of IPC into F ∧ at and in Section 4, as an application, we present an alternative proof for the strong normalization of full IPC considering the standard (β and η) conversions.

Preliminaries
The embedding of full IPC into F ∧ at uses a well-known translation of the connectives ⊥ and ∨ in terms of → and ∀ due to Bertrand Russell [12] and Dag Prawitz [11].For every formula A of the full propositional calculus we define its translation A * into F ∧ at inductively as follows: (P) * :≡ P, for P a propositional constant (⊥) * :≡ ∀X.X where X is a second-order variable which does not occur in A * nor in B * .Note that the Russell-Prawitz translation also allows for the translation of ∧ in terms of → and ∀.In our context we do not need to translate conjunction in such a way because ∧ is a primitive symbol in F ∧ at .The previous translation is, in fact, a sound embedding, i.e., denoting by i provability in the full intuitionistic propositional calculus and by F ∧ at provability in the atomic polymorphic system F ∧ at , we have: The proof can be found in [3,8] 3 and relies in the phenomenon of instantiation overflow.Instantiation overflow ensures that from formulas of the form for any (not necessarily atomic) formula F. The proof of instantiation overflow is given in [3,8] and it yields algorithmic methods for obtaining the two kinds of deductions above.For a recent study on instantiation overflow see [1].Since the (canonical) deductions provided by instantiation overflow are going to be extensively used in sections 3 and 4, we exemplify instantiation overflow with the case of disjunction.More precisely, by induction on the complexity of F, we show how to deduce in For F atomic there is nothing to argue, it is the application of a single rule: ∀E.We just have to analyze the cases in which F is D 1 ∧ D 2 , D 1 → D 2 and ∀XD admitting (by induction hypothesis) that instantiation overflow is available for D 1 , D 2 and D. For where D is the derivation: For F :≡ ∀XD we have: When we refer to the translation of a certain derivation of IPC into F ∧ at , we mean the canonical translation (rule-by-rule) provided by the proof of the embedding of IPC into F ∧ at (see [3,8]).We exemplify the canonical translation with the elimination rule of disjunction: The translation of the IPC rule above into F ∧ at is: where the double line hides the instantiation overflow discussed before.
Next we will observe that the strategy to prove strong normalization for F at presented in [7] also works to prove strong normalization for F ∧ at .By the Curry-Howard isomorphism also known as "formulas-as-types paradigm", F ∧ at can be presented in the (operational) λ-calculus style.Types in F ∧ at are the ones in F at -see [7] Definition 1, page 261, resulting from the atomic types (propositional constants and type variables) by means of two type-forming operations → and ∀ -with an extra type-forming operation ∧, i.e. if A and B are types then A ∧ B is a type.Terms in F ∧ at are defined as the terms in F at (see [7], Definition 2, page 261-262) adding two clauses: Note that in F ∧ at we have the same conversions we have in F at plus the conversions for ∧ which, in the λ-calculus style, are the following two β-conversions: π 1 t, s t π 2 t, s s and the following η-conversion: Remember that the strategy in [7] to prove that F at has the strong normalization property (a simple adaptation of Tait's convertibility technique) proceeds as follows: i) we define by induction on the complexity of the types a class Red of terms of F at ; ii) we prove that all terms in Red are strongly normalizable considering βη-conversions; iii) we prove that all terms in F at are in Red.
Remember also that Red was defined in the following way: For C an atomic type, t ∈ Red C :≡ t is strongly normalizable.
t ∈ Red A→B :≡ for all q, if q ∈ Red A then tq ∈ Red B .
t ∈ Red ∀X.A :≡ for all atomic types C, tC In the context of F ∧ at we only have to add a new clause for conjunction: We say that a term is neutral if it is not of the form t, s or λx.t or ΛX.t. 4he proof of strong normalization for F ∧ at , considering βη-conversions, proceeds as in [7].For the treatment of conjunction see [9] pages 42-46.
3 How do the η-conversions of IPC translate into F ∧ at ?
In (full) intuitionistic propositional calculus we have the following η-conversions: Proposition 1.Consider an η-conversion of (full) IPC.The canonical translation of its redex into F ∧ at reduces, by means of a finite number (at least one) of βη-conversions, into the canonical translation of its contractum into F ∧ at .Proof.The case of the η-conversions for ∧ and → is trivial.Let us study the ηconversion for ∨.In what follows, for ease of notation, we ignore the translations of A and B.
The translation of the redex into F ∧ at , we denote by derivation D, has the form: where the double line hides the proof in F ∧ at that exists by instantiation overflow, and for reasons of space, we write A ∨ B in some points of the derivation to abbreviate ∀X ((A → X) → ((B → X) → X)).This abbreviation for economy of space will also be used in other parts of this proof.Note that no confusion arises from this abuse of notation since all deductions are in the context of F ∧ at where there is no disjunction symbol ∨.
We want to prove that from the derivation D above, applying standard (β and η) conversions of F ∧ at , we obtain the derivation Applying one η-conversion and one β-conversion we obtain: With another η-conversion (and without making use of abbreviations) we have: Thus, with three η-conversion we get 4 Alternative strong normalization proof for IPC considering βη-conversions In [7] it was proved that the β-conversions of IPC translate into βη-conversions of F at .
In this section we start by arguing that the result remains valid for F ∧ at .The β-conversions of IPC are the ones for conjunction and implication (see pages 2 and 3 in the context of F ∧ at ) plus the following ones for disjunction: Proposition 2. Consider a β-conversion of (full) IPC.The canonical translation of its redex into F ∧ at reduces, by means of a finite number (at least one) of βη-conversions, into the canonical translation of its contractum into F ∧ at .
Proof.The only case non trivial is disjunction.It is possible to prove that the derivation by induction on the complexity of the formula F exactly as in [7], Lemma 4, pp.268-271 (for F at ).In the present context of F ∧ at , we only need to analyze a new case (F :≡ D 1 ∧ D 2 ).Take the derivation .
Disclosing a bit the double line we have: Applying the induction hypothesis twice, the derivation reduces to by means of βη-conversions.Note that we have only changed the portion of derivation above D 1 and D 2 .
Applying one η and one β-conversions we obtain .
and, with one more βconversion we obtain .
Finally we are able to present the alternative proof of strong normalization for (full) IPC considering βη-conversions.
of Theorem 1 can no longer be used when considering the IPC commuting conversions.Nevertheless, it remains an open question if the atomic polymorphic framework is able to produce an alternative proof of strong normalization for the full intuitionistic propositional calculus with commuting conversions.
3) The reason we work in the atomic polymorphic calculus (at the price of having to consider instantiation overflow) instead of working directly in system F is twofold.Firstly, note that (as opposed to F ∧ at ) the canonical translation of proofs (rule-by-rule) of IPC into system F (via the Russell-Prawitz translation) does not preserve standard conversions.Take, for instance, the η-conversion for ∨ analysed in the proof of Proposition 1.The canonical translation of its redex into system F is the derivation D (see page 7) with the double line (for instantiation overflow) replaced by a single line.
).] Since D with the modification above has no redexes it can not be βη-reduced to the canonical translation (into system F) of the contractum of the η-conversion for ∨.
Since proofs in F ∧ at are, in particular, proofs in system F we can argue that (although not canonical) the simulation of η-conversions in this paper can be seen as having Girard's system F as our target system.This leads us to our second point.
This paper is part of a line of research that intends to develop an alternative to full intuitionistic propositional calculus free from the defects (bad connectives/commuting conversions) pointed by Girard et al. in [9].Such alternative is the atomic polymorphic framework.As opposed to system F, system F ∧ at is predicative, enjoys the subformula property and allows for an elementary proof of strong normalization (see [4]).Properties in IPC (see [5] for the disjunction property, [7] for the strong β-normalization property and the present paper for the strong βη-normalization property) can be reduced to properties elegantly proved in F ∧ at with no bad connectives nor permutative conversions.4) We think that the algorithms involved in the proof-theoretical analysis via the Russell-Prawitz translation -e.g. the one that underlies the embedding of IPC into the atomic polymorphic system (relying on instantiation overflow) and the algorithms for the simulation of β, η (and commuting) conversions of IPC into βη-conversions of F ∧ at -which are in the base of alternative proofs of strong normalization for IPC, deserve further investigation and comparison with related studies via CPS transformations (see [10] and [2]).