aa r X i v : . [ m a t h . L O ] A ug Gentzen-Mints-Zucker duality
Daniel Murfet, William TroianiAugust 25, 2020
Abstract
The Curry-Howard correspondence is often described as relating proofs (in intu-tionistic natural deduction) to programs (terms in simply-typed lambda calculus).However this narrative is hardly a perfect fit, due to the computational content ofcut-elimination and the logical origins of lambda calculus. We revisit Howard’s workand interpret it as an isomorphism between a category of proofs in intuitionistic se-quent calculus and a category of terms in simply-typed lambda calculus. In ourtelling of the story the fundamental duality is not between proofs and programs butbetween local (sequent calculus) and global (lambda calculus or natural deduction)points of view on a common logico-computational mathematical structure.
Contents
A Background on lambda calculus 62
There may, indeed, be other applications of the system than its use as a logic.A. Church,
Postulates for the foundation of logic Introduction
Sequent calculus and lambda calculus were both invented in the context of logical inves-tigations, the former by Gentzen as a language of proofs [9] and the latter by Church as alanguage of functions [3]. The computational content of these calculi emerged at differenttimes, with the relevance of β -reduction of lambda terms to the emerging theory of com-putation being more quickly realised than the relevance of cut-elimination. By now it isclear that both calculi have logical and computational aspects, and that the two calculiare deeply related to one another. In this paper we revisit this relationship in the formof an isomorphism of categories (Theorem 4.15)(1.1) F Γ : S Γ ∼ = / / L Γ for each sequence Γ of formulas (n´ee types) where S Γ is a category of proofs in intuitionisticsequent calculus (defined in Section 2) and L Γ is a category of simply-typed lambda terms(defined in Section 3). Both categories have the same set of objects, viewed either asthe formulas of intuitionistic propositional logic or simple types. The set of morphisms S Γ ( p, q ) is the set of proofs of Γ ⊢ p ⊃ q up to an equivalence relation ∼ p generated by cut-elimination transformations and commuting conversions together with a small number ofadditional natural relations, while L Γ ( p, q ) is the set of simply-typed lambda terms of type p → q whose free variables have types taken from Γ, taken up to βη -equivalence. We referto this isomorphism of categories and the normal form theorem which refines it (Theorem4.49) as the Gentzen-Mints-Zucker duality between sequent calculus and lambda calculus.The name reflects work by Zucker [35] and Mints [22], elaborated below.A duality consists of two different points of view on the same object [1]. The greaterthe difference between the two points of view, the more informative is the duality whichrelates them. Such correspondences are important because two independent discoveries ofthe same structure is strong evidence that the structure is natural. The above duality isinteresting precisely because sequent calculus proofs and lambda terms are not tautolog-ically the same thing: for example the cut-elimination relations are fundamentally local while the β -equivalence relation is global (see Section 4.3). In this sense sequent calculusand lambda calculus are respectively local and global points of view on a common logico-computational mathematical structure.This duality is related to, but distinct from, the Curry-Howard correspondence. Theprecise relationship is elaborated in Section 1.1 below, but broadly speaking it is capturedby conceptual diagram of Figure 1. The Curry-Howard correspondence gives a bijectionbetween natural deduction proofs and lambda terms, while Gentzen-Mints-Zucker dualityreveals that βη -normal lambda terms are normal forms for sequent calculus proofs moduloan equivalence relation generated by pairs that are well-motivated from the point of viewof the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic proof.2equent calculus Gentzen, Zucker y y rrrrrrrrrrrrrrrrrrrrrrr Mints % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ Natural deduction
Curry, Howard
Lambda calculusFigure 1: The relationship between three logico-computational calculi.
The relationship between proofs and lambda terms (or “programs”) has become widelyknown as the Curry-Howard correspondence following Howard’s work [15], although aninformal understanding of the computational content of intuitionistic proofs has olderroots in the Brouwer-Heyting-Kolmogorov interpretation [31]. The correspondence hasbeen so influential in logic and computer science that the Curry-Howard correspondenceas philosophy now overshadows the Curry-Howard correspondence as a theorem .As a theorem, the Curry-Howard correspondence is the observation that the formulasof implicational propositional logic are the same as the types of simply-typed lambdacalculus, and that there is a surjective map from the set of all proofs of a sequent Γ ⊢ α in“sequent calculus” to the set of all lambda terms of type α with free variables in Γ (due toHoward [15, §
3] building on ideas of Curry and Tait). This map is not a bijection, and assuch does not represent the best possible statement about the relationship between proofsand lambda terms. The problem is that there are two natural continuations of Howard’swork, depending on how one interprets the somewhat vague notion of proof in [15, § § proof means natural deduction proof ; see for example [29, § § § § Curry-Howard correspondence . Despite its philosophical importance,this correspondence is not mathematically of great interest, because natural deduction3nd lambda calculus are so similar that the bijection is close to tautological (in the caseof closed terms it is left as an exercise in one standard text [29, Ex. 4.8]).In the present work we investigate the second natural continuation of [15], which takesseriously the structural rules in Howard’s “sequent calculus” and seeks to give a bijectionbetween sequent calculus proofs and lambda terms. As soon as explicit structural rulesare introduced into proofs, however, there will be multiple proofs that map to the samelambda term, and so for there to be a bijection between proofs and lambda terms, proof must mean equivalence class of preproofs modulo some relation . If this relation is simply“maps to the same lambda term” then what we have constructed is merely a surjectivemap from proofs to lambda terms, which is hardly more than what is in [15]. Hence inthis second line of thought, the identity of proofs becomes a central concern.Consequently one of the contributions of this paper is to give explicit generating rela-tions for a relation ∼ p on preproofs such that π ∼ p π if and only if F Γ ( π ) = F Γ ( π ), andto give a logical justification of these relations independent of the translation to lambdaterms. This establishes S Γ as a mathematical structure in its own right, so that the com-parison to L Γ may be meaningfully referred to as a duality.Given the Curry-Howard correspondence, the identity of proofs is closely related tothe old problem of when two sequent calculus proofs map to the same natural deductionunder the translation defined by defined by Gentzen [9] (see Remark 4.5). This has beenstudied by various authors, most notably Zucker [35], Pottinger [25], Dyckhoff-Pinto [7],Mints [22] and Kleene [17]. The most important results are those obtained by Zucker andMints, and we restrict ourselves here to comments on their work; see also Section 4.4.There is substantial overlap between our main results and those of Zucker and Mints,which we became aware of after this paper had been completed. In [35] Zucker gives a setof generating relations characterising when two sequent calculus proofs map to the samenatural deduction, for a calculus that does not contain weakening and exchange. The maincontent of Theorem 4.15 also lies in identifying an explicit set of generating relations onpreproofs, for the map from sequent calculus proofs to lambda terms in a system of sequentcalculus that is (as far as is possible for a system that must be translated unambiguouslyto lambda terms) as close as possible to Gentzen’s LJ. As far as we know, this paper isthe first place that the generating relations have been established for Gentzen’s LJ withall structural rules.Mints, using ideas of Kleene [17], identifies a set of normal forms of sequent calculusproofs and studies them using the map from proofs to lambda terms. Our proof of themain theorem (Theorem 4.15) relies on the identification of normal forms, which differslightly from those of Mints (see also Theorem 4.49). Again we treat a standard form ofLJ, whereas [22] follows Kleene’s system G [17] in the form of its ( L ⊃ ) rule.Finally, since we must argue that S Γ has an independent existence in order for theduality to be a relationship between equals we are committed to mounting a purely logi-cal defense of all the generating relations of ∼ p . This is not a concern shared by Zucker,4ints or Kleene. The most interesting generating relations are those that we call λ -equivalences (Definition 2.21) which are justified on the grounds that they represent aninternal Brouwer-Heyting-Kolmogorov interpretation, see the discussion preceding Defi-nition 2.21 and Section 4.2. There is an infinite set of atomic formulas and if p and q are formulas then so is p ⊃ q . LetΨ ⊃ denote the set of all formulas. For each formula p let Y p be an infinite set of variablesassociated with p . For distinct formulas p, q the sets Y p , Y q are disjoint. We write x : p for x ∈ Y p and say x has type p . Let P n be the set of all length n sequences of variableswith P := { ∅ } , and P := ∪ ∞ n =0 P n . A sequent is a pair (Γ , p ) where Γ ∈ P and p ∈ Ψ ⊃ ,written Γ ⊢ p . We call Γ the antecedent and p the succedent of the sequent. Given Γ anda variable x : p we write Γ , x : p for the element of P given by appending x : p to the endof Γ. A variable x : p may occur more than once in a sequent.Our intuitionistic sequent calculus is the system LJ of [9, § III] restricted to implication,with formulas in the antecedent tagged with variables and a more liberal set of deductionrules (see Remark 2.5). We follow the convention of [11, § Definition 2.1. A deduction rule results from one of the schemata below by a substitutionof the following kind: replace p, q, r by arbitrary formulas, x, y by arbitrary variables, andΓ , ∆ , Θ by arbitrary (possibly empty) sequences of formulas separated by commas: • the identity group : – Axiom : (ax) x : p ⊢ p – Cut : Γ ⊢ p ∆ , x : p, Θ ⊢ q (cut)Γ , ∆ , Θ ⊢ q • the structural rules : – Contraction : Γ , x : p, y : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q – Weakening : Γ , ∆ ⊢ q (weak)Γ , x : p, ∆ ⊢ q Exchange : Γ , x : p, y : q, ∆ ⊢ r (ex)Γ , y : q, x : p, ∆ ⊢ r • the logical rules : – Right introduction : Γ , x : p, ∆ ⊢ q ( R ⊃ )Γ , ∆ ⊢ p ⊃ q – Left introduction : Γ ⊢ p ∆ , x : q, Θ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ ⊢ r Definition 2.2. A preproof is a finite rooted planar tree where each edge is labelled bya sequent and each node except for the root is labelled by a valid deduction rule. If theedge connected to the root is labelled by the sequent Γ ⊢ p then we call the preproof a preproof of Γ ⊢ p .Observe that the only valid label for a leaf node is an axiom rule, so a preproof readsfrom the leaves to the root as a deduction of Γ ⊢ p from axiom rules. Example 2.3.
Here is the Church numeral 2 in our sequent calculus(ax) x : p ⊢ p (ax) x : p ⊢ p (ax) x : p ⊢ p ( L ⊃ ) y ′ : p ⊃ p, x : p ⊢ p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ p (ctr) y : p ⊃ p, x : p ⊢ p ( R ⊃ ) y : p ⊃ p ⊢ p ⊃ p Remark 2.4.
Multiple occurrences of a deduction rule are communicated in the notationwith doubled horizontal lines. For example if Γ = x : p , . . . , x n : p n then the preproof(ax) y : q ⊢ q (weak)Γ , y : q ⊢ q weakens in every formula in the sequence. The doubled horizontal line therefore standsfor n occurrences of the rule (weak). The preproofs which perform these weakenings ina different order are, of course, not equal as preproofs, so the notation is an abuse. Wewill only use it below in the context of defining generating pairs of equivalence relationsin cases where any reading of this notation leads to the same equivalence relation.6 emark 2.5. A deduction rule is strict if it is an arbitrary (ax) or (ex) rule, or it is oneof the other rules and the occurrence of x : p in the rule is leftmost in the antecedent.A strict preproof is a preproof in which every deduction rule is strict. These are thededuction rules and preproofs of Gentzen’s original sequent calculus [9, § III]. A generaldeduction rule is clearly derivable from the strict rules by exchange, and so we may chooseto view non-strict deduction rules as derived rules; see Lemma 2.16.We adopt the more liberal rules since they make the commuting conversions, cut-elimination transformations and the proof of cut-elimination easier to present. A similarcalculus is adopted, for similar reasons, in [2] and elsewhere.
Remark 2.6.
We follow Gentzen [9, § III] in putting the variable introduced by a ( L ⊃ )rule at the first position in the antecedent. This choice is correct from the point of viewof the relationship between sequent calculus proofs and lambda terms, as may be seen inLemma 4.29 and Section 4.1.When should two preproofs be considered to be the same proof? Clearly some of thestructure of a preproof is logically insignificant, but it is by no means trivial to identify aprecise notion of proof as separate from preproof . Historically, logic has concerned itselfprimarily with the provability of sequents Γ ⊢ p rather than the structure of the set ofall preproofs, but as proof theory has developed the question of the identity of proofs hasacquired increasing importance; see Ungar [34] and Prawitz [27, § ∼ on the set of preproofs satisfies condition (C0) if π ∼ π implies π , π are preproofs of the same sequent. The relation satisfies condition (C1) ifit satisfies (C0) and π ∼ π implies π ′ ∼ π ′ where π ′ , π ′ are the result of applying thesame deduction rule to π , π respectively. For example if ∼ satisfies (C1) and π ∼ π then π ...Γ , ∆ ⊢ p (weak)Γ , x : p, ∆ ⊢ q ∼ π ...Γ , ∆ ⊢ p (weak)Γ , x : p, ∆ ⊢ q Condition (C2) is defined using the following schematics: π i ...Γ ⊢ p ρ i ...∆ , x : p, Θ ⊢ q (cut)Γ , ∆ , Θ ⊢ q π i ...Γ ⊢ p ρ i ...∆ , x : q, Θ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ ⊢ r (2 . ∼ on the set of preproofs satisfies condition (C2) if it satisfies (C0)and whenever π ∼ π and ρ ∼ ρ then also κ ∼ κ where κ i for i ∈ { , } is obtainedfrom the pair ( π i , ρ i ) by application of one of the deduction rules in (2.1). Definition 2.7.
A relation ∼ on preproofs is compatible if it satisfies (C0),(C1),(C2).7n occurrence of x : p in a preproof π is an occurrence in the antecedent Γ of a sequentlabelling some edge of π . Some occurrences are related by the flow of information in thepreproof, and some are not. More precisely: Definition 2.8 (Ancestors).
An occurrence of z : s in a preproof π is an immediatestrong ancestor (resp. immediate weak ancestor ) of an occurrence z : s if there is adeduction rule in π where z : s is in the numerator and z : s is in the denominator, andone of the following holds (referring to the schemata in Definition 2.1):(i) z : s, z : s are in the same position of Γ , ∆ , Θ in the numerator and denominator.(ii) the rule is (ctr), z : s is the first of the two variables being contracted (resp. z : s iseither of the variables being contracted) and z : s is the result of that contraction.(iii) the rule is (ex) and either z : s = x : p, z : s = x : p or z : s = y : p, z : s = y : p .One occurrence z : s is a strong ancestor (resp. weak ancestor ) of another z ′ : s if thereis a sequence z : s = z : s, . . . , z n : s = z ′ : s of occurrences in π with z i : s an immediatestrong (resp. weak) ancestor of z i +1 : s for 1 ≤ i < n .Note that if z : s is a strong ancestor of z : s then z = z but this is not necessarilytrue for weak ancestors. Definition 2.9.
Let ≈ str (resp. ≈ wk ) denote the equivalence relation on the set of variableoccurrences generated by the strong (resp. weak) ancestor relation. Definition 2.10 (Ancestor substitution).
Let x : p be an occurrence of a variablein a preproof π and y : p another variable. We denote by subst str ( π, x, y ) the preproofobtained from π by replacing the occurrence x : p and all its strong ancestors by y . Example 2.11.
In the preproof 2 of Example 2.3 the partition of variable occurrencesaccording to the equivalence relation ≈ str is shown by colours in(ax) x : p ⊢ p (ax) x : p ⊢ p (ax) x : p ⊢ p ( L ⊃ ) y ′ : p ⊃ p, x : p ⊢ p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ p (ctr) y : p ⊃ p, x : p ⊢ p ( R ⊃ ) y : p ⊃ p ⊢ p ⊃ p and the partition according to ≈ wk in(ax) x : p ⊢ p (ax) x : p ⊢ p (ax) x : p ⊢ p ( L ⊃ ) y ′ : p ⊃ p, x : p ⊢ p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ p (ctr) y : p ⊃ p, x : p ⊢ p ( R ⊃ ) y : p ⊃ p ⊢ p ⊃ p
8n our preproofs we have tags, in the form of variables, for hypotheses. Since the precisenature of these tags is immaterial, if the variable is eliminated in a ( R ⊃ ), ( L ⊃ ),(ctr) or(cut) rule the identity of the proof should be independent of the tag. Definition 2.12 ( α -equivalence). We define ∼ α to be the smallest compatible equiva-lence relation on preproofs such thatΓ ⊢ p π ...∆ , x : p, Θ ⊢ q (cut)Γ , ∆ , Θ ⊢ q ∼ α Γ ⊢ p subst str ( π, x, y )...∆ , y : p, Θ ⊢ q (cut)Γ , ∆ , Θ ⊢ q (2 . π ...Γ , x : p, y : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q ∼ α subst str ( π, y, z )...Γ , x : p, z : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q (2 . π ... x : p, Γ ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q ∼ α subst str ( π, x, y )... y : p, Γ ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q (2 . ⊢ p π ...∆ , x : q, Θ ⊢ r ( L ⊃ ) z : p ⊃ q, Γ , ∆ , Θ ⊢ r ∼ α Γ ⊢ p subst str ( π, x, y )...∆ , y : q, Θ ⊢ r ( L ⊃ ) z : p ⊃ q, Γ , ∆ , Θ ⊢ r (2 . π and variables x, y, z of the same type. Remark 2.13.
The generating relation (2.5) of Definiton 2.12 is to be read as a pair ofpreproofs ( ψ, ψ ′ ) ∈ ∼ α where both preproofs have final sequent z : p ⊃ q, Γ , ∆ , Θ ⊢ r andthe branch ending in Γ ⊢ p is any preproof (but it is the same preproof in both ψ and ψ ′ ).To avoid clutter we will not label branches, here or elsewhere, if it is clear how to matchup the branches in the two preproofs involved in the relation.The price for our more liberal deduction rules is the inclusion of τ -equivalences below,which express that two instances of the same deduction rule, operating in different places,are essentially the same. Definition 2.14 ( τ -equivalence). We define ∼ τ to be the smallest compatible equiva-lence relation on preproofs satisfyingΓ ⊢ p ∆ , x : p, y : q, Θ ⊢ q (cut)Γ , ∆ , y : q, Θ ⊢ q ∼ τ Γ ⊢ p ∆ , x : p, y : q, Θ ⊢ q (ex)∆ , y : q, x : p, Θ ⊢ q (cut)Γ , ∆ , y : q, Θ ⊢ q (2 . , x : p, x ′ : p, y : q, Γ ′ ⊢ r (ctr)Γ , x : p, y : q, Γ ′ ⊢ r (ex)Γ , y : q, x : p, Γ ′ ⊢ r ∼ τ Γ , x : p, x ′ : p, y : q, Γ ′ ⊢ r (ex)Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ctr)Γ , y : q, x : p, Γ ′ ⊢ r (2 . , x : p, Γ ′ ⊢ q (weak)Γ , x : p, y : r, Γ ′ ⊢ q (ex)Γ , y : r, x : p, Γ ′ ⊢ q ∼ τ Γ , x : p, Γ ′ ⊢ q (weak)Γ , y : r, x : p, Γ ′ ⊢ q (2 . , x : p, z : r, Γ ′ ⊢ s ( R ⊃ )Γ , x : p, Γ ′ ⊢ r ⊃ s ∼ τ Γ , x : p, z : r, Γ ′ ⊢ s (ex)Γ , z : r, x : p, Γ ′ ⊢ s ( R ⊃ )Γ , x : p, Γ ′ ⊢ r ⊃ s (2 . ⊢ p ∆ , x : q, z : r, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z : r, ∆ ′ ⊢ r ∼ τ Γ ⊢ p ∆ , x : q, z : r, ∆ ′ ⊢ r (ex)∆ , z : r, x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z : r, ∆ ′ ⊢ r (2 . x : p , . . . , x n : p n ⊢ q (ex , σ , . . . , σ r ) x τ : p τ , . . . , x τn : p τn ⊢ q ∼ τ x : p , . . . , x n : p n ⊢ q (ex , ρ , . . . , ρ s ) x τ : p τ , . . . , x τn : p τn ⊢ q (2 . τ is a permutation and σ , . . . , σ r and ρ , . . . , ρ s are sequences of transpositions ofconsecutive positions (one or both lists may be empty) with the property that σ · · · σ r = τ = ρ · · · ρ s in the permutation group. The two preproofs in (2.11) are respectively thesequences of exchanges corresponding to the σ i and ρ j . Remark 2.15.
Note that we only include the τ -equivalences with “right to left” ex-changes, since the other possible relation follows from these and (2.11), for example:Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ctr)Γ , y : q, x : p, Γ ′ ⊢ r (ex)Γ , x : p, y : q, Γ ′ ⊢ r ∼ τ Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ex)Γ , x : p, x ′ : p, y : q, Γ ′ ⊢ r (ex)Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ctr)Γ , y : q, x : p, Γ ′ ⊢ r (ex)Γ , x : p, y : q, Γ ′ ⊢ r ∼ τ Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ex)Γ , x : p, x ′ : p, y : q, Γ ′ ⊢ r (ctr)Γ , x : p, y : q, Γ ′ ⊢ r (ex)Γ , y : q, x : p, Γ ′ ⊢ r (ex)Γ , x : p, y : q, Γ ′ ⊢ r τ Γ , y : q, x : p, x ′ : p, Γ ′ ⊢ r (ex)Γ , x : p, x ′ : p, y : q, Γ ′ ⊢ r (ctr)Γ , x : q, y : q, Γ ′ ⊢ r Lemma 2.16.
Every preproof is equivalent under ∼ τ to a strict preproof.Proof. Left to the reader.We have a strong intuition about the structure of logical arguments which leads tothe expectation that the antecedent of a sequent is an extended “space” disjoint subsetsof which may be the locus of independent operations. This independence is formalisedby commuting conversions, which identify preproofs that differ only by “insignificant”rearranging of deduction rules.
Definition 2.17 (Commuting conversions).
We define ∼ c to be the smallest compat-ible equivalence relation on preproofs generated by the following pairs. We begin withpairs involving two structural rules:Γ , Γ ′ , Γ ′′ ⊢ q (weak)Γ , x : p, Γ ′ , Γ ′′ ⊢ q (weak)Γ , x : p, Γ ′ , y : r, Γ ′′ ⊢ q ∼ c Γ , Γ ′ , Γ ′′ ⊢ q (weak)Γ , Γ ′ , y : r, Γ ′′ ⊢ q (weak)Γ , x : p, Γ ′ , y : r, Γ ′′ ⊢ q (2 . , x : p, x ′ : p, Γ ′ , Γ ′′ ⊢ q (ctr)Γ , x : p, Γ ′ , Γ ′′ ⊢ q (weak)Γ , x : p, Γ ′ , y : r, Γ ′′ ⊢ q ∼ c Γ , x : p, x ′ : p, Γ ′ , Γ ′′ ⊢ q (weak)Γ , x : p, x ′ : p, Γ ′ , y : r, Γ ′′ ⊢ q (ctr)Γ , x : p, Γ ′ , y : r, Γ ′′ ⊢ q (2 . , Γ ′ , x : p, x ′ : p, Γ ′′ ⊢ q (ctr)Γ , Γ ′ , x : p, Γ ′′ ⊢ q (weak)Γ , y : r, Γ ′ , x : p, Γ ′′ ⊢ q ∼ c Γ , Γ ′ , x : p, x ′ : p, Γ ′′ ⊢ q (weak)Γ , y : r, Γ ′ , x : p, x ′ : p, Γ ′′ ⊢ q (ctr)Γ , y : r, Γ ′ , x : p, Γ ′′ ⊢ q (2 . , x : p, y : q, Γ ′ , Γ ′′ ⊢ r (ex)Γ , y : q, x : p, Γ ′ , Γ ′′ ⊢ r (weak)Γ , y : q, x : p, Γ ′ , z : s, Γ ′′ ⊢ r ∼ c Γ , x : p, y : q, Γ ′ , Γ ′′ ⊢ r (weak)Γ , x : p, y : q, Γ ′ , z : s, Γ ′′ ⊢ r (ex)Γ , y : q, x : p, Γ ′ , z : s, Γ ′′ ⊢ r (2 . , Γ ′ , x : p, y : q, Γ ′′ ⊢ r (ex)Γ , Γ ′ , y : q, x : p, Γ ′′ ⊢ r (weak)Γ , z : s, Γ ′ , y : q, x : p, Γ ′′ ⊢ r ∼ c Γ , Γ ′ , x : p, y : q, Γ ′′ ⊢ r (weak)Γ , z : s, Γ ′ , x : p, y : q, Γ ′′ ⊢ r (ex)Γ , z : s, Γ ′ , y : q, x : p, Γ ′′ ⊢ r (2 . , x : p, x ′ : p, Γ ′ , y : q, y ′ : q, Γ ′′ ⊢ r (ctr)Γ , x : p, Γ ′ , y : q, y ′ : q, Γ ′′ ⊢ r (ctr)Γ , x : p, Γ ′ , y : q, Γ ′′ ⊢ r ∼ c Γ , x : p, x ′ : p, Γ ′ , y : p, y ′ : p, Γ ′′ ⊢ r (ctr)Γ , x : p, x ′ : p, Γ ′ , y : p, Γ ′′ ⊢ r (ctr)Γ , x : p, Γ ′ , y : p, Γ ′′ ⊢ r (2 . , x : p, y : q, Γ ′ , z : r, z ′ : r ⊢ s (ex)Γ , y : q, x : p, Γ ′ , z : r, z ′ : r ⊢ s (ctr)Γ , y : q, x : p, Γ ′ , z : r ⊢ s ∼ c Γ , x : p, y : q, Γ ′ , z : r, z ′ : r ⊢ s (ctr)Γ , x : p, y : q, Γ ′ , z : r ⊢ s (ex)Γ , y : q, x : p, Γ ′ , z : r ⊢ s (2 . , z : r, z ′ : r, Γ ′ , x : p, y : q, Γ ′′ ⊢ s (ex)Γ , z : r, z ′ : r, Γ ′ , y : q, x : p, Γ ′′ ⊢ s (ctr)Γ , z : r, Γ ′ , y : q, x : p, Γ ′′ ⊢ s ∼ c Γ , z : r, z ′ : r, Γ ′ , x : p, y : q, Γ ′′ ⊢ s (ctr)Γ , z : r, Γ ′ , x : p, y : q, Γ ′′ ⊢ s (ex)Γ , z : r, Γ ′ , y : q, x : p, Γ ′′ ⊢ s (2 . R ⊃ ):Γ , x : p, y : q, Γ ′ , z : r, Γ ′′ ⊢ s ( R ⊃ )Γ , x : p, y : q, Γ ′ , Γ ′′ ⊢ r ⊃ s (ex)Γ , y : q, x : p, Γ ′ , Γ ′′ ⊢ r ⊃ s ∼ c Γ , x : p, y : q, Γ ′ , z : r, Γ ′′ ⊢ s (ex)Γ , y : q, x : p, Γ ′ , z : r, Γ ′′ ⊢ s ( R ⊃ )Γ , y : q, x : p, Γ ′ , Γ ′′ ⊢ r ⊃ s (2 . , x : p, Γ ′ ⊢ q ( R ⊃ )Γ , Γ ′ ⊢ p ⊃ q (weak)Γ , y : r, Γ ′ ⊢ p ⊃ q ∼ c Γ , x : p, Γ ′ ⊢ q (weak)Γ , y : r, x : p, Γ ′ ⊢ q ( R ⊃ )Γ , y : r, Γ ′ ⊢ p ⊃ q (2 . , x : p, x ′ : p, Γ ′ , y : q, Γ ′′ ⊢ r ( R ⊃ )Γ , x : p, x ′ : p, Γ ′ , Γ ′′ ⊢ q ⊃ r (ctr)Γ , x : p, Γ ′ , Γ ′′ ⊢ q ⊃ r ∼ c Γ , x : p, x ′ : p, Γ ′ , y : q, Γ ′′ ⊢ r (ctr)Γ , x : p, Γ ′ , y : q, Γ ′′ ⊢ r ( R ⊃ )Γ , x : p, Γ ′ , Γ ′′ ⊢ q ⊃ r (2 . ⊢ r ∆ , z : s, x : p, ∆ ′ ⊢ q ( R ⊃ )∆ , z : s, ∆ ′ ⊢ p ⊃ q ( L ⊃ ) y : r ⊃ s, Γ , ∆ , ∆ ′ ⊢ p ⊃ q ∼ c Γ ⊢ r ∆ , z : s, x : p, ∆ ′ ⊢ q ( L ⊃ ) y : r ⊃ s, Γ , ∆ , x : p, ∆ ′ ⊢ q ( R ⊃ ) y : r ⊃ s, Γ , ∆ , ∆ ′ ⊢ p ⊃ q (2 . L ⊃ ) (one of which was considered already above):Γ , Γ ′ ⊢ p (weak)Γ , z : r, Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , z : r, Γ ′ , ∆ , ∆ ′ ⊢ r ∼ c Γ , Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , Γ ′ , ∆ , ∆ ′ ⊢ r (weak) y : p ⊃ q, Γ , z : r, Γ ′ , ∆ , ∆ ′ ⊢ r ∼ c Γ , Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r (weak) z : r, ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , Γ ′ , z : r, ∆ , ∆ ′ ⊢ r (ex) y : p ⊃ q, Γ , z : r, Γ ′ , ∆ , ∆ ′ ⊢ r (2 . , z : r, z ′ : r, Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , z : r, z ′ : r, Γ ′ , ∆ , ∆ ′ ⊢ r (ctr) y : p ⊃ q, Γ , z : r, Γ ′ , ∆ , ∆ ′ ⊢ r ∼ c Γ , z : r, z ′ : r, Γ ′ ⊢ p (ctr)Γ , z : r, Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , z : r, Γ ′ , ∆ , ∆ ′ ⊢ r (2 . ⊢ p ∆ , x : q, z : r, z ′ : r, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z : r, z ′ : r, ∆ ′ ⊢ r (ctr) y : p ⊃ q, Γ , ∆ , z : r, ∆ ′ ⊢ r ∼ c Γ ⊢ p ∆ , x : q, z : r, z ′ : r, ∆ ′ ⊢ r (ctr)∆ , x : q, z : r, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z : r, ∆ ′ ⊢ r (2 . , z : r, z ′ : s, Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , z : r, z ′ : s, Γ ′ , ∆ , ∆ ′ ⊢ r (ex) y : p ⊃ q, Γ , z ′ : s, z : r, Γ ′ , ∆ , ∆ ′ ⊢ r ∼ c Γ , z : r, z ′ : s, Γ ′ ⊢ p (ex)Γ , z ′ : s, z : r, Γ ′ ⊢ p ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , z ′ : s, z : r, Γ ′ , ∆ , ∆ ′ ⊢ r (2 . ⊢ p ∆ , z : r, z ′ : s, x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z : r, z ′ : s, ∆ ′ ⊢ r (ex) y : p ⊃ q, Γ , ∆ , z ′ : s, z : s, ∆ ′ ⊢ r ∼ c Γ ⊢ p ∆ , z : r, z ′ : s, x : q, ∆ ′ ⊢ r (ex)∆ , z ′ : s, z : r, x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , z ′ : s, z : r, ∆ ′ ⊢ r (2 . ⊢ p ∆ ⊢ q Θ , x : r, y : l, Θ ′ ⊢ s ( L ⊃ ) z : q ⊃ l, ∆ , Θ , x : r, Θ ′ ⊢ s ( L ⊃ ) z ′ : p ⊃ r, Γ , z : q ⊃ l, ∆ , Θ , Θ ′ ⊢ s ∼ c ∆ ⊢ q Γ ⊢ p Θ , x : r, y : l, Θ ′ ⊢ s ( L ⊃ ) z ′ : p ⊃ r, Γ , Θ , y : l, Θ ′ ⊢ r ( L ⊃ ) z : q ⊃ l, ∆ , z ′ : p ⊃ r, Γ , Θ , Θ ′ ⊢ s (ex) z ′ : p ⊃ r, Γ , z : q ⊃ l, ∆ , Θ , Θ ′ ⊢ s (2 . ⊢ l ∆ , x : p, ∆ ′ ⊢ q Θ , y : r, Θ ′ ⊢ s ( L ⊃ ) z : q ⊃ r, ∆ , x : p, ∆ ′ , Θ , Θ ′ ⊢ s ( L ⊃ ) z ′ : l ⊃ p, Γ , z : q ⊃ r, ∆ , ∆ ′ , Θ , Θ ′ ⊢ s ∼ c Γ ⊢ l ∆ , x : p, ∆ ′ ⊢ q ( L ⊃ ) z ′ : l ⊃ p, Γ , ∆ , ∆ ′ ⊢ q Θ , y : r, Θ ′ ⊢ s ( L ⊃ ) z : q ⊃ r, z ′ : l ⊃ p, Γ , ∆ , ∆ ′ , Θ , Θ ′ ⊢ s (ex) z ′ : l ⊃ p, Γ , z : q ⊃ r, ∆ , ∆ ′ , Θ , Θ ′ ⊢ s (2 . x : p, y : p that are contracted be logically signifi-cant? We are prevented from identifying contraction on x : p, y : p with contraction on y : p, x : p because the former leaves x : p and the latter y : p , but a sufficient cocom-mutativity principle is expressed by (2.32). Similarly (2.31) expresses that contraction iscoassociative. The rule (2.33) is counitality, which says that we attach no logical meaningto contraction with a variable which has been weakened in. These principles assert thatcontraction is coalgebraic , a point of view further ramified in linear logic. Definition 2.18 ( co -equivalence). We define ∼ co to be the smallest compatible equiv-alence relation on preproofs satisfyingΓ , x : p, y : p, z : p, ∆ ⊢ q (ctr)Γ , x : p, z : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q ∼ co Γ , x : p, y : p, z : p, ∆ ⊢ q (ctr)Γ , x : p, y : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q (2 . π ...Γ , x : p, y : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q ∼ co subst str ( π, y, x )...Γ , x : p, x : p, ∆ ⊢ q (ex)Γ , x : p, x : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q (2 . , x : p, ∆ ⊢ q ∼ co Γ , x : p, ∆ ⊢ q (weak)Γ , x : p, x ′ : p, ∆ ⊢ q (ctr)Γ , x : p, ∆ ⊢ q (2 . emark 2.19. The relation (2.31) appears as the contraction conversion [35, § Remark 2.20.
The rule (2.33) has a left-handed version π ...Γ , x : p, ∆ ⊢ q (weak)Γ , x ′ : p, x : p, ∆ ⊢ q (ctr)Γ , x ′ : p, ∆ ⊢ q (2.32) ∼ subst str ( π, x, x ′ )...Γ , x ′ : p, ∆ ⊢ q (weak)Γ , x ′ : p, x ′ : p, ∆ ⊢ q (ex)Γ , x ′ : p, x ′ : p, ∆ ⊢ q (ctr)Γ , x ′ : p, ∆ ⊢ q ∼ τ subst str ( π, x, x ′ )...Γ , x ′ : p, ∆ ⊢ q (weak)Γ , x ′ : p, x ′ : p, ∆ ⊢ q (ctr)Γ , x ′ : p, ∆ ⊢ q (2.33) ∼ subst str ( π, x, x ′ )...Γ , x ′ : p, ∆ ⊢ q Principles (2.34) and (2.35) below are of profound importance, as they are the internalmanifestation in our system of the Brouwer-Heyting-Kolmogorov interpretation of proofsin intuitionistic logic [31]. Under that interpretation a proof of a hypothesis y : p ⊃ q reads as a transformation of proofs of p to proofs of q . Rule (2.34) expresses that ifthe output proof of q is to be used multiple times the transformation must be employedonce for each copy. Rule (2.35) expresses that if the output is not needed, neither is thetransformation nor any of its inputs. Note that these principles mirror (2.42) and (2.43)and therefore in some sense realise ( L ⊃ ) as an internalised cut . We develop this point ofview more systematically in Section 4.2. Definition 2.21 ( λ -equivalence). We define ∼ λ to be the smallest compatible equiva-lence relation on preproofs satisfying 15 ...Γ ⊢ p π ...∆ , x : q, x ′ : q, ∆ ′ ⊢ r (ctr)∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , ∆ ′ ⊢ r ∼ λ π ...Γ ⊢ p π ...Γ ⊢ p π ...∆ , x : q, x ′ : q, ∆ ′ ⊢ r ( L ⊃ ) y ′ : p ⊃ q, Γ , ∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , y ′ : p ⊃ q, Γ , ∆ , ∆ ′ ⊢ r (ex) y : p ⊃ q, y ′ : p ⊃ q, Γ , Γ , ∆ , ∆ ′ ⊢ r (ctr) y : p ⊃ q, Γ , Γ , ∆ , ∆ ′ ⊢ r (ctr/ex) y : p ⊃ q, Γ , ∆ , ∆ ′ ⊢ r (2 . π ...Γ ⊢ p π ...∆ , ∆ ′ ⊢ r (weak)∆ , x : q, ∆ ′ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , ∆ ′ ⊢ r ∼ λ π ...∆ , ∆ ′ ⊢ r (weak) y : p ⊃ q, ∆ , ∆ ′ ⊢ r (weak) y : p ⊃ q, Γ , ∆ , ∆ ′ ⊢ r (2 . Remark 2.22.
The relation (2.34) appears as contraction conversion [35, § p , should be restricted to atomic formulas. Let Σ Γ q denote the set of preproofs of Γ ⊢ q under our system and Π Γ q the set of preproofs under this system with a restricted axiomrule. Clearly Π Γ q ⊆ Σ Γ q and if Σ Γ q is nonempty then so is Π Γ q . Since the restriction on theaxiom rule does not affect provability we are free to adopt it, either directly by changingthe deduction rules, or indirectly by keeping the deduction rules as given but adoptingan equivalence relation on preproofs which effectively makes the axiom rule on compoundformulas a derived rule: Definition 2.23 ( η -equivalence). We define ∼ η to be the smallest compatible equiva-lence relation on preproofs such that for arbitrary formulas p, q (ax) x : p ⊢ p (ax) y : q ⊢ q ( L ⊃ ) z : p ⊃ q, x : p, ⊢ q ( R ⊃ ) z : p ⊃ q ⊢ p ⊃ q ∼ η (ax) z : p ⊃ q ⊢ p ⊃ q (2 . inversion princi-ple [26, § II] which states that the left introduction rule ( L ⊃ ) is, in a sense, the inverse ofthe right introduction rule ( R ⊃ ). This principle is made manifest in the cut-eliminationtheorem of Gentzen (Theorem 2.29). To make the point in a slightly different way, notethat the (cut) rule asserts that an occurrence of A on the left of the turnstile is precisely asstrong as an occurrence on the right; see [10, § § proper if it is not (cut). Definition 2.24 (Single step cut reduction).
We define → cut to be the smallest com-patible relation (not necessarily an equivalence relation) on preproofs containing: • For any proper deduction rule ( r )(ax) x : p ⊢ p π ... ( r ) y : p, Γ ⊢ q (cut) x : p, Γ ⊢ q → cut subst str ( π, y, x )... ( r ) x : p, Γ ⊢ q (2 . π ... ( r )Γ ⊢ p (ax) x : p ⊢ p (cut)Γ ⊢ p → cut π ... ( r )Γ ⊢ p (2 . • Let ( r ) be a structural rule, ( r ) any proper deduction rule. Then π ...Γ ⊢ p ( r )Γ ′ ⊢ p π ... ( r ) y : p, ∆ ⊢ s (cut)Γ ′ , ∆ ⊢ s → cut π ...Γ ⊢ p π ... ( r ) y : p, ∆ ⊢ s (cut)Γ , ∆ ⊢ s ( r )Γ ′ , ∆ ⊢ s (2 . • ( L ⊃ ) on the left and ( r ) any proper deduction rule17 ...Γ ⊢ p π ...∆ , x : q, Θ ⊢ s ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ ⊢ s π ... ( r ) z : s, Λ ⊢ l (cut) y : p ⊃ q, Γ , ∆ , Θ , Λ ⊢ l → cut π ...Γ ⊢ p π ...∆ , x : q, Θ ⊢ s π ... ( r ) z : s, Λ ⊢ l (cut)∆ , x : q, Θ , Λ ⊢ l ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ , Λ ⊢ l (2 . • For any logical rule ( r ) and structural rule ( r ), where the cut variable x : p wasnot manipulated by ( r ): π ... ( r )Γ ⊢ p π ... x : p, ∆ ⊢ q ( r ) x : p, ∆ ′ ⊢ q (cut)Γ , ∆ ′ ⊢ q → cut π ... ( r )Γ ⊢ p π ... x : p, ∆ ⊢ q (cut)Γ , ∆ ⊢ q ( r )Γ , ∆ ′ ⊢ q (2 . • For any logical rule ( r ): π ... ( r )Γ ⊢ p π ...∆ , y : p, y ′ : p, Θ ⊢ s (ctr)∆ , y : p, Θ ⊢ s (cut)Γ , ∆ , Θ ⊢ s → cut π ... ( r )Γ ⊢ p π ... ( r )Γ ⊢ p π ...∆ , y : p, y ′ : p, Θ ⊢ s (cut)Γ , ∆ , y ′ : p, Θ ⊢ s (cut)Γ , Γ , ∆ , Θ ⊢ s (ctr/ex)Γ , ∆ , Θ ⊢ s (2 . ... ( r )Γ ⊢ p π ...∆ , Θ ⊢ q (weak)∆ , x : p, Θ ⊢ q (cut)Γ , ∆ , Θ ⊢ q → cut π ...∆ , Θ ⊢ q (weak)Γ , ∆ , Θ ⊢ q (2 . • ( R ⊃ ) on the left and ( R ⊃ ) on the right: π ...Γ , x : p, ∆ ⊢ q ( R ⊃ )Γ , ∆ ⊢ p ⊃ q π ... y : p ⊃ q, Θ , z : s ⊢ l ( R ⊃ ) y : p ⊃ q, Θ ⊢ s ⊃ l (cut)Γ , ∆ , Θ ⊢ s ⊃ l → cut π ...Γ , x : p, ∆ ⊢ q ( R ⊃ )Γ , ∆ ⊢ p ⊃ q π ... y : p ⊃ q, Θ , z : s ⊢ l (cut)Γ , ∆ , Θ , z : s ⊢ l ( R ⊃ )Γ , ∆ , Θ ⊢ s ⊃ l (2 . • ( R ⊃ ) on the left and ( L ⊃ ) on the right: π ...Γ , x : p, ∆ ⊢ q ( R ⊃ )Γ , ∆ ⊢ p ⊃ q π ...Θ ⊢ p π ...Λ , x ′ : q, Ω ⊢ s ( L ⊃ ) y : p ⊃ q, Θ , Λ , Ω ⊢ s (cut)Γ , ∆ , Θ , Λ , Ω ⊢ s → cut π ...Θ ⊢ p π ...Γ , x : p, ∆ ⊢ q (cut)Θ , Γ , ∆ ⊢ q π ...Λ , x ′ : q, Ω ⊢ s (cut)Θ , Γ , ∆ , Λ , Ω ⊢ s (ex)Γ , ∆ , Θ , Λ , Ω ⊢ s (2 . • ( R ⊃ ) on the left and ( L ⊃ ) on the right but ( L ⊃ ) does not introduce the variablewhich is involved in the (cut) 19 ... ( R ⊃ )Γ ⊢ p π ...∆ ⊢ q π ... x : p, Θ , y : r, Λ ⊢ s ( L ⊃ ) z : q ⊃ r, ∆ , x : p, Θ , Λ ⊢ s (cut)Γ , z : q ⊃ r, ∆ , Θ , Λ ⊢ s → cut π ...∆ ⊢ q π ... ( R ⊃ )Γ ⊢ p π ... x : p, Θ , y : r, Λ ⊢ s (cut)Γ , Θ , y : r, Λ ⊢ s ( L ⊃ ) z : q ⊃ r, ∆ , Γ , Θ , Λ ⊢ s (ex)Γ , z : q ⊃ r, ∆ , Θ , Λ ⊢ s (2 . π ... ( R ⊃ )Γ ⊢ p π ... x : p, ∆ ⊢ q π ...Θ , y : r, Λ ⊢ s ( L ⊃ ) z : q ⊃ r, x : p, ∆ , Θ , Λ ⊢ s (cut)Γ , z : q ⊃ r, ∆ , Θ , Λ ⊢ s → cut π ... ( R ⊃ )Γ ⊢ p π ... x : p, ∆ ⊢ q (cut)Γ , ∆ ⊢ q π ...Θ , y : r, Λ ⊢ q ( L ⊃ ) z : q ⊃ r, Γ , ∆ , Θ , Λ ⊢ s (ex)Γ , z : q ⊃ r, ∆ , Θ , Λ ⊢ s (2 . Definition 2.25.
We define ∼ cut to be the smallest equivalence relation on preproofscontaining the relation → cut . Definition 2.26 (Proof equivalence).
We define ∼ p to be the smallest compatibleequivalence relation on preproofs containing the union of • α -equivalence (Definition 2.12), • τ -equivalence (Definition 2.14), • Commuting conversions (Definition 2.17), • co -equivalence (Definition 2.18), 20 λ -equivalence (Definition 2.21), • η -equivalence (Definition 2.23), • Cut equivalence (Definition 2.25).A proof is an equivalence class of preproofs under proof equivalence. We say that twopreproofs are equivalent if they are equivalent under ∼ p . Why give yet another proof of cut-elimination? The structure of our proof is similar toGentzen’s [9] but we avoid the “mix” rule by making use of commuting conversions. Weinclude the details so as to make clear which conversions are used. The treatment in theliterature most similar to ours is [2], however there the induction is structured differentlyand the focus is on weakening rather than contraction trees.At a conceptual level, in order to justify the generating rules for proof equivalence,particularly the λ -equivalence rules, we have chosen our cut-elimination transformations(Definition 2.24) to bring out as clearly as possible the parallels between (cut) and ( L ⊃ )(see Remark 4.54). Our proof of cut-elimination reinforces this connection, with some ofthe key steps in eliminating (cut) repeated below to eliminate a subset of ( L ⊃ ) rules inSection 4 (see Lemma 4.18 and Lemma 4.29). Definition 2.27.
The width w ( q ) of a formula q is the number of occurrences of ⊃ . Definition 2.28.
The height of a preproof π , denoted h ( π ), is one less than the numberof deduction rules encountered on the longest path in the underlying tree of the preproof.Note that two preproofs can be equivalent under ∼ p but have different heights. Aproof consisting of an axiom rule has height zero. A preproof which does not contain anoccurrence of the (cut) rule is called cut-free . Theorem 2.29.
Every preproof is equivalent under ∼ p to a cut-free preproof.Proof. Given any preproof π , we can choose an instance of the (cut) rule in π which is atthe greatest possible height, and apply Proposition 2.32 below to the subproof given bytaking this as the root. Iterating this finitely many times yields the result.With reference to the prototype contraction in Definition 2.1 we say that the variables x : p, y : p are involved in that deduction rule. Definition 2.30.
Let π be a preproof. We say that a particular instance of (ctr) in theproof tree is active with respect to an occurrence of a variable x : p in the preproof if theinvolved variables in the contraction are weak ancestors of that occurrence.We begin with an easy special case: 21 emma 2.31. Suppose given a preproof π of the form π ... ( R ⊃ )Γ ⊢ p π ... ( r ) x : p, ∆ ⊢ q (cut)Γ , ∆ ⊢ q where π and π are both cut-free, the cut variable x : p is introduced in π by an axiomrule and π contains no active contractions with respect to the displayed occurrence of x : p . Then π is equivalent under ∼ p to a cut-free preproof.Proof. By induction on the height of π . In the base case π is π ... ( R ⊃ )Γ ⊢ p (ax) x : p ⊢ p (cut)Γ ⊢ p which is equivalent by (2.38) to π . For the inductive step where π has height > r ): • ( r ) is a structural rule. Since x : p is introduced by (ax) and there are no activecontractions in π , the cut variable x : p is not manipulated by ( r ) and so this casefollows by the inductive hypothesis and (2.41). • ( r ) = ( R ⊃ ) by (2.44) and the inductive hypothesis. • ( r ) = ( L ⊃ ) by (2.46) and (2.47) and the inductive hypothesis, using that x : p isnot introduced by ( L ⊃ ).This completes the inductive step and the proof of the lemma. Proposition 2.32.
Any preproof π of the form π ... ( r )Γ ⊢ p π ... ( r ) x : p, ∆ ⊢ q (cut)Γ , ∆ ⊢ q where π and π are both cut-free, is equivalent under ∼ p to a cut-free preproof.Proof. Let P ( w, n ) denote the following statement: any preproof π with cut-free branches π , π and final cut variable x : p (as above) satisfying w ( p ) = w and n = h ( π ) + h ( π )is equivalent under ∼ p to a cut-free preproof. Let P ( w ) denote ∀ nP ( w, n ). We will prove ∀ wP ( w ) by induction on w . Thus we must show P (0) and that if for all v < w P ( v ) then P ( w ). We refer to this as the outer induction . Base case of the outer induction: to prove P (0) (that is, ∀ nP (0 , n )) we proceedby induction on n , which we refer to as the inner induction . In the base case P (0 , r ) , ( r ) are axiom rules, so the claim follows from (2.37),(2.38). Now assume n > P (0 , k ) holds for all k < n . If ( r ) is (ax) then we aredone by (2.37). If ( r ) is a structural rule then the claim follows by applying the innerinductive hypothesis and (2.39). If ( r ) is a logical rule then since w ( x : p ) = 0 it mustbe ( L ⊃ ) and the claim follows from (2.40) and the inner inductive hypothesis. Inductive step of the outer induction : now suppose that w > P ( v )holds for all v < w . To prove P ( w ) (that is, ∀ nP ( w, n )) we proceed by induction on n ,which we again refer to as the inner induction. If n ≤ r ) , ( r ) is (ax) sothe claim follows from (2.37), (2.38). Suppose now that n > P ( w, k ) holds forall k < n . We again divide into cases depending on the final deduction rules ( r ) , ( r ).Some cases follow from the inner inductive hypothesis as in the proof of the base case ofthe outer induction above, and we will not repeat them. The new cases that are easilydispensed with: • ( r ) = ( R ⊃ ) , ( r ) = ( R ⊃ ) follows by (2.44) and the inner inductive hypothesis. • ( r ) = ( R ⊃ ) , ( r ) = ( L ⊃ ) may be divided into two subcases. Either the ( L ⊃ )does not introduce the cut variable x : p , in which case the claim follows by (2.46)and the inner inductive hypothesis, or the ( L ⊃ ) does introduce the cut variable x of type p = r ⊃ s , in which case π is by (2.45) equivalent to a proof of the form π ′ ...Θ ⊢ r π ′ ...Γ ′ , y : r, Γ ′′ ⊢ s (cut)Θ , Γ ′ , Γ ′′ ⊢ s π ′′ ...Λ , z : s, Ω ⊢ s (cut)Θ , Γ ′ , Γ ′′ , Λ , Ω ⊢ s (ex)Γ , ∆ ⊢ q where Γ = Γ ′ , Γ ′′ and ∆ = Θ , Λ , Ω. Since both of these cuts involve types of lowerwidth than p , the claim follows from the outer inductive hypothesis. • ( r ) is logical and ( r ) is one of (weak) , (ex) follow from the inner inductive hypoth-esis and (2.43), (2.6) respectively. • ( r ) is ( L ⊃ ) and ( r ) is (ctr) follows as above in the proof of the base case of theouter induction, by the inner inductive hypothesis and (2.40).The only remaining case is where ( r ) = ( R ⊃ ) and ( r ) = (ctr), which will occupy therest of the proof. In this case π is of the form π ... ( R ⊃ )Γ ⊢ p π ′ ... x : p, x : p, ∆ ⊢ q (ctr) x : p, ∆ ⊢ q (cut)Γ , ∆ ⊢ q x = x . Using τ -equivalence, commuting conversions and co -equivalence wecan manipulate π (meaning π ′ plus the final contraction) so that all the active contrac-tions with respect to the cut variable x : p occur at the bottom of the proof tree (seeLemma 2.36 and Remark 2.37 below). Note that the final deduction rule of π is, byhypothesis, an active contraction. After this step we see that π is equivalent to π ... ( R ⊃ )Γ ⊢ p π ′′ ... ( r ) x , x , . . . , x l , ∆ ⊢ q (ctr) x , x , . . . , x l − , ∆ ⊢ q ... (ctr) x , x , ∆ ⊢ q (ctr) x , ∆ ⊢ q (cut)Γ , ∆ ⊢ q (2 . π ′′ is cut-free and contains no active contractions with respect to x . To reduceclutter we have dropped the types from the variables x i : p . By repeated applications of(2.42) we obtain the following preproof equivalent to π : π ... ( R ⊃ )Γ ⊢ p π ... ( R ⊃ )Γ ⊢ p π ... ( R ⊃ )Γ ⊢ p π ... ( R ⊃ )Γ ⊢ p π ′′ ... ( r ) x , . . . , x l , ∆ ⊢ q (cut)Γ , x , . . . , x l − , ∆ ⊢ q ... (cut)( l − , x , x , ∆ ⊢ q (cut)( l − , x , ∆ ⊢ q (cut) l Γ , ∆ ⊢ q (ctr/ex)Γ , ∆ ⊢ q where r Γ denotes the concatenation of r copies of the sequence Γ. Note that this proofcontains no active contractions for the final cut variable x : p = x : p .The variable x i is introduced inside π ′′ by an instance ( r i ) of a deduction rule whichis (weak) , ( L ⊃ ) or (ax). Possibly using (2.32) to rearrange the ordering, we may assumethat there is an integer 1 ≤ m ≤ l such that for all 1 ≤ i ≤ m the variable x i is introducedby either (weak) or ( L ⊃ ) and for i > m it is introduced by (ax). First we deal withthe cases 1 ≤ i ≤ m . Using commuting conversions ( r i ) may be commuted downwards in π ′′ past the rule ( r ). Further by (2.41), (2.46) the rule ( r i ) may be commuted past notonly the (cut) directly below ( r ) but every cut down to the one that is actually against It is possible that π contains other contractions on variables of type p , perhaps even the variable x : p but which are not weak ancestors of the cut variable; these we all ignore. Note that the ordering on the x i has no meaning, and we do not require ( r i ) to be in any sense“above” or “below” ( r j ) if i < j . x i : p introduced by ( r i ). Here we use in an essential way that the activecontractions have been accounted for in the the previous step.At the end of this process we see that π is equivalent to a preproof, roughly of thesame shape as above, with l copies of π being cut against the “trunk” of the tree at the“crown” of which is a preproof π ′′′ of x m +1 , . . . , x l , ∆ ′ ⊢ q derived from π ′′ . The first m of these copies of π are cut against variables x , . . . , x m introduced immediately beforethe cut, and the final l − m copies of π are cut against a proof of x m +1 , . . . , x i , ∆ ′ ⊢ q forsome m + 1 ≤ i ≤ l . These final l − m cuts may be eliminated using Lemma 2.31 (thisdoes not use either the inner or outer inductive hypothesis) noting that in the notation ofthat lemma, any variable in ∆ introduced by an (ax) in π is still introduced by an (ax)in the cut-free proof produced which is equivalent to π , so that the lemma may be appliedmultiple times. The remaining cuts on x , . . . , x m may then be sequentially eliminatedusing either (2.43) or (2.45) and the outer inductive hypothesis. The end result is acut-free preproof equivalent to π . Remark 2.33.
Note that the proof of cut-elimination (including the proof of the exis-tence of contraction normal from in Lemma 2.36) only uses τ -equivalence, co -equivalence,commuting conversions and the cut-elimination transformations (2.37)-(2.47) of Defini-tion 2.24 (note that all of these cut-elimination transformations are used). So the cut-elimination theorem holds without λ -equivalence or η -equivalence.In the rest of the section we develop the notion of a contraction normal form, whichwas used in the proof of cut-elimination. To avoid conflicting with the notation for thepreproof π there we denote the subject of following by ϕ . Definition 2.34.
Let ϕ be a cut-free preproof of x : p, ∆ ⊢ q . The contraction tree of( ϕ, x : p ) is the labelled oriented graph whose vertices are the final occurrence of x : p together with all weak ancestors of x : p in ϕ , where we draw an edge y : p → z : p if z : p is an immediate weak ancestor of y : p in ϕ . We label each edge with the correspondingdeduction rule. The final occurrence of x : p is the root of the tree.A slack vertex of ( ϕ, x : p ) is a trivalent vertex z : p in the contraction tree where theincoming edge y : p → z : p is labelled by any rule other than a contraction active withrespect to the final occurrence of x : p . The slack of ( ϕ, x : p ) is the number of slackvertices. We say ( ϕ, x : p ) is in contraction normal form if it has a slack of zero.25 xample 2.35. The contraction tree of the pair 2 , y : p ⊃ p of Example 2.11 is GFED@ABC y ′ ?>=<89:; y GFED@ABC y ′ ( L ⊃ ) O O ?>=<89:; y (ctr) d d ■■■■■■■■■■■■■ (ctr) : : ttttttttttttt ?>=<89:; y ( R ⊃ ) O O The pair 2 , y : p ⊃ p therefore has a slack of 1. Using (2.22) we see that 2 is equivalentunder ∼ p to the following proof 2 ′ in which the weak ancestors of the final y : p ⊃ p areagain marked in blue: (ax) x : p ⊢ p (ax) x : p ⊢ p (ax) x : p ⊢ p ( L ⊃ ) y ′ : p ⊃ p, x : p ⊢ p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ p ( R ⊃ ) y : p ⊃ p, y ′ : p ⊃ p ⊢ p ⊃ p (ctr) y : p ⊃ p ⊢ p ⊃ p The contraction tree of (2 ′ , y : p ⊃ p ) is GFED@ABC y ′ ?>=<89:; y GFED@ABC y ′ ( L ⊃ ) O O ?>=<89:; y ( R ⊃ ) O O GFED@ABC y ′ ( R ⊃ ) O O ?>=<89:; y (ctr) d d ■■■■■■■■■■■■■ (ctr) : : ttttttttttttt which has slack zero, so (2 ′ , y : p ⊃ p ) is in contraction normal form. Lemma 2.36.
Any cut-free preproof ϕ of x : p, ∆ ⊢ q is equivalent under ∼ p to a cut-freepreproof in contraction normal form.Proof. Consider a slack vertex y : p in ( ϕ, x : p ) with incoming edge labelled by the rule( r ). If ( r ) is (ex) using (2.7), (2.18),(2.19), or ( r ) is (weak) using (2.13), or ( r ) is ( R ⊃ )26sing (2.22), or ( r ) is ( L ⊃ ) using (2.25), (2.26), or ( r ) is an (ctr) which is not active forthe final occurrence of x : p by (2.17), we have an equivalence of preproofs ϕ ∼ p ϕ ′ underwhich the contraction tree is changed around y : p as follows: ?>=<89:; y GFED@ABC y ′ ?>=<89:; y (ctr) ^ ^ ❂❂❂❂❂❂❂❂❂ (ctr) @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ( r ) O O → ?>=<89:; y ( r ) K S GFED@ABC y ′ ( r ) K S ?>=<89:; y (ctr) ^ ^ ❂❂❂❂❂❂❂❂❂ (ctr) @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) The doubled arrows reflect the fact that in the case (2.7) there are two edges labelled (r)rather than one. Note that if ( r ) is ( L ⊃ ) then the contraction cannot be as in (2.34)because this contraction cannot be active with respect to the final occurrence of x : p . Let S = S ( ϕ, x : p ) be the set of vertices in the contraction tree with two outgoingedges, or what is the same, the set of active contractions for x : p in ϕ . We define the depth d ( y : p ) of such a vertex to be the number of rules ( r ) on the unique path from thatvertex to the root which are not active contractions for the final occurrence of x : p . Theproof of the lemma is by induction on the integer n ( ϕ, x : p ) = X y : p ∈ S d ( y : p ) . In the base case n = 0 the pair ( ϕ, x : p ) is already in contraction normal form and thereis nothing to prove. Given ( ϕ, x : p ) with n ( ϕ, x : p ) > y : p in ϕ and we let ϕ ∼ p ϕ ′ be the corresponding transformation as constructed above. Thereis a canonical bijection f : S ( ϕ, x : p ) −→ S ( ϕ ′ , x : p )and by inspection of the proof transformations d ( f ( z : p )) ≤ d ( z : p ) for every z : p in S ( ϕ, x : p ). By construction d ( f ( y : p )) < d ( y : p ) so that n ( ϕ ′ , x : p ) < n ( ϕ, x : p ) andthe claim follows by the inductive hypothesis. Remark 2.37.
In a cut-free preproof in contraction normal form, all the active contrac-tions appear the bottom of the tree but the pattern of these contractions is arbitrary. Inthe proof of Proposition 2.32, specifically in (2.48), we assume that the contractions maybe organised such that only the rightmost two ancestors in the list are ever contracted;this is possible by (2.31) and (2.32). Note that the transformation from ϕ to ϕ ′ may act nontrivially on other parts of the contractiontree: for instance if ϕ at y : p is as in (2.25) then there are two occurrences of ∆ (which, if ∆ containsa weak ancestor of x : p will contribute two vertices in the contraction tree) whereas ϕ ′ contains threeoccurrences of ∆. .2 The category of proofs Under the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic [31] a proofof Γ ⊢ p ⊃ q is viewed as a transformation from proofs of p to proofs of q . Thus it isnatural to view such proofs as morphisms from p to q in a category where objects areformulas, morphisms are proofs and composition is (cut). Throughout this section Γ is asequence of variables. Let Ψ ⊃ denote the set of formulas. Definition 2.38.
For a formula p we denote by Σ Γ p the set of preproofs of Γ ⊢ p . Definition 2.39.
Given a preproof π of Γ ⊢ p ⊃ q and x : p let π { x } denote π ...Γ ⊢ p ⊃ q (ax) x : p ⊢ p (ax) y : q ⊢ q ( L ⊃ ) z : p ⊃ q, x : p ⊢ q (cut)Γ , x : p ⊢ q (2 . ∼ p of y : p, z : q by (2.5) and (2.2). Lemma 2.40.
Any preproof π of Γ ⊢ p ⊃ q is equivalent under ∼ p to π { x } ... Γ , x : p ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q (2 . Proof.
By Theorem 2.29 we may assume π is cut-free. Consider walking the tree under-lying the preproof π starting from the root, and taking the right hand branch at every( L ⊃ ) rule. This walk must eventually encounter a ( R ⊃ ) rule. Take the first such ruleand by commuting conversions (2.20),(2.21),(2.22),(2.23) move this rule down so that itis the final rule in a preproof of the form ψ ...Γ , x : p ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q (2 . π under ∼ p . Now observe that π { x } is equivalent under ∼ p to ψ ...Γ , x : p ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q (ax) x : p ⊢ p (ax) y : q ⊢ q ( L ⊃ ) z : p ⊃ q, x : p ⊢ q (cut)Γ , x : p ⊢ q which is by (2.45) equivalent to 28ax) x : p ⊢ p ψ ...Γ , x : p ⊢ q (cut) x : p, Γ ⊢ q (ax) y : q ⊢ q (cut) x : p, Γ ⊢ q (ex)Γ , x : p ⊢ q which is equivalent by (2.37),(2.38),(2.11) to ψ which completes the proof. Definition 2.41.
The category S Γ has objects Ψ ⊃ ∪ { } and morphisms S Γ ( p, q ) = Σ Γ p ⊃ q / ∼ p S Γ ( , q ) = Σ Γ q / ∼ p with special cases S Γ ( p, ) = {∗} , S Γ ( , ) = {∗} . For formulas p, q, r composition S Γ ( q, r ) × S Γ ( p, q ) → S Γ ( p, r )sends the pair ( ψ, π ) to the proof ψ ◦ π given by π { x } ...Γ , x : p ⊢ q ψ { y } ...Γ , y : q ⊢ r (cut)Γ , x : p, Γ ⊢ r (ex / ctr)Γ , x : p ⊢ r ( R ⊃ )Γ ⊢ p ⊃ r (2 . p, q the map S Γ ( p, q ) × S Γ ( , p ) → S Γ ( , q ) sends ( ψ, π ) to π ...Γ ⊢ p ψ { x } ...Γ , x : p ⊢ q (cut)Γ , Γ ⊢ q (ex / ctr)Γ ⊢ q (2 . S Γ ( , q ) × S Γ ( p, ) → S Γ ( p, q ) sends ( π, ∗ ) to π ...Γ ⊢ q (weak)Γ , x : p ⊢ q ( R ⊃ )Γ ⊢ p ⊃ q (2 . S Γ ( , p ) × S Γ ( , ) → S Γ ( , p ) is the projection.29ote that the composition ψ ◦ π depends as a preproof on the choices of intermediatevariables x : p, y : q but is independent of these choices by (2.4) and (2.2). The identitymorphism 1 p : p −→ p in S Γ for a formula p is the proof(ax) x : p ⊢ p (weak)Γ , x : p ⊢ p ( R ⊃ )Γ ⊢ p ⊃ p We define a category L whose objects are the types of simply-typed lambda calculus,and whose morphisms are the terms of that calculus. The natural desiderata for sucha category are that the fundamental algebraic structure of lambda calculus, functionapplication and lambda abstraction, should be realised by categorical algebra.We assume familiarity with simply-typed lambda calculus; some details are recalled inAppendix A. Following Church’s original presentation our lambda calculus only containsfunction types and Φ → denotes the set of simple types. We write Λ σ for the set of α -equivalence classes of lambda terms of type σ . Definition 3.1 (Category of lambda terms).
The category L has objectsob( L ) = Φ → ∪ { } and morphisms given for types σ, τ ∈ Φ → by L ( σ, τ ) = Λ σ → τ / = βη L ( , σ ) = Λ σ / = βη L ( σ, ) = { ⋆ }L ( , ) = { ⋆ } , where ⋆ is a new symbol. For σ, τ, ρ ∈ Φ → the composition rule is the function L ( τ, ρ ) × L ( σ, τ ) −→ L ( σ, ρ )( N, M ) λx σ . ( N ( M x )) , where x / ∈ FV( N ) ∪ FV( M ). We write the composite as N ◦ M . In the remaining specialcases the composite is given by the rules L ( τ, ρ ) × L ( , τ ) −→ L ( , ρ ) , N ◦ M = ( N M ) , L ( , ρ ) × L ( , ) −→ L ( , ρ ) , N ◦ ⋆ = N , L ( , ρ ) × L ( σ, ) −→ L ( σ, ρ ) , N ◦ ⋆ = λt σ . N , where in the final rule t / ∈ FV( N ). All other cases are trivial. Note that these functions,which have been described using a choice of representatives from a βη -equivalence class,are nonetheless well-defined. 30or terms M, N the expression M = N always means equality of terms (that is, up to α -equivalence) and we write M = βη if we want to indicate equality up to βη -equivalence(for example as morphisms in the category L ). Since the free variable set of a lambdaterm is not invariant under β -reduction, some care is necessary in defining the category L Q below. Let ։ β denote multi-step β -reduction [29, Definition 1.3.3]. Lemma 3.2. If M ։ β N then FV( N ) ⊆ FV( M ) . Definition 3.3.
Given a term M we defineFV β ( M ) = \ N = β M FV( N )where the intersection is over all terms N which are β -equivalent to M .Clearly if M = β M ′ then FV β ( M ) = FV β ( M ′ ). Lemma 3.4.
Given terms M : σ → ρ and N : σ we have FV β (( M N )) ⊆ FV β ( M ) ∪ FV β ( N ) . Proof.
We may assume
M, N β -normal, in which case there is a chain of β -reductions( M N ) ։ β \ ( M N ) whence we are done by Lemma 3.2.By the same argument
Lemma 3.5.
Given M : σ → ρ and N : τ → σ we have (3.1) FV β ( M ◦ N ) ⊆ FV β ( M ) ∪ FV β ( N ) . Given a set Q of variables we write Λ Qσ for the set of lambda terms M of type σ withFV( M ) ⊆ Q . Let = βη denote the induced relation on this subset of Λ σ . Lemma 3.6.
For any type σ and set Q of variables the image of the injective map (3.2) Λ Qp / = βη −→ Λ p / = βη is the set of equivalence classes of terms M with FV β ( M ) ⊆ Q .Proof. Since the simply-typed lambda calculus is strongly normalising [29, Theorem 3.5.1]and confluent [29, Theorem 3.6.3] there is a unique normal form c M in the β -equivalenceclass of M , and FV β ( M ) = FV( c M ). Hence if FV β ( M ) ⊆ Q then FV( c M ) ⊆ Q and so M is in the image of (3.2). 31 efinition 3.7. For a set of variables Q we define a subcategory L Q ⊆ L byob( L Q ) = ob( L ) = Φ → ∪ { } and for types σ, ρ L Q ( σ, ρ ) = { M ∈ L ( σ, ρ ) | FV β ( M ) ⊆ Q } , L Q ( , σ ) = { M ∈ L ( , σ ) | FV β ( M ) ⊆ Q } , L Q ( σ, ) = L ( σ, ) = { ⋆ } , L Q ( , ) = L ( , ) = { ⋆ } . Note that the last two lines have the same form using the convention that FV β ( ⋆ ) = ∅ .The fact that L Q is a subcategory follows from Lemma 3.5. Remark 3.8.
We sketch how function application and lambda abstraction in the simply-typed lambda calculus are realised as natural categorical algebra in L . Function appli-cation is composition, and lambda abstraction is given by a universal property involvingfactorisation of morphisms in L through morphisms in L Q .To explain, let M ∈ L ( σ, ρ ) be a morphism and q : τ a variable. We can consider theset of all commutative diagrams in L of the form(3.3) σ M / / f (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ ρκ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ where q / ∈ FV β ( f ). Taking f = λq.M gives the universal such factorisation. Remark 3.9.
In the standard approach to associating a category to the simply-typedlambda calculus, due to Lambek and Scott [19, § I.11], one extends the lambda calculus toinclude product types and the objects of the category C → , × are the types of the extendedcalculus (which includes an empty product ) and the set C → , × ( σ, ρ ) is a set of equivalenceclasses of pairs ( x : σ, M : ρ ) where x is a variable and M is a term with FV( M ) ⊆ { x } .The relation to the approach given above is as follows: for Q finite L Q may be viewedas a polynomial category over L ∅ and if we write L = ∅ ⊆ L ∅ for the subcategory whoseobjects are types Φ → there is an equivalence of categories C → ∼ = L = ∅ where C → denotesthe full subcategory of C → , × whose objects are elements of the set Φ → . We have defined a category of formulas and proofs S Γ in intuitionistic sequent calculus(Definition 2.41) for any finite sequence Γ of variables, and a category of types and terms32 Q in simply-typed lambda calculus (Definition 3.7) for any set of variables Q . In logicwe have associated variables to formulas and in lambda calculus to types, but identifyingatomic formulas with atomic types and ⊃ with → gives a bijection between the set Ψ ⊃ of formulas and the set Φ → of types, and we now make this identification.Given a sequence Γ of variables we denote by [Γ] the underlying set of variables[ x : p , . . . , x n : p n ] = { x : p , . . . , x n : p n } . We prove that S Γ ∼ = L [Γ] if Γ is repetition-free. To define the precise translation from proofsto lambda terms, we have to pay close attention to the variables annotating hypothesesin proofs, and this requires some preliminary comments.Given a preproof π of Γ ⊢ p an equivalence class x of ≈ str (Definition 2.9) can bewritten as a sequence x = ( x , . . . , x n ) of copies x i of a variable x : p , with x introducedin one of (ax) , (weak) , ( L ⊃ ) and x n either in the antecedent of the final sequent (labellingthe root node of π ) or eliminated in (cut) , (ctr) , ( R ⊃ ) or ( L ⊃ ). If x n is in the antecedentof the final sequent we say x is a boundary class otherwise it is an interior class . Definition 4.1.
A preproof π is well-labelled if for any interior class x of occurrences ofa variable x : p in π , the only occurrences of x : p in π are the ones in x . Lemma 4.2.
Every preproof is equivalent under ∼ p to a well-labelled preproof.Proof. Using α -equivalence. Example 4.3.
The preproof 2 of Example 2.11 is not well-labelled, but it is equivalentunder ∼ α to the following well-labelled preproof:(ax) x : p ⊢ p (ax) x ′ : p ⊢ p (ax) x ′′ : p ⊢ p ( L ⊃ ) x ′ : p, y ′ : p ⊃ p ⊢ p ( L ⊃ ) x : p, y : p ⊃ p, y ′ : p ⊃ p ⊢ p (ctr) x : p, y : p ⊃ p ⊢ p ( R ⊃ ) y : p ⊃ p ⊢ p ⊃ p In the following Γ is a sequence of variables, possibly empty, with Q = [Γ]. Given asequent Γ ⊢ p we let Σ Γ p denote the set of all preproofs of that sequent, and given a finiteset Q of variables we denote by Λ Qp the subset of Λ p consisting of terms with free variablescontained in Q . Below we make use of the substitution operation of Definition A.1. Definition 4.4 (Translation).
We let(4.1) f Γ p : Σ Γ p −→ Λ Qp denote the function defined on well-labelled preproofs by annotating the succedent of thededuction rules of Definition 2.1 with lambda terms so that each preproof may be readas a construction of a term: 33ax) x : p ⊢ x : p (4 . ⊢ N : p ∆ , x : p, Θ ⊢ M : q (cut)Γ , ∆ , Θ ⊢ M [ x := N ] : q (4 . , x : p, y : p, ∆ ⊢ M : q (ctr)Γ , x : p, ∆ ⊢ M [ y := x ] : q (4 . , ∆ ⊢ M : q (weak)Γ , x : p, ∆ ⊢ M : q (4 . , x : p, y : q, ∆ ⊢ M : r (ex)Γ , y : q, x : p, ∆ ⊢ M : r (4 . , x : p, ∆ ⊢ M : q ( R ⊃ )Γ , ∆ ⊢ λx.M : p ⊃ q (4 . ⊢ N : p ∆ , x : q, Θ ⊢ M : r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ ⊢ M [ x := ( y N )] : r (4 . π annotated as above, f Γ p ( π ) is the lambda term annotatingthe succedent on the root of π . If π is not well-labelled, we first α -rename as necessaryusing (2.2), (2.3), (2.4), (2.5) any interior equivalence class under ≈ str to obtain a preproof π ′ which is well-labelled and define f Γ p ( π ) := f Γ p ( π ′ ). This term is independent of choicesmade during α -renaming. We refer to f Γ p ( π ) as the translation of π . Remark 4.5.
The function from sequent calculus proofs to derivations in natural deduc-tion is implicit in Gentzen [9] as the concatenation of a translation from sequent calculusLJ to the Hilbert-style system LHJ [9, § V.5] and a translation from LHJ to natural de-duction NJ [9, § V.3]. The map from LJ to NJ is also discussed very briefly by Prawitz[26, p.90-91]. The translation to natural deduction appears explicitly in Zucker [35] andthe translation to lambda terms in Mints [22]. For a textbook treatment of the formersee [32, § § Remark 4.6.
The constraint that π is well-labelled is necessary for Definition 4.4 tocapture the intended translation from proofs to lambda terms. For example if there areadditional occurrences of y in the part of the antecedent labelled ∆ in the numerator ofthe contraction rule which are not in the same ≈ str -equivalence class as the occurrence y being contracted, then the substitution M [ y := x ] will rewrite these other occurrences to x , which is not what we intend. Definition 4.7.
We define ∼ o to be the smallest compatible equivalence relation on pre-proofs containing the union of α -equivalence, τ -equivalence, commuting conversions, co -equivalence and λ -equivalence. Lemma 4.8.
Let π, π ′ be preproofs of Γ ⊢ p . Then i) If π ∼ o π ′ then f Γ p ( π ) = f Γ p ( π ′ ) .(ii) If π ∼ p π ′ then f Γ p ( π ) = βη f Γ p ( π ′ ) .Proof. By inspection of the generating relations.
Remark 4.9.
A more precise statement than Lemma 4.8 is that if π, π ′ are related by anyof the generating relations for proof equivalence other than (2.36), (2.45) then f Γ p ( π ) = f Γ p ( π ). The translation of (2.36) is η -equivalence(ax) x : p ⊢ x : p (ax) y : q ⊢ y : q ( L ⊃ ) z : p ⊃ q, x : p ⊢ ( z x ) : q ( R ⊃ ) z : p ⊃ q ⊢ λx. ( z x ) : p ⊃ q ∼ η (ax) z : p ⊃ q ⊢ z : p ⊃ q (4 . β -reduction. Example 4.10.
The lambda term associated to the well-labelled 2 from Example 4.3 is(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p (ax) x ′′ : p ⊢ x ′′ : p ( L ⊃ ) y ′ : p ⊃ p, x ′ : p ⊢ ( y ′ x ′ ) : p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ ( y ′ ( y x )) : p (ctr) y : p ⊃ p, x : p ⊢ ( y ( y x )) : p ( R ⊃ ) y : p ⊃ p ⊢ λx. ( y ( y x )) : p ⊃ p Lemma 4.11.
For any sequence Γ there is a functor F Γ : S Γ −→ L Q which is the identityon objects and which is defined on morphisms for formulas p, q by F Γ ( p, q ) = f Γ p ⊃ q : S Γ ( p, q ) −→ L Γ ( p, q ) ,F Γ ( , q ) = f q : S Γ ( , q ) −→ L Γ ( , q ) . Proof.
For any formula p it is clear that F Γ (1 p ) = 1 p . If p, q, r are formulas we need toshow that the diagram(4.10) S Γ ( q, r ) × S Γ ( p, q ) / / f Γ q ⊃ r × f Γ p ⊃ q (cid:15) (cid:15) S Γ ( p, r ) f Γ p ⊃ r (cid:15) (cid:15) L Q ( q, r ) × L Q ( p, q ) / / L Q ( p, r )commutes. Let a pair of preproofs ψ, π of Γ ⊢ q ⊃ r and Γ ⊢ p ⊃ q respectively be given.We may assume by Lemma 2.40 that ψ, π are obtained respectively by ( R ⊃ ) rules frompreproofs ψ { y } , π { x } of sequents Γ , y : q ⊢ r and Γ , x : p ⊢ q . If the translations of thesepreproofs are M, N respectively then the following annotated proof tree35 { x } ...Γ , x : p ⊢ N : q ψ { y } ...Γ , y : q ⊢ M : r (cut)Γ , x : p, Γ ⊢ M [ y := N ] : r (ex / ctr)Γ , x : p ⊢ M [ y := N ] : r ( R ⊃ )Γ ⊢ λx.M [ y := N ] : p ⊃ r (4 . f Γ p ⊃ r ( ψ ◦ π ) = λx.M [ y := N ]. The other way around (4.10) gives f Γ q ⊃ r ( ψ ) ◦ f Γ p ⊃ q ( π ) = ( λy.M ) ◦ ( λx.N )= λx. ( λy.M ( λx.N x ))= β λx. ( λy.M N )= β λx.M [ y := N ]as required. The remaining special cases are left to the reader. Lemma 4.12. If π is cut-free then f Γ p ( π ) is a β -normal form.Proof. We may assume that π is well-labelled. Without the cut rule the only occurrencesof applications are those introduced by (4.8) which have the form ( y N ) with y a variableand so f Γ p ( π ) contains no β -redexes.The translation from preproofs to lambda terms is not well-behaved if the sequent Γcontains repetitions, as the following example shows: Example 4.13.
The lambda term associated to the following well-labelled preproof 001is (part of) the standard representation in lambda calculus of the binary integer 001:(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p (ax) x ′′ : p ⊢ x ′′ : p (ax) x ′′′ : p ⊢ x ′′′ : p ( L ⊃ ) y ′′ : p ⊃ p, x ′′ : p ⊢ ( y ′′ x ′′ ) : p ( L ⊃ ) y ′ : p ⊃ p, x ′ : p, y ′′ : p ⊃ p ⊢ ( y ′′ ( y ′ x ′ )) : p ( L ⊃ ) y : p ⊃ p, x : p, y ′ : p ⊃ p, y ′′ : p ⊃ p ⊢ ( y ′′ ( y ′ ( y x ))) : p (ex) y : p ⊃ p, y ′ : p ⊃ p, x : p, y ′′ : p ⊃ p ⊢ ( y ′′ ( y ′ ( y x ))) : p (ctr) y : p ⊃ p, x : p, y ′′ : p ⊃ p ⊢ ( y ′′ ( y ( y x ))) : p Note that the following preproof, denoted 001 ′ is well-labelled and differs only in the vari-able annotations chosen to be introduced by one of the ( L ⊃ ) rules (shown highlighted):(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p (ax) x ′′ : p ⊢ x ′′ : p (ax) x ′′′ : p ⊢ x ′′′ : p ( L ⊃ ) y : p ⊃ p, x ′′ : p ⊢ ( y x ′′ ) : p ( L ⊃ ) y ′ : p ⊃ p, x ′ : p, y : p ⊃ p ⊢ ( y ( y ′ x ′ )) : p ( L ⊃ ) y : p ⊃ p, x : p, y ′ : p ⊃ p, y : p ⊃ p ⊢ ( y ( y ′ ( y x ))) : p (ex) y : p ⊃ p, y ′ : p ⊃ p, x : p, y : p ⊃ p ⊢ ( y ( y ′ ( y x ))) : p (ctr) y : p ⊃ p, x : p, y : p ⊃ p ⊢ ( y ( y ( y x ))) : p ′ of the same sequent as 001 ′ which is not equivalent under ∼ p to 001 ′ but whose translationis the same lambda term. The clash of variables is innocuous in sequent calculus becausewe have enough additional information to disambiguate the role of the two variables, butthis information is not present in the lambda term. This shows that the map from ∼ p -equivalence classes of preproofs to βη -equivalence classes of terms is not injective if Γ hasmultiple occurrences of variables.Another, simpler, example of this phenomenon is the following pair of preproofs wherethe variable introduced by the weakening is highlighted:(ax) z : s ⊢ z : s (weak) z : s, z : s ⊢ z : s (ax) z : s ⊢ z : s (weak) z : s, z : s ⊢ z : s (4 . z : s we see that theycannot be equivalent under ∼ p but their translations are the same lambda term. Definition 4.14.
We say that Γ is repetition-free it for any variable x : p the sequence Γcontains at most one occurrence of x : p . Theorem 4.15 (Gentzen-Mints-Zucker duality) . If Γ is repetition-free then the transla-tion functor F Γ : S Γ −→ L Q is an isomorphism of categories.Proof. The functor is a bijection on objects, so we have to show that it is fully faithfuland this follows immediately from Proposition 4.16 below, using Lemma 3.6.
Proposition 4.16.
For any sequent Γ ⊢ p with Γ repetition-free there is a bijection (4.13) Σ Γ p / ∼ p ∼ = / / Λ Qp / = βη induced by the function f Γ p . We prove the proposition in a series of lemmas. Recall that by combining Theorem 2.29and Lemma 4.2 any preproof is equivalent under ∼ p to a cut-free well-labelled preproof.Recall from Definition 2.30 the notion of a contraction rule active for a variable occurrence. Definition 4.17.
A preproof π of Γ ⊢ p is called a ( L ⊃ ) -normal form if(i) it is cut-free and well-labelled(ii) no contraction is active for a variable occurrence eliminated in a ( L ⊃ ) rule.(iii) no variable occurrence introduced by a (weak) rule is equivalent under the relation ≈ str to a variable occurrence eliminated by a ( L ⊃ ) rule. Lemma 4.18.
Every preproof π is equivalent under ∼ p to a ( L ⊃ ) -normal form. roof. The proof parallels the proof of cut-elimination in Proposition 2.32. We mayassume π is cut-free and well-labelled. Call a ( L ⊃ ) rule in π defective if either condition(ii) or (iii) of Definition 4.17 fails for that particular rule. Applying the following reasoningto each defective ( L ⊃ ) rule in π from greatest to lowest height, it suffices to consider thecase where π is π ...∆ ⊢ p π ... x : q, Θ ⊢ s ( L ⊃ ) y : p ⊃ q, ∆ , Θ ⊢ s (4 . L ⊃ )-normal forms π and π . By Lemma 2.36 we can put the pair ( π , x : q ) incontraction normal form, so that π equivalent under ∼ p to a preproof of the form π ...∆ ⊢ p π ′ ... x , ..., x l , Θ ⊢ s (ctr) x , ..., x l − , Θ ⊢ s ... (ctr) x , x , Θ ⊢ s (ctr) x , Θ ⊢ s ( L ⊃ ) y : p ⊃ q, ∆ , Θ ⊢ s where x = x and we drop the formula q from the notation. Considering the algorithmimplicit in the proof of Lemma 2.36 we see that we may assume π ′ to be a ( L ⊃ )-normalform containing no active contractions for x . By zero or more applications of (2.34) andcommuting conversions we obtain a preproof equivalent to π of the form π ...∆ ⊢ p π ...∆ ⊢ p π ...∆ ⊢ p π ′ ... x , . . . , x l , Θ ⊢ s ( L ⊃ ) y l , ∆ , x , . . . , x l − , Θ ⊢ s ... y , ∆ , . . . , y l , ∆ , x , x , Θ ⊢ s ( L ⊃ ) y , ∆ , y , ∆ , . . . , y l , ∆ , x , Θ ⊢ s ( L ⊃ ) y , ∆ , . . . , y l , ∆ , Θ ⊢ s (ex / ctr) y , ∆ , Θ ⊢ s (4 . y i is a variable of type p ⊃ q and y = y . Note that no contraction in thispreproof is active for any of the variables eliminated in a ( L ⊃ ) rule. For 1 ≤ i ≤ l therule which introduces x i in π ′ must be either (ax) or (weak). If x i is introduced by (weak)then this rule can be moved, using commuting conversions, down to the corresponding( L ⊃ ) rule and then eliminated with (2.35). Repeating this finitely many times yields a( L ⊃ )-normal form π ′ equivalent to π . 38 emma 4.19. Given a preproof π ... ∆ ⊢ p π ... x : q, Θ ⊢ s ( L ⊃ ) y : p ⊃ q, ∆ , Θ ⊢ s (4 . which is a ( L ⊃ ) -normal form, the variable x occurs as a free variable in f x : q, Θ s ( π ) .Proof. By induction on the height of π . The base case is where π is an axiom rule,which is clear. Suppose the height of π is positive. If x : q is introduced in an axiomrule, then no subsequent rule can remove it. If x : q is introduced by a ( L ⊃ ) rule thenby the inductive hypothesis that ( L ⊃ ) rule eliminates a variable z : s which occurred asa free variable in the translation R of its right hand branch, yielding a term R [ z := ( x L )]which contains an occurrence of x as a free variable. Lemma 4.20.
Suppose π is a preproof of Γ ⊢ p which satisfies • π is a ( L ⊃ ) -normal form • π contains no variable occurrences introduced by (weak) which are ≈ str -equivalentto occurrences in the final sequent Γ .Then with M = f Γ p ( π ) we have [Γ] = FV( M ) .Proof. By construction of the translation FV( M ) ⊆ [Γ], so we need only argue that anyvariable x : q in Γ occurs as a free variable in M . This is clear if a strong ancestor of x : q is introduced by (ax). If a strong ancestor of x : q is introduced by ( L ⊃ ) then it followsfrom Lemma 4.19.Recall from Definition 4.7 the relation ∼ o which is weaker than ∼ p . Lemma 4.21. If π is a preproof of Γ ⊢ p which is a ( L ⊃ ) -normal form and f Γ s ( π ) is avariable z : s then π is equivalent under ∼ o to (ax) z : s ⊢ s (weak)Γ ⊢ s (4 . We call a preproof such as (4.17) a variable normal form .Proof. We deduce by inspection of the translation in Definition 4.4 that the only possiblerules which appear in π are (ax) , (ctr) , (weak) , (ex), or a ( L ⊃ ) rule in which the eliminatedvariable does not occur, so that no substitution takes place. But by Lemma 4.19 no such( L ⊃ ) rule can occur in π from which we deduce that π must be ( L ⊃ )-free.The preproof π of Γ ⊢ s contains precisely one (ax) rule and otherwise consists entirelyof structural rules. Consider an occurrence of (ctr) in π which contracts x : p, x ′ : p . Astrong ancestor of either x : p or x ′ : p must be introduced by (weak) and using commuting39onversions we may move this rule down the tree and eliminate it with the contractionusing (2.33) and Remark 2.20. Thus π is equivalent under ∼ o to a preproof containing nocontractions, and similarly one may also using commuting conversions and (2.11),(2.8) toeliminate all exchanges. The resulting preproof is of the desired form. Note that Γ maycontain multiple occurrences of z : s . Lemma 4.22. If π is a preproof of Γ ⊢ p which is cut-free and well-labelled and f Γ q ⊃ r ( π ) is an abstraction λx.N then π is equivalent under ∼ o to a preproof of the form ψ ... ∆ , x : q, ∆ ′ ⊢ r ( R ⊃ )Γ ⊢ q ⊃ r (4 . where f ∆ ,x : q, ∆ ′ r ( ψ ) = N . We call such a preproof an abstraction normal form .Proof. By the proof of Lemma 2.40 we see π is equivalent under ∼ o to a preproof (4.18)using relations that do not change the translated term, so that the translation of (4.18)is still λx.N . From this the claim follows. Definition 4.23.
Given a lambda term M let FV seq ( M ) denote the sequence of distinctfree variables in M ordered by first occurrence. Example 4.24.
Let M = ( y ′′ ( y ( y x ))) : p be as in Example 4.13. Then FV seq ( M ) is thesequence y ′′ : p ⊃ p, y : p ⊃ p, x : p . Definition 4.25. A ladder is a sequence of rules of the formΓ , x : p , . . . , x n : p n , y : q, ∆ ⊢ q (ex)Γ , x : p , . . . , y : q, x n : p n , ∆ ⊢ q ... (ex)Γ , y : q, x : p , . . . , x n : p n , ∆ ⊢ p (4 . tail index of a ladder is the position of y : q in Γ , y : q, x : p , . . . , x n : p n , ∆ ⊢ p . Aladder is maximal in a preproof π if there is no larger ladder in π containing it. We write(lad) i for a maximal ladder with tail index i . Definition 4.26. A derived contraction is a sequence of rules of the formΓ , x : p, ∆ , y : p, Θ ⊢ q (ex)Γ , x : p, y : p, ∆ , Θ ⊢ q (ctr)Γ , x : p, ∆ , Θ ⊢ p (4 . tail index of a derived contraction is the position of x : p in Γ , x : p, ∆ , Θ and the head index is the position of y : p in Γ , x : p, ∆ , y : p, Θ. We write (dctr) i,j to standfor a derived contraction with tail index i and head index j . A derived contraction in apreproof π is maximal if there is no larger derived contraction in π containing it.40 efinition 4.27. The index of a weakening rule, with reference to the rule schemata ofDefinition 2.1, is the position of x : p in Γ , x : p, ∆. We write (weak) i for a weakening rulewith index i . Definition 4.28.
A preproof π of Γ ⊢ p is well-ordered if Γ = FV seq ( M ) where M = f Γ p ( π ).It is more difficult to give a normal form for preproofs whose translation is an applica-tion. Note that technically speaking we should require that Γ contains no variable fromthe canonical series (4.22). Lemma 4.29. If Γ is repetition-free and π is a preproof of Γ ⊢ p which is a ( L ⊃ ) -normalform and f Γ p ( π ) is an application ( M M ) with M : r ⊃ p and M : r then π is equivalentunder ∼ o to a preproof of the form τ b ... Γ b ⊢ L b : p b τ b − ... Γ b − ⊢ L b − : p b − τ ... Γ ⊢ L : p ζ ... ∆ ⊢ R : s (ax) x : p ⊢ x : p ( L ⊃ ) y, ∆ ⊢ ( y R ) : p ( L ⊃ ) y , Γ , ∆ ⊢ (( y L ) R ) : p ... y b − , Γ b − , . . . , Γ , ∆ ⊢ p ( L ⊃ ) y b − , Γ b − , . . . , Γ , ∆ ⊢ p ( L ⊃ ) y b , Γ b , . . . , Γ , ∆ ⊢ p (dctr)Θ ⊢ p (lad)Θ ′ ⊢ p (weak)Γ ⊢ ( M M ) : p with the following properties:(i) ζ and τ j are ( L ⊃ ) -normal forms for ≤ j ≤ b .(ii) ζ and τ j are well-ordered for ≤ j ≤ b .(iii) No variable occurrence in Γ has more than one weak ancestor in ∆ , and no variableoccurrence in Γ has more than one weak ancestor in Γ j for ≤ j ≤ b .(iv) The series of derived contractions y b , Γ b , . . . , Γ , ∆ ⊢ p (dctr)Θ ⊢ p (4 . is of the form (dctr) a ,b , (dctr) a ,b , . . . , (dctr) a m ,b m with ( a , b ) ≤ ( a , b ) ≤ · · · ≤ ( a m , b m ) in the lexicographic order. v) Let Λ( u ) be the sequence obtained from the numerator in (4.21) by deleting from y b , Γ b , . . . , Γ , ∆ any variable occurrence which is either not of type u or which is astrong ancestor of an occurrence in Θ . Then for any u we require that Λ( u ) is equalto an initial segment (possibly empty) of some fixed “canonical series” of variables (4.22) ℵ u : u, ℵ u : u, ℵ u : u, . . . (vi) The series of weakenings Θ ′ ⊢ p (weak)Γ ⊢ p (4 . is of the form (weak) d , (weak) d , . . . , (weak) d m with d < d < · · · < d m .(vii) The series of maximal ladders Θ ⊢ p (lad)Θ ′ ⊢ p (4 . is of the form (lad) c , (lad) c , . . . , (lad) c n with c < c < . . . < c n .This representation is unique, in the following sense: any other such representation in-volves the same index b , the same sequents Γ j ⊢ p j and the same lambda terms R and L j for ≤ j ≤ b . We call such a preproof an application normal form .Proof. Walk the tree underlying the preproof π starting from the root, taking the righthand branch at every ( L ⊃ ) rule, and stop at the first instance of the ( L ⊃ ) rule whichsatisfies the following property: the preproof constituting the right hand branch has forits translation under Definition 4.4 a variable x : p and this is the variable eliminated bythe ( L ⊃ ) rule. Note that a ( L ⊃ ) rule satisfying this property will be encountered on thewalk, because M is an application. By Lemma 4.21 the preproof π is therefore equivalentunder ∼ o to a preproof of the form ζ ...∆ ⊢ R : s (ax) x : p ⊢ x : p ( L ⊃ ) y : s ⊃ p, ∆ ⊢ ( y R ) : p ...Γ ⊢ ( M M ) : p (4 . L ⊃ ) rule.We refer to the sequence of deduction rules connecting the root of the preproof to thedisplayed ( L ⊃ ) rule as the porch (note that the preproof may contain other branches that42eet the displayed preproof as left hand branches at deduction rules within the porch).The porch may contain (ctr), (ex), (weak) and ( L ⊃ ) rules. Since π is a ( L ⊃ )-normalform none of these weakenings or contractions are relevant to the variables eliminated by( L ⊃ ) rules in the porch, so we may use commuting conversions to ensure that the ( L ⊃ )rules are all above any of these other rules.We index the ( L ⊃ ) rules on the porch, from top to bottom, by indices ατ α ...Γ α ⊢ p α ... t α : q α , Λ α ⊢ p ( r α ) y α : p α ⊃ q α , Γ α , Λ α ⊢ p We now migrate ( L ⊃ ) rules on the porch into ζ and the τ α branches. The variable t either has a strong ancestor in ζ or its strong ancestor is the y : s ⊃ p introduced by the( L ⊃ ) displayed in (4.25). In the former case, we can by (2.30) move the rule ( r ) up into ζ . In the latter case, we do nothing. Now assume that α > β < α the variable t β is introduced by one of the previous ( L ⊃ ) rules on the porch. If t α isintroduced by one of the previous ( L ⊃ ) rules on the porch then we do nothing, otherwiseif t α is introduced in ζ (resp. τ β for β < α ) then we use (2.29), (2.30) to move ( r α ) into ζ (resp. τ β ). These applications of (2.29), (2.30) may introduce (ex) rules onto the porch,which may either be absorbed into ( L ⊃ ) rules by (2.10) or moved to the bottom of theporch as above. Proceeding in this way through all the indices α ∈ { , . . . , b } in increasingorder completes the migration.This migration procedure shows that we may as well have assumed from the beginningthat the only ( L ⊃ ) rules on the porch are those in which t α is introduced by the previous( L ⊃ ) rule on the porch. We now make this assumption. Using commuting conversions wecan move any (weak) rules in ζ (resp. any τ j ) which introduce variables equivalent under ≈ str to an occurrence in ∆ (resp. Γ j ) down to the bottom of the porch. This shows that π is equivalent under ∼ o to a preproof of the form given in the statement of the lemmawhere ζ and all the τ j are ( L ⊃ )-normal forms satisfying the hypotheses of Lemma 4.20so that [∆] = FV( R ) and [Γ j ] = FV( L j ). Using (2.31),(2.32) and commuting conversionswe may also assume that the condition (iii) is satisfied by moving contractions up intothe branches.Now we use for the first time the hypothesis that Γ is repetition-free. If any repetitionsoccurred in ∆ or one of the Γ j ’s then this would have to be corrected by a contractionon the porch, which by (iii) is impossible. So ∆ and all the Γ j are also repetition-free.Without loss of generality we may therefore assume, possibly inserting exchanges into ζ and τ j that ∆ = FV seq ( R ) and Γ j = FV seq ( L j ) which is condition (ii). Condition (iv) canbe arranged using (2.31), (2.32), (2.33). In the notation of (v) observe that for any u thesequence Λ( u ) consists of variable occurrences which are eliminated in contraction ruleswithin (4.21) and so by α -equivalence (2.3) we can rename them as we wish, providedthe result is well-labelled. In particular we can rename them according to the specified43ules with respect to a predetermined canonical series. This completes the proof of theexistence of an application normal form and it only remains to prove uniqueness.Considering the translation of the normal form we see that M = ( M M ) is obtainedfrom(4.26) (cid:0) ( · · · (( y b L b ) L b − ) · · · L ) R (cid:1) by some number of contractions. Thus the index b and the types p , . . . , p b , s in the normalform can be read off from the term M . Suppose that(4.27) ( M M ) = (cid:0) ( · · · (( y b L ′ b ) L ′ b − ) · · · L ′ ) R ′ (cid:1) . Now consider the sequence(4.28) y b , FV seq ( L ′ b ) , . . . , FV seq ( L ′ ) , FV seq ( R ′ )and perform the following operation: if any variable z : u is repeated in this sequence thenreplace all but the first occurrence by a special symbol • u associated to u but independentof z . This is done for every type u and every variable of type u before the next step. Inthe next step, for each type u replace all the occurrences of • u in order by variables takenfrom the canonical series (4.22) for u . By conditions (ii),(iii),(iv),(v) the result of thisoperation is the sequence y b , Γ b , . . . , Γ , ∆ which is therefore determined by M and isindependent of any choices made above. Suppose that free variables z : u , . . . , z k : u k in L ′ j are replaced by this procedure with ℵ u t , . . . , ℵ u k t k . Then L j = L ′ j [ z := ℵ u t , . . . , z k := ℵ u k t k ]and similarly for R , which completes the proof of the uniqueness statement.Actually the application normal form is unique in a much stronger sense, but we returnto this in Section 4.1. We note that b = 0 is allowed in the definition of an applicationnormal form, in which case there is a single ( L ⊃ ) rule with left branch ζ , followed byexchanges, contractions and weakenings as above. Lemma 4.30.
Let π be an application normal form in which the rule series (4.23) , (4.24) are empty. Then π is well-ordered.Proof. Let π be an application normal form as in the statement of Lemma 4.29. ByLemma 4.20 we have [Γ] = FV( M ). Suppose that z : u, z ′ : u ′ appear in this order withinΓ so that their strong ancestors appear in the same order within y b , Γ b , . . . , Γ , ∆. If z = y b then it is clear that the first free occurrence of z ′ : u ′ in M appears after the first freeoccurrence of z : u . Otherwise there are two cases: in the first case z : u, z ′ : u ′ bothappear within the same Γ j or both within ∆, and in this case the variables appear inthe same order within FV seq ( M ) by condition (ii) of an application normal form. In thesecond case z : u is in Γ j for some j and z ′ : u ′ is in Γ j ′ for j ′ < j or is in ∆. In this caseby inspection of (4.26), (4.27) it is clear that z : u appears before z ′ : u ′ in FV seq ( M ).44 roposition 4.31. If Γ is repetition-free and π , π are preproofs of Γ ⊢ p that are ( L ⊃ ) -normal forms then f Γ p ( π ) = f Γ p ( π ) implies π ∼ o π .Proof. To be clear f Γ p ( π ) = f Γ p ( π ) means equality of terms (that is, α -equivalence ofpreterms). We set M i := f Γ p ( π i ) for i ∈ { , } so that by hypothesis M = M as terms.We proceed by induction on the length of the term M = M = M . In the base case M is a variable, and Lemma 4.21 shows that π i is equivalent under ∼ o to(ax) z : s ⊢ s (weak)∆ i , z : s, Θ i ⊢ s for some decomposition Γ = ∆ i , z : s, Θ i . Since Γ is repetition-free there is only oneoccurrence of z : s in Γ so ∆ = ∆ , Θ = Θ and this variable normal form is the samefor both π , π . Hence π ∼ o π as required.Next, suppose that M = λx.N is an abstraction where p = q ⊃ r . By Lemma 4.22 each π i is equivalent under ∼ o to an abstraction normal form π ′ i . Let ψ i denote the preproofobtained from π ′ i by deleting the final ( R ⊃ ) rule, which we may assume eliminates avariable x : q in both π ′ and π ′ which does not occur in Γ and which is leftmost in theantecedent. Then f Γ ,x : qr ( ψ ) = N = f Γ ,x : qr ( ψ )so by the inductive hypothesis ψ ∼ o ψ from which we deduce π ∼ o π .Finally suppose that M is an application ( M M ) : p with M : r ⊃ p and M : r .By Lemma 4.29 each π i is equivalent under ∼ o to an application normal form π ′ i . Theproof of the lemma shows that the types p , . . . , p b , s , sequences Γ b , . . . , Γ , ∆ , y b and terms L b , . . . , L , R may be read off from M and therefore coincide in the normal forms for π , π .Let τ ij , ζ i denote the preproofs involved in the normal form for π i . We deduce f Γ j p j ( τ j ) = f Γ j p j ( τ j ) 1 ≤ j ≤ b and f ∆ s ( ζ ) = f ∆ s ( ζ ). Since ∆ and Γ j for 1 ≤ j ≤ b are repetition-free it follows from theinductive hypothesis that τ j ∼ o τ j for 1 ≤ j ≤ b and ζ ∼ o ζ and hence π ∼ o π whichcompletes the proof of the inductive step. Definition 4.32.
Let π be a preproof of Γ ⊢ p which is a ( L ⊃ )-normal form. A η -pattern in π is a configuration of rules within π of the form ζ ...∆ ⊢ s θ ...Θ , z : p, Θ ′ ⊢ p ( L ⊃ ) y : s ⊃ p, ∆ , Θ , Θ ′ ⊢ p ...Γ , x : s, Γ ′ ⊢ p ( R ⊃ )Γ , Γ ′ ⊢ s ⊃ p (4 . R ⊃ ) rule to the displayed ( L ⊃ ) rule takes only theright hand branch of any intermediate ( L ⊃ ) rule and contains no ( R ⊃ ) rules.(ii) f ∆ s ( ζ ) and f Θ ,z : p, Θ ′ p ( θ ) are both variables.(iii) The contraction tree of the occurrence of x : s eliminated by the ( R ⊃ ) rule con-tains as leaves one occurrence introduced by an axiom in ζ and all other leaves areoccurrences introduced by weakenings. Example 4.33.
The prototypical example of an η -pattern is (4.9). However the readershould be aware that weakenings can complicate this picture:(ax) x ′ : p ⊢ x ′ : p (weak) x : p, x ′ : p ⊢ x ′ : p (ctr) x : p ⊢ x : p (ax) y : q ⊢ y : q ( L ⊃ ) z : p ⊃ q, x : p ⊢ ( z x ) : q ( R ⊃ ) z : p ⊃ q ⊢ λx. ( z x ) : p ⊃ q (4 . Definition 4.34.
Let π be a preproof of Γ ⊢ p . We say that π is a special ( L ⊃ ) -normalform if it is a ( L ⊃ )-normal form which contains no η -pattern.Recall that an η -redex in a lambda term M is a subterm of the form λx. ( N x ) in which x does not occur as a free variable in N . Lemma 4.35.
A preproof π of Γ ⊢ p which is a ( L ⊃ ) -normal form contains an η -patternif and only if f Γ p ( π ) contains an η -redex.Proof. Suppose that π is an ( L ⊃ )-normal form which contains an η -pattern (4.29). ThenLemma 4.19 shows that z : p occurs as a free variable in f Θ ,z : p, Θ ′ p ( θ ) which must thereforebe equal to z : p . The translation of the part of the η -pattern ending at the ( L ⊃ )rule is therefore ( y x ′ ) where x ′ : s = f ∆ s ( ζ ). Since π is well-labelled there is preciselyone occurrence of x ′ : s in ∆ which is a weak ancestor of x : s but not necessarily astrong ancestor. Since this occurrence cannot be a weak ancestor both of x : s and of anoccurrence eliminated in a ( L ⊃ ) rule, we see that the translation of the η -pattern is ofthe form λx. ( M x ) for some term M .This term M is constructed from ( L ⊃ ) rules within the η -pattern starting with y and the only way for x to appear as a free variable in M is for some weak ancestorof x to appear in the antecedent of the left hand branch of one of these ( L ⊃ ) rules.But by condition (iii) of a special ( L ⊃ )-normal form such weak ancestors must all beintroducing by weakenings, from which we conclude that x is not free in M . This showsthat the translation of the η -pattern is an η -redex, which survives in the translation of π .Conversely, suppose that f Γ p ( π ) contains an η -redex λx. ( M x ) where x : s, M : s ⊃ p .Then η contains, since it is well-labelled, precisely one ( R ⊃ ) rule that eliminates an46ccurrence of x : s and we may assume it is as displayed in (4.29). Follow the treeupwards from this rule taking the right hand branch at every ( L ⊃ ) rule until an ( L ⊃ )rule is encountered for which the translation of the right hand branch θ is a variable z : p and an occurrence of this variable is eliminated by the ( L ⊃ ) rule. Since the translationof the tree above the ( R ⊃ ) rule is ( M x ) this walk encounters no ( R ⊃ ) rule and isguaranteed to encounter an ( L ⊃ ) rule of the specified kind. The left hand branch ζ ofthis ( L ⊃ ) rule must similarly have for its translation a variable.Now consider the contraction tree of x : s . It is clear that it contains one leaf corre-sponding to a weak ancestor introduced by (ax) in ζ . Suppose that there were anotherweak ancestor introduced by ( L ⊃ ) or (ax). By the proof of Lemma 4.21 we know that ζ , θ contain no ( L ⊃ ) rules so this other weak ancestor must be introduced between ( R ⊃ )and ( L ⊃ ) in the η -pattern or in one of the left hand branches of one of the intermediate( L ⊃ ) rules and therefore occurs as a free variable in M , which is a contradiction. Hence π contains an η -pattern. Lemma 4.36.
Suppose that π , π are ( L ⊃ ) -normal forms with π ∼ o π . If π is aspecial ( L ⊃ ) -normal form then so is π .Proof. Immediate from Lemma 4.8(i) and Lemma 4.35.
Lemma 4.37. If π is a special ( L ⊃ ) -normal form then f Γ p ( π ) is a βη -normal form.Proof. Immediate from Lemma 4.12 and Lemma 4.35.
Lemma 4.38.
Every preproof π is equivalent under ∼ p to a special ( L ⊃ ) -normal form.Proof. We may by Lemma 4.18 assume π is a ( L ⊃ )-normal form. Consider an η -pattern(4.29) within π . By Lemma 4.21 there is a preproof equivalent under ∼ p to π in whichthe branch of the proof given by the η -pattern is replaced by(ax) x ′ : s ⊢ x ′ : s (ax) z : p ⊢ z : p ( L ⊃ ) y : s ⊃ p, x ′ : s ⊢ ( y x ′ ) : p ...Γ , x : s, Γ ′ ⊢ M : p ( R ⊃ )Γ , Γ ′ ⊢ λx. ( M x ) : s ⊃ p (4 . L ⊃ ). Using(2.33) and Remark 2.20 we may eliminate all weak ancestors of x : s in π except for thedisplayed x ′ : s , yielding a preproof in which the topmost occurrence of x : s is the strongancestor of occurrence eliminated in the ( R ⊃ ) rule:(ax) x : s ⊢ x : s (ax) z : p ⊢ z : p ( L ⊃ ) y : s ⊃ p, x : s ⊢ ( y x ) : p ...Γ , x : s, Γ ′ ⊢ M : p ( R ⊃ )Γ , Γ ′ ⊢ λx. ( M x ) : s ⊃ p (4 . R ⊃ ) and ( L ⊃ ) in (4.32) are either structural rulesor ( L ⊃ ) rules and by (2.20),(2.21),(2.22) and (2.23) we may commute the ( R ⊃ ) withall of these rules, until we obtain a preproof equivalent to π under ∼ p with the original η -pattern branch replaced by (ax) x : s ⊢ x : s (ax) z : p ⊢ z : p ( L ⊃ ) y : s ⊃ p, x : s ⊢ ( y x ) : p ( R ⊃ ) y : s ⊃ p ⊢ λx. ( y x ) : s ⊃ p ...which is by (2.36) equivalent to (ax) y : s ⊃ p ⊢ y : s ⊃ p ...Applying the above reasoning to all η -patterns in π from greatest to lowest height (mea-suring the height at the ( R ⊃ ) rule) completes the proof. Proof of Proposition 4.16.
Let Γ be repetition-free and let SL Σ Γ p denote the set of pre-proofs of Γ ⊢ p which are special ( L ⊃ )-normal forms. Let ∼ p denote the induced relationon SL Σ Γ p noting that two elements may be equivalent via intermediate preproofs that arenot special ( L ⊃ )-normal forms. The inclusion SL Σ Γ p ⊆ Σ Γ p induces by Lemma 4.38 abijection(4.33) SL Σ Γ p / ∼ p ∼ = / / Σ Γ p / ∼ p Recall that Q = [Γ]. Now consider the translation map f Γ p restricted to special ( L ⊃ )-normal forms and the induced map on the quotients f Γ p : SL Σ Γ p / ∼ p −→ Λ Qp / = βη . We have a commutative diagram(4.34) Σ Γ p (cid:15) (cid:15) SL Σ Γ p (cid:15) (cid:15) inc o o f Γ p / / (cid:15) (cid:15) Λ Qp (cid:15) (cid:15) Σ Γ p / ∼ p SL Σ Γ p / ∼ p ∼ = o o f Γ p / / Λ Qp / = βη in which the vertical arrows are the canonical maps to the quotient. It clearly suffices toprove that f Γ p is a bijection.To prove it is injective, let π , π ∈ SL Σ Γ p be such that f Γ p ( π ) = βη f Γ p ( π ). Since bothof these terms are βη -normal forms by Lemma 4.37 it follows from a standard result inthe theory of lambda calculus [28, Corollary 4.3] that f Γ p ( π ) = f Γ p ( π ) in Λ Qp . Since Γ isassumed to be repetition-free Proposition 4.31 then implies π ∼ p π as required.48o prove surjectivity of f Γ p we prove surjectivity of the map(4.35) f Γ p : SL Σ Γ p −→ N Λ Qp where N Λ Qp denotes the set of βη -normal forms. The proof is by induction of the proposi-tion P ( n ) which says that for any repetition-free Γ and formula p any βη -normal lambdaterm M of length n is in the image of (4.35) where Q = [Γ]. By appending exchangesand weakenings we may assume without loss of generality that Γ is the set of distinct freevariables of M , in order of appearance. The base case is clear by inspection of (4.17). If M = λx.N ∈ N Λ Qp is an abstraction with p = q ⊃ r, x : q and N : r and x / ∈ Q then N ∈ N Λ Q ∪{ x : q } r so by the inductive hypothesis there is a special ( L ⊃ )-normal form ψ with f Γ ,x : qr ( ψ ) = N and by appending a ( R ⊃ ) rule to π as in (4.18) we construct a special( L ⊃ )-normal form π with f Γ p ( π ) = M . If M ∈ N Λ Qp is an application then since M is βη -normal it must be of the form (4.27) that is(4.36) M = (cid:0) ( · · · (( y b L ′ b ) L ′ b − ) · · · L ′ ) R ′ (cid:1) for some formulas p , . . . , p b , s and βη -normal terms L ′ j : p j and R ′ : s and variable y b .Possibly b = 0 in which case M = ( yR ′ ). As in the proof of Lemma 4.29 we constructfrom this data a sequence of formulas y b , Γ b , . . . , Γ , ∆ and terms R : s and L j : p j for1 ≤ j ≤ b . By the inductive hypothesis we have special ( L ⊃ )-normal forms τ j and ζ such that f Γ j p j ( τ j ) = L j and f ∆ s ( ζ ) = R . From these preproofs and the contraction patternthat produces y b , L ′ , . . . , L ′ b , R ′ from y b , L , . . . , L b , R we construct an application normalform π as given in the statement of Lemma 4.29 with f Γ p ( π ) = M . By construction π isa special ( L ⊃ )-normal form so the proof is complete.Let N Λ Qp denote the subset of βη -normal forms in Λ Qp . What the proof of Proposition4.16 actually shows is that there is a bijection(4.37) SL Σ Γ p / ∼ o ∼ = / / N Λ Qp . This is still not satisfactory. For example, we cannot rule out a priori that there are somespecial ( L ⊃ )-normal forms π , π that are related by ∼ o but every chain of generatingrelations between them involves intermediate preproofs which are not special ( L ⊃ )-normal forms. The methods already developed suffice to prove a much stronger statement,which we treat systematically in Section 4.1. The cut-elimination theorem of Gentzen [9] is the first step in the direction of establishinga normal form for sequent calculus proofs, but as there remain many cut-free proofs insequent calculus that are “the same” this can hardly be called a normal form. The workof Mints [22] building on Kleene’s work on permutative conversions [17] is the first to49stablish a true normal form result for sequent calculus proofs, albeit in a system that isnot quite standard LJ. In this section we revisit the topic of such normal forms.The guiding principle behind our normal form for sequent calculus proofs is the conceptof encapsulation . Consider a preproof of the form ζ ...∆ ⊢ R : s (ax) x : p ⊢ x : p ( L ⊃ ) y, ∆ ⊢ ( y R ) : p ...The left hand branch of the ( L ⊃ ) rule supplies a term R that may be viewed as eitherdata or a subroutine. This subroutine is well encapsulated if it is possible to apprehendits role in the broader proof entirely by inspecting the branch itself, that is, if the meetingpoint between ζ and the rest of the proof at this ( L ⊃ ) rule serves as a boundary acrosswhich there is minimal information flow. These are vague statements; to be more precise,we identify two kinds of boundary violation which break this principle of encapsulation.There are other kinds of boundary violations that one may imagine, but these are alreadyimpossible in special ( L ⊃ )-normal forms so we do not elaborate them.In the following π denotes a preproof of Γ ⊢ p and we assume Γ is repetition-free. Iftwo variable occurrences are introduced above a boundary and contracted below it, thenthis creates a boundary violation of contraction type: Definition 4.39. A boundary violation of (ctr) type in π is a pair consisting of a ( L ⊃ )rule and a (ctr) rule, with the latter below the former as in...Λ ⊢ s ... ( L ⊃ )...Γ , x : p, x ′ : p, Γ ′ ⊢ q (ctr)Γ , x : p, Γ ′ ⊢ q (4 . x : p has at least two distinct weak ancestors in Λ.If a variable occurrence is introduced above a boundary and eliminated by a ( L ⊃ )rule below it, this creates a boundary violation of ( L ⊃ )-type: Definition 4.40. A boundary violation of ( L ⊃ ) type in π is a pair consisting of two ( L ⊃ )rules as in ...Γ ⊢ p ...Λ ⊢ s ... ( L ⊃ )...∆ , x : q, Θ ⊢ r ( L ⊃ ) y : p ⊃ q, Γ , ∆ , Θ ⊢ r (4 . x : q has a strong ancestor in Λ.Recall the notation (dctr) i,j for derived contractions from Definition 4.26. A rule pair in π is a pair of rules ( r ) , ( r ′ ) adjacent in the underlying tree of π with ( r ′ ) occurringimmediately after ( r ) on the path from ( r ) to the root. Definition 4.41.
A preproof π of Γ ⊢ p is called well-structured if it is a special ( L ⊃ )-normal form and further satisfies the following conditions:(a) There are no boundary violations of (ctr) type.(b) There are no boundary violations of ( L ⊃ ) type.(c) The only (weak) rules occur in pairs (weak) , ( R ⊃ ) with the second rule eliminatingthe variable occurrence introduced by the first, which is leftmost in the antecedent.(d) There is no rule pair ( r ) , ( L ⊃ ) with ( r ) structural on the right branch.(e) There is no rule pair ( R ⊃ ) , ( r ) where ( r ) is structural.(f) There is no rule pair ( R ⊃ ) , ( L ⊃ ) with the ( R ⊃ ) on the right branch.(g) There is no pair (dctr) a,b , (dctr) a ′ ,b ′ of consecutive maximal derived contractions with( a ′ , b ′ ) < ( a, b ) in the lexicographic ordering.(h) Every (ex) rule occurs as part of a derived contraction.Recall from Definition 4.28 the notion of a well-ordered preproof. Definition 4.42.
A preproof π is normal if it is of the form ψ ...Γ ′′ ⊢ p (lad)Γ ′ ⊢ p (weak)Γ ⊢ p where ψ is well-ordered and well-structured, and the ladders and weakening rules are(lad) c , (lad) c , . . . , (lad) c n (weak) d , (weak) d , . . . , (weak) d m with c < c < · · · < c n and d < d < · · · < d m (using the notation of Definition 4.25and Definition 4.27). One or both of these series of rules may be empty.51 emark 4.43. Note that by (c), (h) no well-structured preproof can end with exchanges orweakenings, so that the subproof ψ of Definition 4.42 can be unambiguously recovered fromthe normal preproof π . The sequence Γ ′′ is by the hypothesis of well-ordering determinedby the term f Γ p ( π ) = f Γ ′′ p ( ψ ) and so from this term and Γ the ladders and weakening rulesand their order are completely determined. Lemma 4.44. If π is well-structured then any subproof of π not ending in (weak) or (ex) is also well-structured.Proof. Left to the reader.
Proposition 4.45.
Let π be a preproof of Γ ⊢ p . Then(I) If M is a variable then π is well-structured if and only if it is an axiom rule.(II) If M is an abstraction then π is well-structured if and only it is equivalent under ∼ α to an abstraction normal form (4.18) where there is no η -pattern involving thefinal ( R ⊃ ) rule and the subproof ψ of (4.18) is either well-structured, or is a well-structured proof followed by a single (weak) rule with the introduced variable leftmostin the antecedent and eliminated by the final rule in π .(III) If M is an application then π is well-structured if and only if it is equivalent under ∼ α to an application normal form as in Lemma 4.29 in which ζ and τ j for ≤ j ≤ b are well-structured and the rule series (4.23) , (4.24) are empty.Proof. (I) If M is a variable and π is well-structured, consulting the proof of Lemma 4.21we see that by condition (c) of the well-structured property there are no (ctr) or (weak)rules in π . There are no (ex) rules by (h). So π is an axiom rule. Conversely, it is clearthat an axiom rule is well-structured.(II) Suppose M is an abstraction and π is well-structured. We must show the finalrule in π is ( R ⊃ ). If we walk the tree from the root taking only right branches of ( L ⊃ )rules we eventually encounter a ( R ⊃ ) rule. The only rules that may precede the first( R ⊃ ) on this walk are ( L ⊃ ) and structural rules, and these are impossible by (e),(f) so π must end in ( R ⊃ ) and we are done. The reverse implication in (II) is also clear.(III) For the reverse direction in (III) observe that if π is a well-labelled applicationnormal form satisfying the conditions then it is a special ( L ⊃ )-normal form. Conditions(a), (b) follow respectively from condition (iii) of application normal form and the shapeof the normal form proof tree, together with the assumption that the branches are well-structured. Any (weak) rule in π either occurs in the τ j or ζ or at the bottom of π , andthe latter is explicitly ruled out, so (c) is satisfied. Similarly for conditions (d)-(h), notingthat (g) uses condition (iv) of an application normal form.Finally suppose M is an application and that π is well-structured. Consulting theproof of Lemma 4.29 the structural rules can only occur at the bottom of the porch by(d). Nothing needs to be done in the migration phase by (b). By (c) there are no (weak)rules in ζ , τ j that need to be moved to the bottom of the porch, and nothing needs to52e done to satisfy (iii) by (a). By (c),(h) the only structural rules on the porch are partof derived contractions which satisfy (iv) by condition (g). We are free to change π upto α -equivalence so we may assume (v) is satisfied, and (vi),(vii) are vacuous. So it onlyremains to prove (ii).To do this we first prove Corollary 4.46 below, which requires only the part of (III)that we have already proven. Suppose for a contradiction that a well-structured preproofexists which is not well-ordered, and let ρ be an example with L = f Λ r ( ρ ) of minimallength. By (I) this term L cannot be a variable, since an axiom rule is well-ordered. If L were an application then by the part of (III) already proven ρ is equivalent under ∼ α toa preproof which is an application normal form but for the possible failure of (ii); but ifany of the branches failed to be well-ordered this would contradict minimality of L , andif they are all well-ordered then (ii) is satisfied and ρ would therefore be well-ordered byLemma 4.30, a contradiction. So the only possibility is that L is an abstraction. By (II)then ρ is equivalent under ∼ α to an abstraction normal form ψ ...∆ , x : q, ∆ ′ ⊢ N : r ( R ⊃ )∆ , ∆ ′ ⊢ λx.N : q ⊃ r By hypothesis ∆ , ∆ ′ = FV seq ( λx.N ). If x : q is introduced by (weak) then this contradictsminimality of L , and if not then by minimality ∆ , x : q, ∆ ′ = FV seq ( N ) which contradicts∆ , ∆ ′ = FV seq ( λx.N ). This completes the proof of the corollary.Returning now to the proof of the theorem proper, the branches ζ , τ j cannot end in(weak) by (c) and cannot end in (ex) by (h) so by Lemma 4.44 they are well-structured andhence by Corollary 4.46 they are well-ordered, which shows condition (ii) of an applicationnormal form and completes the proof. Corollary 4.46. If ρ is a well-structured preproof then it is well-ordered. In particular, a well-structured preproof is precisely a normal preproof in which theseries of ladders and weakenings at the bottom are empty. We may now prove a strength-ening of Proposition 4.31:
Proposition 4.47. If Γ is repetition-free and π , π are normal preproofs of Γ ⊢ p then f Γ p ( π ) = f Γ p ( π ) implies π ∼ α π .Proof. If π , π are normal and f Γ p ( π ) = f Γ p ( π ) then by Remark 4.43 the ladders andweakenings at the bottom of π , π agree and writing ψ , ψ for the well-structured sub-proofs as in Definition 4.42 we have f Γ p ( π ) = f Γ p ( π ) ⇐⇒ f Γ ′′ p ( ψ ) = f Γ ′′ p ( ψ )(4.40) π ∼ α π ⇐⇒ ψ ∼ α ψ (4.41)The proof is similar to Proposition 4.31 and is again by induction on the length of theterm M = f Γ p ( π ) = f Γ p ( π ). In the base case M is a variable, and by what we have just53aid and Proposition 4.45 (I) it is immediate that π ∼ α π . If M is an abstraction λx.N then by Proposition 4.45 (II) both ψ , ψ end in ( R ⊃ ) rules and we let ψ ′ , ψ ′ denote thesubproofs of ∆ , x : q, ∆ ′ ⊢ q and ∆ , x : q, ∆ ′ ⊢ q respectively obtained by deleting thesefinal rules. These are either both well-structured (let us call this the first case) or bothwell-structured after deleting a final (weak) rule which introduces x : q in the leftmostposition (call this the second case) hence from Corollary 4.46 we deduce that∆ , x : q, ∆ ′ = ∆ , x : q, ∆ ′ and this sequence is in the first case FV seq ( N ) and in the second case x : q, FV seq ( N ). Inthe first case let ψ ′′ i = ψ ′ i and in the second case let ψ ′′ i be obtained from ψ ′ i by deleting thefinal (weak), for i ∈ { , } . Then by Corollary 4.46 the preproofs ψ ′′ i are normal preproofsof the same sequent Θ ⊢ q and f Θ q ( ψ ′′ ) = N = f Θ q ( ψ ′′ )so by the inductive hypothesis ψ ′′ ∼ α ψ ′′ from which it follows that ψ ′ ∼ α ψ ′ and hence π ∼ α π . If M is an application then by Proposition 4.45 (III) both ψ , ψ are equivalentunder ∼ α to application normal forms, in which the τ j and ζ are well-structured, and soby the inductive hypothesis equivalent under ∼ α , hence π ∼ α π . Lemma 4.48. If Γ is repetition-free then every preproof π is equivalent under ∼ p to anormal preproof.Proof. We may by Lemma 4.38 prove the lemma for special ( L ⊃ )-normal forms π , inwhich case the proof is by induction on the length of M = f Γ p ( π ). By Lemma 4.21, Lemma4.22 and Lemma 4.29 π is equivalent under ∼ o to one of the three types of normal forms π ′ . In the base case M is a variable, and the claim follows from Proposition 4.45 (I).For the inductive step, if π ′ is an abstraction normal form, we may assume by theinductive hypothesis that the subproof ψ obtained by deleting the final ( R ⊃ ) rule isnormal, and after moving exchanges and weakenings below the ( R ⊃ ) rule we may assume ψ satisfies the hypotheses of Proposition 4.45 (II), so that π ′ is normal. If π ′ is anapplication normal form then by the inductive hypothesis we may assume τ j for 1 ≤ j ≤ b and ζ are normal. By condition (ii) of an application normal form these branches cannotend in (weak) rules. Let κ denote one of the τ j or ζ and suppose that κ ends in a seriesof (ex) rules. The well-structured subproof κ ′ of κ obtained by deleting these rules iswell-ordered and has the same translation as κ , which is also well-ordered, so the seriesof (ex) rules implement the identity permutation and may be deleted using (2.11). Hencewe may assume without loss of generality that τ j for 1 ≤ j ≤ b and ζ are not just normal,but well-structured. Hence by Proposition 4.45 (III) the preproof π ′ is normal.Recall that N Λ Qp denotes the subset of βη -normal forms in Λ Qp . We let N Σ Γ p denotethe set of normal preproofs of Γ ⊢ p in the sense of Definition 4.42.54 heorem 4.49. If Γ is repetition-free there is a commutative diagram (4.42) Σ Γ p / ∼ p ∼ = / / Λ Qp / = βη N Σ Γ p / ∼ α ∼ = O O ∼ = / / N Λ Qp ∼ = O O in which the rows are bijections induced by the function f Γ p and the columns are bijectionsinduced by the inclusions N Σ Γ p ⊆ Σ Γ p and N Λ Qp ⊆ Λ Qp .Proof. There is clearly a commutative diagram of this form, and the first row is a bijectionby Proposition 4.16. The second column is a bijection by the existence and uniqueness of βη -normal forms [28, Corollary 4.3]. Surjectivity of the first column is Lemma 4.48 so itsuffices to prove the second row is injective, which is Proposition 4.47. Example 4.50.
The well-labelled Church numeral 2 of Example 4.10 is not normal, sinceit contains a boundary violation of ( L ⊃ ) type, highlighted below:(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p (ax) x ′′ : p ⊢ x ′′ : p ( L ⊃ ) y ′ : p ⊃ p, x ′ : p ⊢ ( y ′ x ′ ) : p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ ( y ′ ( y x )) : p (ctr) y : p ⊃ p, x : p ⊢ ( y ( y x )) : p ( R ⊃ ) y : p ⊃ p ⊢ λx. ( y ( y x )) : p ⊃ p The algorithm of the proof of Lemma 4.29 eliminates this boundary violation as part ofthe “migration” phase, which consists in this case of an application of (2.30) resulting inthe ∼ o -equivalent preproof(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p ( L ⊃ ) y : p ⊃ p, x : p ⊢ ( y x ) : p (ax) x ′′ : p ⊢ x ′′ : p ( L ⊃ ) y ′ : p ⊃ p, y : p ⊃ p, x : p ⊢ ( y ′ ( y x )) : p (ex) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ ( y ′ ( y x )) : p (ctr) y : p ⊃ p, x : p ⊢ ( y ( y x )) : p ( R ⊃ ) y : p ⊃ p ⊢ λx. ( y ( y x )) : p ⊃ p This is still not normal, but applying (2.32) the above preproof is ∼ o -equivalent to(ax) x : p ⊢ x : p (ax) x ′ : p ⊢ x ′ : p ( L ⊃ ) y ′ : p ⊃ p, x : p ⊢ ( y ′ x ) : p (ax) x ′′ : p ⊢ x ′′ : p ( L ⊃ ) y : p ⊃ p, y ′ : p ⊃ p, x : p ⊢ ( y ( y ′ x )) : p (ctr) y : p ⊃ p, x : p ⊢ ( y ( y x )) : p ( R ⊃ ) y : p ⊃ p ⊢ λx. ( y ( y x )) : p ⊃ p (4 . Remark 4.51.
The translation function f Γ p induces by Theorem 4.49 a bijection between α -equivalence classes of normal preproofs and βη -normal lambda terms. The inverse map(4.44) g Γ p : N Λ Qp −→ N Σ Γ p / ∼ α is implicit in Proposition 4.45 and we now make this explicit. Given a βη -normal lambdaterm M with free variables contained in Q and writing ∆ = FV seq ( M ), the preproof g Γ p ( M )is the preproof g ∆ p ( M ) followed by the ladders and weakenings uniquely determined bythe pair ∆ , Γ as explained in Remark 4.43. It therefore suffices to define a well-structuredpreproof g Γ p ( M ) in the special case where Γ = FV seq ( M ), which we denote by g ( M ).The preproof g ( M ) has the following inductive definition: • If M = x : p is a variable then g ( M ) is an axiom rule. • If M = λx.N is an abstraction with x : q and N : r then g ( M ) is g ( N ) followed by, if x / ∈ FV( N ), a rule pair (weak) , ( R ⊃ ) respectively introducing and then eliminatingan occurrence of x : q , or if x ∈ FV( N ), a ( R ⊃ ) rule eliminating x : q . • If M = ( M M ) is an application then it is of the form(4.45) (cid:0) ( · · · (( y b L b ) L b − ) · · · L ) R (cid:1) where y b is a variable and R and L j for 1 ≤ j ≤ b are βη -normal forms. Then g ( M )is the application normal form with branches g ( R ) and g ( L j ) for 1 ≤ j ≤ b endingin the uniquely determined derived contractions.For example, with M = λx. ( y ( y x )) the normal preproof g ( M ) is (4.43). Remark 4.52.
It is convenient to treat the relationship between our sequent calculus andZucker’s via the Curry-Howard correspondence. Zucker has defined a surjective functionwhich maps from the set of derivations in his sequent calculus S [35, § N [35, § ϕ : S → N [35, § ∼ on S is defined [35, § S / ∼−→ N is a bijection.The sequent calculus S differs from the one considered in this paper in that it omits theweakening and exchange rules; the absence of exchange is compensated by the system S having a set of formulas as the antecedent of a sequent, and the absence of weakening iscompensated by a special form of the ( R ⊃ ) rule.Let Γ be a repetition-free sequence of variables and γ a set of indexed formulas in thesense of Zucker [35, § Q = [Γ], by Theorem 4.15, [35, Theorem 1] andthe Curry-Howard correspondence [28, § N Σ Γ p / ∼ α ∼ = / / N Λ Qp ∼ = / / N γp ϕ − ∼ = / / S γp / ∼ N γp is the set of natural deduction derivations of p from γ (a set of assumptionclasses) and S γp is the set of proofs in S of γ ⊢ p , following the convention of [35, § π ∈ N Σ Γ p , erases the final weakeningsand replaces all (weak) , ( R ⊃ ) pairs by Zucker’s special ( R ⊃ ) rule, erases all exchangesand replaces the antecedent Γ in every sequent by an appropriate set of indexed formulas.Next we compare the Mints normal form of [22] with ours; see also [30]. For claritywe refer to the sequent calculus proofs of Mints as derivations . Remark 4.53.
In the sequent calculus system GJ of Mints a normal derivation (in thesense of [22, Definition 4]) whose translation is the Church numeral λx. ( y ( y x )) is(ax) p ⊢ p (ax) p ⊢ p (weak) p, p ⊢ p ( L ⊃ ) p ⊃ p, p ⊢ p (ax) p ⊢ p (weak) p, p ⊃ p, p ⊢ p ( L ⊃ ) p ⊃ p, p ⊃ p, p ⊢ p ( R ⊃ ) p ⊃ p, p ⊃ p ⊢ p ⊃ p (ctr) p ⊃ p ⊢ p (4 . L ⊃ ) rule and introduce global structure into the tree (since the weakening onthe right branch of p ⊃ p reflects the appearance of this formula on the left branch). Theonly other difference to (4.43) is the order of the ( R ⊃ ) , (ctr) rules.In general, in a Mints normal form contraction rules take place as late as possible (thatis, as close as possible to the bottom of the proof tree) whereas in our normal form theserules take place as early as possible. This encapsulation means that our normal forms arecomposable in a way that Mints normal forms are not. For example, denoting by 2 thenormal preproof of (4.43), the preproof2... y : p ⊃ p ⊢ M : p ⊃ p (ax) z : q ⊢ q ( L ⊃ ) t : ( p ⊃ p ) ⊃ q, y : p ⊃ p ⊢ ( t M ) : q is normal. However, appending a similar ( L ⊃ ) rule (with attendant weakenings) to (4.47)results in a derivation that is not in Mints normal form; to obtain the normal form thecontractions must be brought down past the ( L ⊃ ) rule.A normal derivation is cut-free, W-normal, C-normal and M-normal. A normal pre-proof is W-normal and M-normal but not necessarily C-normal (as the above discussionshows). Hence, apart from the differences between LJ and GJ, the only difference betweenour notion of normality and that of Mints lies in the arrangement of contractions. A literal reading of [22, Definition 4] would suggest the above derivation is not M-normal, but thisseems to be due to a lack of precision in loc.cit. , which should read “a main formula of an inference ruleor axiom, with only weakenings intervening” see also [22, Example 1]. .2 Internal BHK What is the intuitionist logical reading of the ( L ⊃ ) rule in sequent calculus? Let us firstrecall that the Brouwer-Heyting-Kolmogorov (BHK) interpretation of intuitionistic propo-sitional logic, as given by Heyting in [14, § § § ⊃ . The following quoteis from [14, § p → q can be asserted, if and only if we possess a construc-tion r , which, joined to any construction proving p (supposing the latter beeffected), would automatically effect a construction proving q . In other words,a proof of p , together with r , would form a proof of q .”The justification of the deduction rules of natural deduction by the BHK-interpretation isgiven for example in [31, § § § q repeatedly appealing to assumption p . This means that we have shown how to construct aproof of q from hypothetical proof of p ; thus on the BHK-interpretation this means that wehave established the implication p ⊃ q and this justifies the ( R ⊃ ) rule of sequent calculusby the same argument justifying introduction for ⊃ in natural deduction. Consider nowthe following simplified form of the ( L ⊃ ) rule in sequent calculus ⊢ p x : q ⊢ r ( L ⊃ ) y : p ⊃ q ⊢ r (4 . p , and a proof of r from a hypo-thetical proof of q . Then we can we may construct a proof of r from a hypothetical proofof p ⊃ q according to the following recipe. Given the earlier justification of the ( R ⊃ )rule, to prove p ⊃ q we must possess a construction of a proof of q from a hypotheticalproof of p . Enact this construction on the given proof of p , and enact on the resultingproof of q the construction which produces from such an object a proof of r .With this intuitionist reading of ( L ⊃ ) in hand let us now consider the logical statusof the following simplified forms of the rules (2.34) and (2.35): ⊢ p x : q, x ′ : q ⊢ r (ctr) x : q ⊢ r ( L ⊃ ) y : p ⊃ q ⊢ r ∼ λ ⊢ p ⊢ p x : q, x ′ : q ⊢ r ( L ⊃ ) x : q, y ′ : p ⊃ q ⊢ r ( L ⊃ ) y : p ⊃ q, y ′ : p ⊃ q ⊢ r (ctr) y : p ⊃ q ⊢ r (4 . ⊢ p ⊢ r (weak) x : q ⊢ r ( L ⊃ ) y : p ⊃ q ⊢ r ∼ λ ⊢ r (weak) y : p ⊃ q ⊢ r (4 . r from two hypothetical58roofs of formulas q, q ′ that just happen to be the same, that is q = q ′ . Joining thiswith a construction of a proof of q and a construction of a proof of q ′ certainly effects aconstruction of a proof of r . The question is: does simply stating q = q ′ and showing asingle construction of a proof of q suffice as a construction of a proof of r ? One possibleanswer is “yes it suffices, because we can simply run the construction of a proof of q andcopy the result” and another is “yes it suffices, because however many copies are required,we can repeat the construction that number of times (necessarily entailing the repetitionof earlier constructions that feed into this one)”. Principle (4.49) and the correspondingcut-elimination rule (2.42) correspond to the endorsement of the second possible reading:there is no fundamental operation of “copying” when it comes to constructions of proofs.This is another logical principle (the first being coalgebraic structure, see Definition 2.18)that is emphasised by linear logic.The intuitionist logical reading of (weak) is that any construction of a proof of r is alsoa construction of a proof of r from a hypothetical proof of q , which is “ignored” duringthe construction. The question here: is ignoring a hypothetical proof of q , constructedfrom a proof of p by a hypothetical proof of p ⊃ q , the same as ignoring the hypotheticalproof of p ⊃ q ? One possible answer is “no, because in the former case more informationis discarded than in the latter” and another is “yes it is the same, I do not believe in alogical distinction between ignoring a machine and ignoring all of its outputs”. Principle(4.50) and the corresponding cut-elimination rule (2.43) endorse the second reading. Remark 4.54 (Internal vs external composition).
Composition in the category S (recall that this means S Γ with Γ empty) which we henceforth refer to as external compo-sition, is effected via the (cut) rule. The internal composition, in the sense of the theoryof Cartesian closed categories, is a morphism κ ∈ S ( p, ( p ⊃ q ) ⊃ q ) given by x : p ⊢ p y : q ⊢ q ( L ⊃ ) x : p, z : p ⊃ q ⊢ q ( R ⊃ ) x : p ⊢ ( p ⊃ q ) ⊃ q ( R ⊃ ) ⊢ p ⊃ (( p ⊃ q ) ⊃ q ) (4 . L ⊃ ) rule determines structure on the category S which internalisescomposition, and thus the (cut) rule. To make this connection fully precise, let us comparethe rules for (cut) with those for ( L ⊃ ) in our sequent calculus system: • (2.39) for (cut) vs (2.24),(2.25),(2.27) for ( L ⊃ ). • (2.40) for (cut) vs (2.30) for ( L ⊃ ). • (2.41) for (cut) vs (2.24),(2.26),(2.28) for ( L ⊃ ). • (2.42) for (cut) vs (2.34) for ( L ⊃ ). • (2.43) for (cut) vs (2.35) for ( L ⊃ ). 59 (2.44) for (cut) vs (2.23) for ( L ⊃ ). • (2.45) for (cut) has no analogue for ( L ⊃ ). • (2.46) for (cut) vs (2.29) for ( L ⊃ ). • (2.47) for (cut) vs (2.30) for ( L ⊃ ). As elaborated in the introduction, Gentzen-Mints-Zucker duality is interesting becausesequent calculus proofs and lambda terms are different. The principal difference is thatthe structure of sequent calculus is local while that of lambda calculus is global .Let us first collect some preliminary comments. Theorem 4.15 can be read as sayingthat the “true” proof objects are βη -equivalence classes of lambda terms (or via the Curry-Howard correspondence, natural deduction proofs) since there is, up to α -equivalence, aunique such object representing every morphism in S Γ . From this point of view sequentcalculus is a system that enables us to work on these objects [10, p.39] and a proof in se-quent calculus “can be looked upon as an instruction on how to construct a correspondingnatural deduction” [26, § A.2] (although see [35, § x : p may occur multiple times as a free variable in a term M , and hence β -reduction involvesglobal coordination: reducing ( λx.M ) N to M [ x := N ] may make arbitrarily many “simul-taneous” substitutions. This global rewriting is the principal reason that time complexityis difficult to analyse directly in lambda calculus. If π is a well-labelled preproof with f Γ p ( π ) = M then π contains, in the form of the contraction tree of the occurrence x : p inΓ, a specification for how any two occurrences of x : p in M are equal and it is by use ofthis information that cut-elimination is able to present a refinement of β -reduction whichis local, in the precise sense that the relation (2.42) represents copying a term only once.The advantage of the sequent calculus proof is that it provides the “missing” structuralrules (ctr) , (weak) and (ex) that allow global β -reduction steps to be replaced by morelocal transformations.The generating relations of proof equivalence for sequent calculus preproofs also involveglobal changes to preproof trees, so this dichotomy between local and global needs to beunderstood in the proper sense. The minor examples are α -equivalence and the strongancestor substitution in (2.37). The more important instances are the generating relations(2.34) and (2.42) which copy a branch and (2.35) and (2.43) which delete a branch; variousother relations rearrange branches. Apart from α -equivalence and this copying, deletingand rearranging of branches, the changes in the proof tree are localised to a small group ofnearby vertices, edges and their labels and in this sense the generating relations of proof60quivalence are local. For further discussion of “locality” in the context of differencesbetween sequent calculus and natural deduction see [24, §
3] and [23].
We have already discussed in some detail the relation of our work to that of Zucker [35]and Mints [22], see Remark 2.19, Remark 2.22, Remark 4.52 and Remark 4.53. In thissection we contrast our approach to that of Dyckhoff-Pinto [7] and Pottinger [25].The most important differences between sequent calculus and natural deduction arethe explicit structural rules in the former, and the fact that sequent calculus has a leftintroduction rule for ⊃ whereas natural deduction has an elimination ruleΓ ⊢ p ⊃ q Γ ⊢ p ( ⊃ E )Γ ⊢ q (4 . synchronised . This allows for a formof contraction in natural deduction, as shown in the following example. Example 4.55.
Compare the Church numeral 2 in sequent calculus (Example 2.3) to thenatural deduction (ax) f Γ ⊢ p ⊃ p (ax) f Γ ⊢ p ⊃ p (ax) x Γ ⊢ p ( ⊃ E )Γ ⊢ p ( ⊃ E ) f : p ⊃ p, x : p ⊢ p ( ⊃ I ) x f : p ⊃ p ⊢ p ⊃ p ( ⊃ I ) f ⊢ ( p ⊃ p ) ⊃ ( p ⊃ p ) (4 . { f : p ⊃ p, x : p } . Here we follow natural deduction as presented in [28, § § ⊃ elimination rule allows for a formof contraction on f at the cost of introducing global structure (the axiom rules (ax) f mustinclude x in the antecedent, and the rule (ax) x must include f ).There are a variety of systems which are “in between” sequent calculus and naturaldeduction in the sense that they either modify the ( L ⊃ ) rule of sequent calculus to bemore like ⊃ elimination, or they omit some or all of the structural rules; see [23] and [24, § § § L ⊃ ) rule to have synchronised antecedent and drops exchange but keepsweakening and contraction, Zucker has a standard ( L ⊃ ) rule but omits weakening andexchange, Dyckhoff-Pinto [7] consider a system like Kleene’s G but without any structuralrules, and Pottinger [25] revisits the work of Zucker for a sequent calculus system withoutstructural rules but with a standard ( L ⊃ ) rule; see [25, p.331].61he motivation for these modifications appears to be primarily technical: it is easierto analyse the relationship between “sequent calculus” and natural deduction (or lambdacalculus) if the former is redefined to be more similar to the latter. There are also applica-tions in logic programming and proof search [7] where the simplified systems are sufficient.However these modifications come at the price of introducing global structure into proofs:the synchronised antecedent of the ( L ⊃ ) rule of Kleene’s G necessitates changes to theChurch numeral 2 along the lines of the natural deduction version (4.53) (see Remark4.53) and omitting structural rules works against the local nature of cut-elimination asdiscussed in Section 4.3. Since in our view it is the duality between local and global thatmakes the comparison of sequent calculus and natural deduction interesting, it seems de-sirable to avoid these compromises.Another line of development relating sequent calculus to lambda calculus due to Her-belin [12] builds on the work of Zucker [35] by considering a restricted set of cut-freeproofs in sequent calculus and showing that this is isomorphic to a form of lambda cal-culus with explicit substitutions. This yields a close alignment between cut-eliminationand β -reduction. It is not the purpose of the present paper to study such alignment. A Background on lambda calculus
In the simply-typed lambda calculus [29, Chapter 3] there is an infinite set of atomictypes and the set Φ → of simple types is built up from the atomic types using → . Let Λ ′ denote the set of untyped lambda calculus preterms in these variables, as defined in [29,Chapter 1]. We define a subset Λ ′ wt ⊆ Λ ′ of well-typed preterms, together with a function t : Λ ′ wt −→ Φ → by induction: • all variables x : σ are well-typed and t ( x ) = σ , • if M = ( P Q ) and
P, Q are well-typed with t ( P ) = σ → τ and t ( Q ) = σ for some σ, τ then M is well-typed and t ( M ) = τ , • if M = λx . N with N well-typed, then M is well-typed and T ( M ) = t ( x ) → t ( N ).We define Λ ′ σ = { M ∈ Λ ′ wt | t ( M ) = σ } and call these preterms of type σ . Next we observethat Λ ′ wt ⊆ Λ ′ is closed under the relation of α -equivalence on Λ ′ , as long as we understand α -equivalence type by type, that is, we take λx . M = α λy . M [ x := y ]as long as t ( x ) = t ( y ). Denoting this relation by = α , we may therefore define the sets of well-typed lambda terms and well-typed lambda terms of type σ , respectively:Λ wt = Λ ′ wt / = α (A.1) Λ σ = Λ ′ σ / = α . (A.2) 62ote that Λ wt is the disjoint union over all σ ∈ Φ → of Λ σ . We write M : σ as a synonymfor [ M ] ∈ Λ σ , and call these equivalence classes terms of type σ . Since terms are, bydefinition, α -equivalence classes, the expression M = N henceforth means M = α N unless indicated otherwise. We denote the set of free variables of a term M by FV( M ). Definition A.1.
The substitution operation on lambda terms is a family of functions (cid:8) subst σ : Y σ × Λ σ × Λ wt −→ Λ wt (cid:9) σ ∈ Φ → We write M [ x := N ] for subst σ ( x, N, M ) and this term is defined inductively (on thestructure of M ) as follows: • if M is a variable then either M = x in which case M [ x := N ] = N , or M = x inwhich case M [ x := N ] = M . • if M = ( M M ) then M [ x := N ] = (cid:0) M [ x := N ] M [ x := N ] (cid:1) . • if M = λy.L we may assume by α -equivalence that y = x and that y does not occurin N and set M [ x := N ] = λy.L [ x := N ].Note that if x / ∈ FV( M ) then M [ x := N ] = M . References [1] M. Atiyah,
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