A proof of Cut-elimination for Linear Logic∗ Luca Tranchini

Abstract The Cut-elimination theorem is proved for a one-sided sequent calculus formulation of Linear Logic (without units). The demonstration is articulated in two results: in the first one, concerning “permutations”, lays the originality of the work: in particular, it is defined in full detail a procedure, by means of which every proof is “transformed” in regular form (that is, in a proof in which every Cut-formula is the principal formula of an application of a logical rule). The second result shows how to eliminate all the applications of the Cut-rule from a regular proof. The whole proof works without the introduction of Gentzen’s “fusion” or of the Cut!-rule used, for example, in [Lincoln et alia 1994] and [Bra¨ uner de Paiva 1996], that is, Cut-rules that “cut” more occurrences of the Cut-formulas at the same time. Finally, the resource sensitive character of Linear Logic suggested formulations in term of occurrences of formulas, instead of one in term of formulas.

Introduction We present a proof of the Cut-elimination theorem for a one-sided version of Propositional Linear Logic without units. Although a Cut-elimination theorem follows from the normalization one for the Proof Nets formulation of the same calculus (see [Girard 1987a]), a full detailed proof of the result can be useful for many purposes. What is presented here is modification of the proof offered in the appendix A of [Lincoln et alia 1994]. What appears more clearly here is the “indeterminacy” role played by the additives connectives, & in particular. The idea of the proof is analogous to that of Gentzen and Takeuti. In their proof of the Cut-elimination theorem for LK, the main lemma shows how to eliminate the Cut from a proof containing only one application of it. The lemma is proved by induction on the rank of the Cut. The rank ρ is a couple hm, ni, where m is the number of sequents ⇒ Γi (1 ≤ i ≤ m) such that 1) each ⇒ Γi (i > 1) is the conclusion of the rule of which the ∗ Published in the Siena University Mathematics Department preprint series “Rapporti interni del dipartimento di Matematica R. Magari” n. 483, October 2006

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sequent ⇒ Γi−1 is the premise; 2) One of the formulas in ⇒ Γm is the left Cut-formula of the Cut; 3) Each ⇒ Γi contains the Cut-formula1 . The same on right holds for n. Of course the rank of a Cut is always ≥ 2. The proof of the lemma deals with two cases. The ρ = 2 case is by induction on the degree of the Cut, that is, (roughly) on the number of logical operators in the Cut-formulas. Each case shows how to replace the Cut with another of lower degree (reduction procedure). The ρ > 2 shows how to reduce the rank, by “pushing up” the Cut in the proof (permutation procedure). In this proof, the two parts of the Gentzen-Takeuti lemma are separated. This is done in order to make clearer the procedure by means of which the rank of the Cut decrees: while in LK this is very easy to work out (thanks to the structural rules), for LL everything is harder. So first we transform the derivations into regular form (that is, in a derivation in which all the applications of the Cut-rule have ρ = 2): this is done in section 2. We encounter two complications w.r.t. the ρ > 2 case of the main lemma induction in the original work. 1. We want to define a procedure for every derivation: so we can’t restrict ourselves to derivations containing only one application of the Cutrule. The consequence is the following: we may have to deal with derivation patterns like the following: .. .. .πH .πH ∗ .. R1 R ∗ ⇒ ∆1 , H, A ⇒ ∆2 , H ∗ 2 .π2 Cut ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ Without loss of generality, we will suppose that the rank of the Cut, whose Cut-formulas are H and H ∗ , is equal to two. Nonetheless the form of the regular derivation will depend on the forms of H and H ∗ , that is, we will have different sub-cases according to the form of H. 2. The original lemma deals with the Fusion-rule, instead of the Cutrule. The fusion is a Cut-rule that “cuts” more occurrences of the Cut-formulas at the same time. Also the other famous proof of the Cut-elimination theorem, the one due to Girard and Tait2 , works with something similar to this new rule: although their main lemma is formulated without referring to the Cut-rule, it asserts the existence of a derivation of ⇒ Γ01 , Γ02 given two derivations of ⇒ Γ1 and ⇒ Γ2 , where Γ01 and Γ02 are obtained from Γ1 and Γ2 by eliminating some 1 The notion of formula must be defined via a suitable equivalence relation defined on occurrences. 2 See the second chapter of [Girard 1987b].

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occurrences (possibly all) of (respectively) A and A∗ . The fusion is introduced to deal with contractions, or, better, not to deal with contractions! In the proof for LL, the introduction of the Cut!-rule (a rule analogous to the gentzenian fusion) seems much more ad hoc than in the proof for LK. This is due to the fact that only interrogated formulas can be contracted, hence the only fusions we have are those on ?A and !A: in particular, the notion of principal formula loses its intuitive clarity, just as the proof of the theorem itself: the “reduction” procedure involves not only the couples of dual rules (⊗ versus F, & versus ⊕ and ! versus ?, ?W and ?C), but also the following unfair couples: ! versus ⊗, Cut and !. So we tried, successfully, to treat contracted formulas as non principal and, consequently, contractions as permutation cases. Finally, because of the restrictions on the structural rules, it seemed to us more natural to formulate all the definitions in term of occurrences of formulas, instead of formulas. The unfair consequence of this choice is an odd terminology (for example, Cut-occurrence and principal occurrences instead of Cut-formulas and principal formulas); nonetheless this choice makes clearer the primitive role of occurrences instead of formulas.

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Definitions

The vocabulary of the language consists in: • An infinite set of atomic formulas both of form A; B; . . . and of form A⊥ ; B ⊥ ; . . .; • the following unary operators: ! and ?; • the following binary operators: ⊗; F; &; ⊕; • the following punctuation marks: ⇒; ,; (; ). We define the notion of formula in the usual way. Definition 1 Atomic formulas are formulas; if A is a formula then both !A and ?A are formulas; if A and B are formulas then A ⊗ B; AFB; A&B; A ⊕ B are formulas. The negation of a formula A (briefly A∗ ) is defined in the metalanguage in the following way. Definition 2 If A is an atomic formula of form B, then A∗ is B ⊥ ; if A is an atomic formula of form B ⊥ , then A∗ is B; 3

if if if if if if

A A A A A A

is is is is is is

!B then A∗ is ?B ∗ ; ?B then A∗ is !B ∗ B ⊗ C then A∗ is B ∗ FC ∗ ; BFC then A∗ is B ∗ ⊗ C ∗ ; B&C then A∗ is B ∗ ⊕ C ∗ ; B ⊕ C then A∗ is B ∗ &C ∗ .

Given a formula sequence Ξ, we define the notion of derivation of the sequent ⇒ Ξ by induction3 . Definition 3 For every atomic formula A ⇒ A, A⊥

Ax

is a derivation of ⇒ A, A⊥ ; if π1 is a derivation of ⇒ Γ, A and π2 a derivation of ⇒ ∆, A∗ then .. .. .π2 .π1 ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ is a derivation of ⇒ Γ, ∆; if π1 is a derivation of ⇒ Γ, A, B then .. .π1 ⇒ Γ, A, B Ex ⇒ Γ, B, A is a derivation of ⇒ Γ, B, A; if π1 is a derivation of ⇒ Γ, A; π2 of ⇒ Γ, B; π3 of ⇒ ∆, B and π4 of ⇒ Γ, A, B then .. .. .π1 .π3 ⇒ Γ, A ⇒ ∆, B ⊗ ⇒ Γ, ∆, A ⊗ B .. .π

.. .π1 2 ⇒ Γ, A ⇒ Γ, B & ⇒ Γ, A&B

.. .π1 ⇒ Γ, A ⊕ ⇒ Γ, A ⊕ B 1

.. .π4 ⇒ Γ, A, B F ⇒ Γ, AFB .. .π

2

⇒ Γ, B ⊕ ⇒ Γ, A ⊕ B 2

are derivations of (respectively) ⇒ Γ, ∆ A ⊗ B; ⇒ Γ, AFB; ⇒ Γ, A&B; ⇒ Γ, A ⊕ B; if π1 is a derivation of ⇒ Γ, A; π2 of ⇒ Γ; π3 of ⇒ Γ, ?A, ?A and π4 of ⇒?Γ, A then 3

The Cut-rule is given as schema, since its formulation makes use of the metalinguistic negation operator.

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.. .π1 ⇒ Γ, A ? ⇒ Γ, ?A

.. .π3 ⇒ Γ, ?A ?A ?C ⇒ Γ, ?A

.. .π2 ⇒ Γ, ?W ⇒ Γ, ?A

.. .π4 ⇒?Ω, A ! ⇒?Ω, !A

are derivations of (respectively) ⇒ Γ, ?A and ⇒?Ω, !A. In what follows we will speak of left and right premises, conclusions and applications of rules in derivations without giving full detailed definitions of such notions. By doing this, we hope not to lose clarity at all. Furthermore, we will consider, as occurrences of a formula A in a derivation π, only those occurrences such that, for some sequent ⇒ A1 , . . . , An in π and for some i, A = Ai . For example in the sequent ⇒ A, ?A, A ⊗ B, ?A&?B, A⊥ there are (in the sense specified) only one occurrence of A and one of ?A. We now define the notions of principal, auxiliary and contextual occurrences of a formula. Definition 4 The formula occurrences, in an application of a rule, appearing in the conclusions but not in the premises are said principal. The applications of the Cut-rule and of the ?C-rule have no principal occurrences of formulas. The formulas occurrences, in an application of a rule, appearing in the premises but not in the conclusions are said auxiliary. The applications of the Ax-rule, of the ?W -rule and of the ?C-rule have no auxiliary occurrences. The auxiliary occurrences of the applications of the Cut-rule are said Cut-occurrences. All the other occurrences are said contextual. Note that in the applications of the Ex-rule and of the ?C all the occurrences are contextual. We will call the occurrence of ?A in the conclusion of an application of the ?C-rule pseudo-principal4 . We define the notion of parent, ancestor and family of a contextual occurrence. Definition 5 If a contextual occurrence of A appears at the i-th place in Γ (∆) in the conclusion of a rule then the occurrence(s) of the same formula appearing at the i-th place in Γ (∆) in the premise(es) is(are) said parent( s) of the occurrence of A in the conclusion. 4 The term pseudo-principal is in analogy with the one used by Brauener and de Paiva in their proof of the Cut-elimination theorem for FILL in [Bra¨ uner de Paiva 1996]. In that work they used to refer to the reduction procedure cases, with which the number of Cuts of maximal degree decrees, as key cases. Although it is formally similar to these one, the case involving ?C does not reduce the number of Cuts of maximal degree. To underscore this difference the authors called it pseudo-key case, and deserved a special lemma for it in order to specify a procedure by means of which the number of Cuts does decrees. Here, instead, we will treat the case as a permutation one, hence we will actually deal with it in lemmas 4 and 5.

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Note that if the rule is Ex, the following holds: the occurrence of A at the second place in the premise is the parent of the occurrence of A at the first place in the conclusion. The analogue of this holds for B. Note that if the rule is ?C, the following holds: the two occurrences of ?A at the last two places in the premise are parents of the occurrence of ?A at the last place in the conclusion. We call ancestor the reflexive and transitive closure of the relation parent. Given an occurrence of a formula A, we call family of this occurrence the set of the its ancestor. Note: the family of a formula occurrence is a finite binary tree, whose root is the formula occurrence itself and whose leaves are occurrences of the same formula that are the principal occurrences of some application of a rule. The notion of family of a formula occurrence can be extend in a straightforward way to that of family of an occurrence of a sequence of formulas. We define the notion of degree of a formula A, d(A) and of height of a derivation π, h(π). Definition 6 For atomic formulas of form A, A⊥ d(A) = d(A⊥ ) = 1 Given two formulas A and B of degree d(A) and d(B) d(!A) = d(?A) = d(A) + 1 d(A&B) = d(A ⊗ B) = d(A ⊕ B) = d(AFB) = d(A) + d(B) By degree of an occurrence of a formula we mean the degree of the formula. By degree of an application of the Cut-rule we mean the degree of its Cut-occurrences5 . Given a derivation π, we define the degree of π, briefly dπ , in the following way: dπ = sup(d(A)) where A ranges over all Cut-occurrences in π. Definition 7 If π is an Axiom then h(π) = 1; if π is In each application of the Cut-rule, the two Cut-occurrences A and A∗ have always the same degree: this can be easily proved by induction on the degree of A, with the help of the defining clauses of definition 2. 5

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.. .. .. π1 .. π2 ⇒Γ ⇒∆ R ⇒Θ then h(π) = sup(h(π1 ), h(π2 )) + 1; if π is .. .. π1 ⇒Γ R ⇒∆ then either h(π) = h(π1 ) + 1 (if R 6= Ex) or h(π) = h(π1 ) (if R = Ex). Finally, we introduce the key notion of the proof. Definition 8 An application of the Cut-rule is said regular if both its Cutoccurrences are principal occurrences of an application of some rule. A Cut is irregular on the left (or on the right), briefly left\right irregular, if its left (right) Cut-occurrence is not principal. By Cut irregular at least on the left (right) we mean a Cut which is irregular on the left (right), but that can be irregular on the right\left as well. By Cut irregular at most on the left (right) we mean a Cut that is either regular or only irregular on the left (right). A derivation is regular if all its application of the Cut rule are regular.

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Regular derivations

In this section we will show how each derivation can be “transformed” in regular form. During the proof of the main lemma (lemma 4), we will make use of some properties of the derivations of sequents containing exclaimed formulas. In order to simplify the formulations of the lemmas concerning such properties, we define the notion of bounded set of subderivations relevant for a formula. Definition 9 Given a derivation π of ⇒ Γ. A set of subderivations bounded b ) if and only if it is by b is relevant for a formula occurrence A (in short RA a non-empty set of subderivation πi of π such that: 1. The number of the subderivations is at most b, that is, for some r such that r ≤ b, 1 ≤ i ≤ r. 2. Each πi ends with an application of a logical rule R (the same for all πi ) such that the principal occurrence of each application of R is one of the leaf occurrences of the family of the occurrence of A in the conclusion of π. 7

We start with lemma 1, which warrants the existence of a set of subderivations relevant for !A in each derivation of a sequent containing an exclaimed formula !A. The set is bounded by the number n of applications of the &rule in the whole derivation: if there are n applications of the &-rule in the whole derivation, than there are, in the set, at most n + 1 subderivations of the given form. n+1 Lemma 1 Given a derivation π of ⇒ Γ, !A. In π there is a set R!A , where n is the number of applications of the &-rule in π.

Proof By induction on h(π): h(π) = 1 Then π cannot be a derivation of ⇒ Γ, !A, whichever Γ may be6 . h(π) > 1 We have different cases, depending on which is the last rule R of π: • R is ⊕: π is

.. .λ ⇒ Γ, C, !A ⊕ ⇒ Γ, C ⊕ D, !A

n+1 The induction hypothesis on λ warrants the existence of a set R!A , where n is the number of applications of the &-rule in λ. Hence in π there is a set n+1 R!A = {πi | for every i, πi = λi }

• R is F: π is

.. .λ ⇒ Γ, C, D, !A F ⇒ Γ, CFD, !A

n+1 The induction hypothesis on λ warrants the existence of a set R!A , where n is the number of applications of the &-rule in λ. Hence in π there is a set n+1 R!A = {πi | for every i, πi = λi }

• R is &: π is

.. 0 .. 00 .λ .λ ⇒ Γ, C, !A ⇒ Γ, D, !A & ⇒ Γ, C&D, !A

The induction hypothesis on λ0 and λ00 warrants the existence of two n1 +1 sets R!A (of subderivations λ0i , with 1 ≤ i ≤ r0 and r0 ≤ n1 ) and n2 +1 R!A (of subderivations λ00i , with 1 ≤ i ≤ r00 and r00 ≤ n2 ), where n1 is the number of applications of the &-rule in λ0 and n2 is the number 6

Remember that we have only atomic “axioms”.

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of applications of the &-rule in λ00 . Since the number of applications of the &-rule in π is n1 + n2 + 1 in π there is a set n+1 R!A = {πi | for every i ≤ r0 , πi = λ0i and for every r0 < i ≤ r0 +r00 , πi = λ00i }

• R is ⊗: π is

.. 0 .. 00 .λ .λ ⇒ Γ, C, !A ⇒ Γ, D ⊗ ⇒ Γ, C ⊗ D, !A

n+1 The induction hypothesis on λ0 warrants the existence of a set R!A , 0 where n is the number of applications of the &-rule in λ . Hence in π there is a set n+1 R!A = {πi | for every i, πi = λ0i }

• R is a Cut: the case is an analogue of the previous one. • R is ?: π is

.. .λ ⇒ Γ, B, !A ? ⇒ Γ, ?B !A

n+1 The induction hypothesis on λ warrants the existence of a set R!A , where n is the number of applications of the &-rule in λ. Hence in π there is a set n+1 R!A = {πi | for every i, πi = λi }

• R is ?W : π is

.. .λ ⇒ Γ, !A ?W ⇒ Γ, ?B !A

n+1 The induction hypothesis on λ warrants the existence of a set R!A , where n is the number of applications of the &-rule in λ. Hence in π there is a set n+1 R!A = {πi | for every i, πi = λi }

• R is ?C: π is

.. .λ ⇒ Γ, ?B, ?B, !A ?C ⇒ Γ, ?B !A

n+1 The induction hypothesis on λ warrants the existence of a set R!A , where n is the number of applications of the &-rule in λ. Hence in π there is a set n+1 R!A = {πi | for every i, πi = λi }

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• R is !: π is7

.. .λ ⇒?Ω, A ! ⇒?Ω, !A

In this case n+1 R!A = {π}

that is, π itself is the only subderivation of ⇒ Γ, !A (?Ω stands for Γi ). The occurrence of !A is principal. It is the only leaf occurrence of the family of the occurrence of !A in the conclusion of π (of course the family contains only the occurrence itself). The lemma obtains.  Now we will introduce the notions of derivation with hypotheses and of noble family. Both notions have no theoretical relevance: their introduction renders the development of the proof more intuitive. Then we will prove another property of the derivations of sequents containing exclaimed formulas. Definition 10 We extend the notion of derivation to the one of derivation with hypotheses, by substituting the Ax clause in definition 1 with the following: Hyp ⇒Θ is a derivation ⇒ Θ with hypotheses ⇒ Θ. Note: the Ax-rule is considered as an instance of the Hyp-rule. Definition 11 The family of an occurrence of a sequence of formulas appearing in the conclusive sequent of a derivation π is called noble if all the leaf occurrences of the family belong to the hypotheses of π. We indicate the presence of a noble family of an occurrence of a sequence of formulas Γ in a derivation π with π(Γ). n+1 Given a derivation π of ⇒ Γ, !A, by lemma 1 there is in π a set R!A of subderivations λi (where n is the number of applications of the &-rule in π). We will call λ the derivation of ⇒ Γ, !A with hypotheses ⇒?∆i , !A obtained by cutting off from π the subderivations ending with ⇒?∆i , !A. A sketch of the structure of π follows:

.. .λi ! Ax Ax ⇒?∆i , !A . . .⇒. Λ1 . . . ⇒ Λm .. λ . ⇒ Γ, !A 7

In this case all the formula occurrences of Γ are interrogated, that is Γ =?Ω.

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where all the Λj , if any at all (that is, if 1 ≤ j ≤ m, for some m ≥ 1), are instances of the Ax-rule. Note that the presence of the contextual occurrence of !A, at each step of λ, doesn’t allow any application of the !-rule in λ. Consequently, during the induction in the following proof, there won’t be an inductive step involving the !-rule. Lemma 2 In the derivation λ of ⇒ Γ, !A with hypotheses ⇒?∆i , !A; ⇒ Λ1 ; . . . ; ⇒ Λm (with 1 ≤ i ≤ r, for some r ≤ n + 1, where n is the number of applications of the &-rule in π; m ≥ 0) the family of the occurrence of !A in the conclusion is noble; in particular all the leaf occurrences of the family belong to one of the ⇒?∆i , !A hypotheses of λ. Proof By induction on h(λ): h(λ) = 1 Then λ consists of a single hypothesis ⇒?∆, !A

Hyp

that is, in this case, Γ =?∆. The noble family consists of this single occurrence of !A. h(λ) > 1 We have different cases, depending on which is the last rule R of λ • R is &: then λ has the following form .. 0 .. 00 .λ .λ ⇒ Γ1 , !A ⇒ Γ2 , !A & ⇒ Γ, !A λ0 is a subderivation of ⇒ Γ1 , !A with hypotheses ⇒?∆1 , !A; . . . ; ⇒?∆p , !A; ⇒ Λ1 ; . . . ; ⇒ Λq (p ≤ r and q ≤ m); by induction hypothesis we get that in this subderivation, the family of the occurrence of !A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ ∆i , A hypotheses of λ0 (1 ≤ i ≤ p). λ00 is a subderivation of ⇒ Γ2 , !A with hypotheses ⇒?∆p+1 , !A; . . . ; ⇒?∆r , !A (r ≤ n + 1, where n is the number of applications of the &-rule in λ) ⇒ Λq+1 ; . . . ; ⇒ Λm (m ≥ 0); by induction hypothesis we get that in this subderivation, the family of the occurrence of !A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒?∆i , A hypotheses of λ00 (p + 1 i ≤ r). Hence the family of the !A occurrence in the conclusion of λ is noble as well, since it is made up of the two noble families of occurrences of !A in both λ0 and λ00 , together with the occurrence of !A in the conclusion of λ. 11

• R is ⊗ or Cut: then λ has the following form .. 0 .. 00 .λ .λ ⇒ Γ1 , !A ⇒ Γ2 ⊗\Cut ⇒ Γ, !A λ0 is a subderivation of ⇒ Γ1 , !A with hypotheses ⇒?∆1 , !A; . . . ; ⇒?∆r , !A; ⇒ Λ1 ; . . . ; ⇒ Λq (q ≤ m); by induction hypothesis we get that in this subderivation, the family of the occurrence of !A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ ∆i , A hypotheses of λ0 (1 ≤ i ≤ r). Hence the family of the !A occurrence in the conclusion of λ is noble as well, since it is made up of the noble family of occurrences of !A in λ0 together with the occurrence of !A in the conclusion of λ. • R is ⊕; F; ?; ?W ; ?C: then λ has the following form .. 0 .λ ⇒ Γ0 , !A ∗ ⇒ Γ, !A λ0 is a subderivation of ⇒ Γ0 , !A with hypotheses ⇒?∆1 , !A; . . . ; ⇒?∆r , !A; ⇒ Λ1 ; . . . ; ⇒ Λm ; by induction hypothesis we get that in this subderivation, the family of the occurrence of !A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ ∆i , A hypotheses of λ0 (1 ≤ i ≤ r). Hence the family of the !A occurrence in the conclusion of λ is noble as well, since it is made up of the noble family of occurrences of !A in λ0 together with the occurrence of !A in the conclusion of λ.  In order to prove the main lemma we need a last demonstration. During the proof of the lemma, we will use the following Proposition Given a derivation with hypotheses ⇒ Θ1 ; . . . ; ⇒ Θn of ⇒ Γ and n derivations of ⇒ Θi (with 1 ≤ i ≤ n), then there is a derivation (without hypotheses) of ⇒ Γ. Lemma 3 Given a derivation π of ⇒ Γ, A with hypotheses ⇒ Θ1 , A; . . . ; ⇒ Θn , A; ⇒ Λ1 ; . . . ; ⇒ Λm , in which the family of the occurrence of A in the conclusion is noble (that is, each leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π): by substituting all the occurrences of the noble family with occurrences of the formula sequence ∆ we get a derivation π 0 of 12

⇒ Γ, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π) = h(π 0 ) and d(π) = d(π 0 ). In the derivation obtained, the family of the occurrence of the formula sequence ∆ in the conclusion is noble; the family is such that • each occurrence of the family is one of the occurrences introduced by substitution; • each leaf occurrence belongs to one of the ⇒ Θi , ∆ hypotheses. The general validity of the lemma holds under the following restriction: if the derivation contains applications of the !-rule8 , then all the formulas in ∆ must be interrogated (i.e. ∆ = ?Σ). Proof By induction on h(π): h(π) = 1 Then π and π 0 are respectively ⇒ Γ, A

Hyp

⇒ Γ, ∆

Hyp

The lemma is satisfied in an obvious way. h(π) > 1 We have different cases, depending on which is the last rule R of π: • R is &: then π and π 0 are respectively .. .. .π1 .π2 ⇒ Γ, B, A ⇒ Γ, C, A & ⇒ Γ, B&C, A

.. 0 .. 0 .π1 .π2 ⇒ Γ, B, ∆ ⇒ Γ, C, ∆ & ⇒ Γ, B&C, ∆

π1 is a subderivation of ⇒ Γ, B, A with hypotheses ⇒ Θ1 , A; . . . ; ⇒ Θp , A; ⇒ Λ1 ; . . . ; ⇒ Λq (1 ≤ p ≤ n, q ≤ m); in this subderivation, the family of the occurrence of A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π1 (1 ≤ i ≤ p). π2 is a subderivation of ⇒ Γ, C, A with hypotheses ⇒ Θp+1 , A; . . . ; ⇒ Θn , A; ⇒ Λq+1 ; . . . ; ⇒ Λm ; in this subderivation, the family of the occurrence of A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π2 (p + 1 ≤ i ≤ n). The induction hypothesis can be applied to π1 : it warrants the existence of a derivation π10 of ⇒ Γ, B, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θp , ∆; ⇒ Λ1 ; ⇒ Λq such that h(π1 ) = h(π10 ) and d(π1 ) = d(π10 ); in this derivation, the family of the occurrence of ∆ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ∆ hypotheses of π10 (1 ≤ i ≤ p). 8

Note that in this case A must be an interrogated formula, since A is a contextual occurrence in the applications of the !-rule (i.e. A =?E).

13

The induction hypothesis can be applied to π2 : it warrants the existence of a derivation π20 of ⇒ Γ, C, ∆ with hypotheses ⇒ Θp+1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λq+1 ; . . . ; ⇒ Λm such that h(π2 ) = h(π20 ) and d(π2 ) = d(π20 ); in this derivation, the family of the occurrence of ∆ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ∆ hypotheses of π20 (p + 1 i ≤ n). Thus π 0 is a derivation of ⇒ Γ, B&C, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π 0 ) = sup(h(π10 ), h(π20 )) + 1 = sup(h(π1 ), h(π2 ) + 1 = h(π) d(π 0 ) = sup(d(π10 ), d(π20 )) = sup(d(π1 ), d(π2 )) = d(π) In this derivation the family of the occurrence of ∆ in the conclusion is noble, since it consists of the union of the two noble families of ∆ belonging to π10 and to π20 together with the occurrence in the conclusion of π 0 . All the leaf occurrence of such a family either belongs to one of the ⇒ Θi , ∆ hypotheses of π10 (1 ≤ i ≤ p), or to one of the ⇒ Θi , ∆ hypotheses of π20 (p + 1 ≤ i ≤ n): that is, they belong to one of the ⇒ Θi , ∆ hypotheses of π 0 itself (1 ≤ i ≤ n). So the lemma obtains. • R is ⊗: then π and π 0 are respectively .. .. .π2 .π1 ⇒ Γ1 , B, A ⇒ Γ2 , C ⊗ ⇒ Γ, B ⊗ C, A

.. 0 .. .π1 .π2 ⇒ Γ1 , B, ∆ ⇒ Γ2 , C ⊗ ⇒ Γ, B ⊗ C, ∆

π1 is a subderivation of ⇒ Γ1 , B, A with hypotheses ⇒ Θ1 , A; . . . ; ⇒ Θn , A; ⇒ Λ1 ; . . . ; ⇒ Λq (q ≤ m); in this subderivation, the family of the occurrence of A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π1 (1 ≤ i ≤ n). π2 is a subderivation of ⇒ Γ2 , C with hypotheses ⇒ Λq+1 ; . . . ; ⇒ Λm . The induction hypothesis can be applied to π1 : it warrants the existence of a derivation π10 of ⇒ Γ1 , B, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; ⇒ Λq such that h(π1 ) = h(π10 ) and d(π1 ) = d(π10 ); in this derivation, the family of the occurrence of ∆ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ∆ hypotheses of π10 (1 ≤ i ≤ n); Thus π 0 is a derivation of ⇒ Γ, B ⊗ C, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π 0 ) = sup(h(π10 ), h(π2 )) + 1 = sup(h(π1 ), hπ2 ) + 1 = h(π)

14

d(π 0 ) = sup(d(π10 ), d(π2 ) = sup(d(π1 ), d(π2 )) = d(π) In this derivation the family of the occurrence of ∆ in the conclusion is noble, since it consists of the union of the noble family of ∆ belonging to π10 together with the occurrence in the conclusion of π 0 . All the leaf occurrence of such a family belongs to one of the ⇒ Θi , ∆ hypotheses of π10 : that is, they belong to one of the ⇒ Θi , ∆ hypotheses of π 0 itself (1 ≤ i ≤ n). So the lemma obtains. • R is a Cut: then π and π 0 are respectively .. .. .π2 .π1 ⇒ Γ1 , B, A ⇒ Γ2 , B ∗ Cut ⇒ Γ, A

.. 0 .. .π1 .π2 ⇒ Γ1 , B, ∆ ⇒ Γ2 , B ∗ Cut ⇒ Γ, ∆

π1 is a subderivation of ⇒ Γ1 , B, A with hypotheses ⇒ Θ1 , A; . . . ; ⇒ Θn , A; ⇒ Λ1 ; . . . ; ⇒ Λq (q ≤ m); in this subderivation, the family of the occurrence of A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π1 (1 ≤ i ≤ n). π2 is a subderivation of ⇒ Γ2 , B ∗ with hypotheses ⇒ Λq+1 ; . . . ; ⇒ Λm . The induction hypothesis can be applied to π1 : it warrants the existence of a derivation π10 of ⇒ Γ1 , B, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; ⇒ Λq such that h(π1 ) = h(π10 ) and d(π1 ) = d(π10 ); in this derivation, the family of the occurrence of ∆ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ∆ hypotheses of π10 (1 ≤ i ≤ n). Thus π 0 is a derivation of ⇒ Γ, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π 0 ) = sup(h(π10 ), h(π2 )) + 1 = sup(h(π1 ), hπ2 ) + 1 = h(π) and either d(π 0 ) = sup(d(π10 ), d(π2 ) = sup(d(π1 ), d(π2 )) = d(π) or (π 0 ) = d(π) = d(B) In this derivation the family of the occurrence of ∆ in the conclusion is noble, since it consists of the union of the noble family of ∆ belonging to π10 together with the occurrence in the conclusion of π 0 . All the leaf occurrence of such a family belongs to one of the ⇒ Θi , ∆ hypotheses of π10 : that is, they belong to one of the ⇒ Θi , ∆ hypotheses of π 0 itself (1 ≤ i ≤ n). So the lemma obtains. 15

• R is ⊕; F; ?; ?W or ?C: then π and π 0 are respectively .. .π1 ⇒ Γ, A ∗ ⇒ Γ0 , A

.. 0 .π1 ⇒ Γ, ∆ ∗ ⇒ Γ0 , ∆

π1 is a subderivation of ⇒ Γ, A with hypotheses ⇒ Θ1 , A; . . . ; ⇒ Θn , A; ⇒ Λ1 ; . . . ; ⇒ Λm ; in this subderivation, the family of the occurrence of A in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , A hypotheses of π1 (1 ≤ i ≤ n). The induction hypothesis can be applied to π1 : it warrants the existence of a derivation π10 of ⇒ Γ, ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; ⇒ Λm such that h(π1 ) = h(π10 ) and d(π1 ) = d(π10 ); in this derivation, the family of the occurrence of ∆ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ∆ hypotheses of π10 (1 ≤ i ≤ n); Thus π 0 is a derivation of ⇒ Γ0 , ∆ with hypotheses ⇒ Θ1 , ∆; . . . ; ⇒ Θn , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π 0 ) = h(π10 ) + 1 = h(π1 ) + 1 = h(π) d(π 0 ) = d(π10 ) = (d(π1 ) = d(π) In this derivation the family of the occurrence of ∆ in the conclusion is noble, since it consists of the union of the noble family of ∆ belonging to π10 together with the occurrence of ∆ in the conclusion of π 0 . All the leaf occurrence of such a family belongs to one of the ⇒ Θi , ∆ hypotheses of π10 : that is, they belong to one of the ⇒ Θi , ∆ hypotheses of π 0 itself (1 ≤ i ≤ n). So the lemma obtains. • R is !: then π and π 0 are respectively9 .. .π1 ⇒?Ω, B, ?E ! ⇒?Ω, !B, ?E

.. 0 .π1 ⇒?Ω, B, ?Σ ! ⇒?Ω, !B, ?Σ

π1 is a subderivation of ⇒?Ω, B, ?E with hypotheses ⇒ Θ1 , ?E; . . . ; ⇒ Θn , ?E; ⇒ Λ1 ; . . . ; ⇒ Λm ; in this subderivation, the family of the occurrence of ?E in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ?E hypotheses of π1 (1 ≤ i ≤ n). 9 As noted above, in this case A must be interrogated, that is A =?E; all the formulas in Γ must be so as well, that is Γ =?Ω.

16

The induction hypothesis can be applied to π1 : it warrants the existence of a derivation π10 of ⇒?Ω, B, ?Σ with hypotheses ⇒ Θ1 , ?Σ; . . . ; ⇒ Θn , ?Σ; ⇒ Λ1 ; ⇒ Λm such that h(π1 ) = h(π10 ) and d(π1 ) = d(π10 ); in this derivation, the family of the occurrence of ?Σ in the conclusion is noble and all its leaf occurrences belong to one of the ⇒ Θi , ?Σ hypotheses of π10 (1 ≤ i ≤ n); Thus π 0 is a derivation of ⇒?Ω, !E, ?Σ with hypotheses ⇒ Θ1 , ?Σ; . . . ; ⇒ Θn , ?Σ; ⇒ Λ1 ; . . . ; ⇒ Λm such that h(π 0 ) = h(π10 ) + 1 = h(π1 ) + 1 = h(π) d(π 0 ) = d(π10 ) = d(π1 ) = d(π) In this derivation the family of the occurrence of ?Σ in the conclusion is noble, since it consists of the union of the noble family of ?Σ belonging to π10 together with the occurrence of ?Σ in the conclusion of π 0 . All the leaf occurrence of such a family belongs to one of the ⇒ Θi , ?Σ hypotheses of π10 : that is, they belong to one of the ⇒ Θi , ?Σ hypotheses of π 0 itself (i ≤ n). So the lemma obtains.  We are now ready to prove the main lemma, which will be directly used in the proof of the permutation-like theorem. To simplify the formulation of the lemma, we introduce the notion of bounded set Cut-relevant for a formula. This notion is analogous to the one of bounded set relevant for a formula occurrence in many aspects. Definition 12 Given a derivation π of ⇒ Γ. A set of subderivations bounded b ) if and only if it is a nonby b is Cut-relevant for a formula A (in short CA empty set of subderivation πi of π such that: 1. The number of the subderivations is at most b, that is, for some r such that r ≤ b, 1 ≤ i ≤ r. 2. Each πi ends with an application of the Cut-rule with A and A∗ as Cut-occurrences. Lemma 4 (Main Lemma) Given a derivation π of ⇒ Ξ such that: 1. The last rule applied in π is a Cut which is at least irregular on the right. Its Cut-occurrences are occurrences of A and A∗ and the subderivations of its two premises are π1 and π2 ; 2. Every other Cut in π is regular; 17

then there is another derivation π 0 of ⇒ Ξ such that: d(π) = d(π 0 ) and if in π there are no Cuts of degree d, so in π 0 ; furthermore we distinguish two cases: • if the irregular Cut in π wasn’t left irregular, then π 0 is regular; n+1 • otherwise in π 0 there is a set CA (n the number of applications of the &-rule in π1 ) of subderivations σi which is Cut-relevant for A. Each n+1 σi ∈ CA is such that:

1. Its conclusive Cut is irregular at most on the left. 2. All its other applications of the Cut-rule are regular. Finally π 0 is such that ♠ The family of each occurrence in the conclusion of π 0 does not contain more pseudo-principal occurrences then the family of the corresponding occurrence in the conclusion of π. Proof The general form of π is the following: .. .. .π1 .π2 R ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ The structure of the proof is the following: the main induction is on the number cπ of pseudo-principal occurrences in the family of the occurrence of A∗ in the conclusive sequent of π1 . This is just a trick in order to keep the contraction case apart form all the other ones. Hence the core of the proof is represented by the cπ = 0 case. In order to prove it we make an induction on the height of π1 . The complexity of the proof is increased by what seems to be a further induction required to deal with the case in which R is an application of the Cut-rule. The induction is on the degree of the Cut-occurrences H and H ∗ . Nevertheless we will not need to make use of the induction hypothesis on the degree. So the third nested induction is more properly just a list of sub-cases. As the reader can easily check case by case, the condition ♠ holds trivially; nonetheless it is required in order to carry out the proof for the cπ > 0 inductive case. So we decided to specify it in the formulation of the lemma, although we won’t explicitly check it during the proof. cπ = 0 We give a proof of this case by induction on h(π1 ): h(π1 ) = 1 In this case the Cut is not irregular on the right. h(π1 ) > 1 We have to note that if the principal occurrence of R is the relevant occurrence of A∗ , then π 0 is π itself. Otherwise we distinguish different cases depending on which is the rule R: 18

• R is ⊕ or F: then π is .. .πA∗ .. ⇒ ∆, A∗ .π2 ⊕\F ⇒ Γ, A ⇒ ∆0 , A∗ Cut ⇒ Γ, ∆0 First we produce a proof µ of the following form: .. .. .πA∗ .π2 ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ ⊕\F ⇒ Γ, ∆0 We call µ1 the subderivation of ⇒ Γ, ∆. We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule, which is irregular at least on the right, that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πA∗ is lower than that of π1 : in particular h(π1 ) = h(πA∗ ) + 1 > h(πA∗ ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left irregular, then n ∗ +1 µ01 is regular; otherwise it contains a set CAA (nA∗ is the number n ∗ +1 of applications of the &-rule in πA∗ ): each τi ∈ CAA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .µ 1

⇒ Γ, ∆ ⊕\F ⇒ Γ, ∆0 Note that if in π there are no Cuts of degree n, neither in π 0 there are. n ∗ +1 n+1 Furthermore either π 0 is regular or in π 0 there is a set CA = CAA , n+1 since n = nA∗ : each σi ∈ CA satisfies the requirements of the lemma, nA∗ +1 as each τi ∈ CA does. • R is &: then π is .. .. .πB .πC .. ∗ ⇒ ∆, A , B ⇒ ∆, A∗ , C .π2 & ∗ ⇒ ∆, A , B&C ⇒ Γ, A Cut ⇒ Γ, ∆, B&C 19

First we produce a proof µ of the following form: .. .. .. .. .πB .πC .π2 .π2 ∗ ⇒ Γ, A ⇒ ∆, A , B ⇒ Γ, A ⇒ ∆, A∗ , C Cut Cut ⇒ Γ, ∆, B ⇒ Γ, ∆, C & ⇒ Γ, ∆, B&C We call µ1 the subderivation of ⇒ Γ, ∆, B and µ2 the subderivation of ⇒ Γ, ∆, C. Note that d(µ1 ) = d(µ2 ) = d(π), or d(µ2 ) ≤ d(µ1 ) = d(π) or d(µ1 ) ≤ d(µ2 ) = d(π). We can apply the induction hypothesis to both µ1 and µ2 , since 1) there is an application of the Cut-rule, which is irregular at least on the right, that concludes the subderivations; 2) all the other applications of the Cut-rule in the derivations are regular; 3) the height of πB and πC is lower than that of π1 : in particular h(π1 ) = sup(h(πB ), h(πC )) + 1 > h(πB ) h(π1 ) = sup(h(πB ), h(πC )) + 1 > h(πC ) The induction hypothesis warrants the existence of two derivation µ01 and µ02 such that: d(µ01 ) = d(µ1 ) and d(µ02 ) = d(µ2 ); if in µ1 (µ2 ) there are no Cuts of degree d, so in µ01 (µ02 ). Furthermore, if the irregular Cut in π wasn’t left irregular, then both µ01 and µ02 are regular; otherwise nB +1 nC +1 both contain a set (respctively) CA and CA (mB and mC are the numbers of applications of the &-rule (respectively) in πB and πC ): nB +1 nC +1 each τj ∈ CA and each τz ∈ CA are such that their conclusive Cuts are irregular at most on the left and all their other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivations µ01 and µ02 to the subderivations µ1 and µ2 (the proposition warrants the correctness of such operation) .. 0 .. 0 .µ2 .µ1 ⇒ Γ, ∆, B ⇒ Γ, ∆, B & ⇒ Γ, ∆, B&C Note that if in π there are no Cuts of degree n, so in π 0 . Furthermore nB +1 nC +1 n+1 either π 0 is regular or in π 0 there is a CA = CA ∪ CA , since n+1 n = nB + nC + 1; each σi ∈ CA satisfies the requirements of the nB +1 nC +1 lemma, as each τj ∈ CA and each τz ∈ CA do. • R is ⊗: then π is .. .. .πB .πC .. ∗ ⇒ Θ, A , B ⇒ Λ, C .π2 ⊗ ⇒ Γ, A ⇒ ∆, A∗ , B ⊗ C Cut ⇒ Γ, ∆, B ⊗ C 20

First we produce a proof µ of the following form: .. .. .πB .π2 .. ⇒ Γ, A ⇒ Θ, A∗ , B .πC Cut ⇒ Γ, Θ, B ⇒ Λ, C ⊗ ⇒ Γ, ∆, A ⊗ B We call µ1 the subderivation of ⇒ Γ, Θ, B. We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule, which is irregular at least on the right, that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πB is lower than that of π1 : in particular h(π1 ) = sup(h(πB ), h(πC )) + 1 > h(πB ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left irregular, then nB +1 µ01 is regular; otherwise it contains a set CA (nB is the number nB +1 of applications of the &-rule in πB ): each τi ∈ CA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .. .µ1 .πC ⇒ Γ, Θ, B ⇒ Λ, C ⊗ ⇒ Γ, ∆, A ⊗ B Note that d(π 0 ) = sup(d(µ01 , d(πC )) = sup(sup(d(π2 ), d(πB ), d(A)), d(πC )) = d(π) = sup(d(π2 ), sup(d(πB ), d(πC ), d(A))) = If in π there are no Cuts of degree n, so in π 0 . Furthermore either π 0 nB +1 n+1 is regular or in π 0 there is a set CA = CA , since n ≥ nB : each nB +1 n+1 σi ∈ CA satisfies the requirements of the lemma, as each τi ∈ CA does. • R is ?: then π

.. .πA∗ .. ⇒ ∆, A∗ , B .π2 ? ⇒ Γ, A ⇒ ∆, A∗ , ?B Cut ⇒ Γ, ∆, ?B 21

First we produce a proof µ of the following form: .. .. .πA∗ .π2 ⇒ Γ, A ⇒ ∆, A∗ , B Cut ⇒ Γ, ∆, B ? ⇒ Γ, ∆, ?B We call µ1 the subderivation of ⇒ Γ, ∆, B. Note that d(µ1 ) = d(π). We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule, which is irregular at least on the right, that concludes the subderivation; 2) all other applications of the Cutrule in the derivation are regular; 3) the height of πA∗ is lower than that of π1 : in particular h(π1 ) = h(πA∗ ) + 1 > h(πA∗ ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left irregular, then n ∗ +1 µ01 is regular; otherwise it contains a set CAA (nA∗ is the number n ∗ +1 of applications of the &-rule in πA∗ ): each τi ∈ CAA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) µ01 ⇒ Γ, ∆, B ? ⇒ Γ, ∆, ?B Note that if in π there are no Cuts of degree n, so in π 0 . Furthermore n ∗ +1 n+1 either π 0 is regular or in π 0 there is a set CA = CAA , since n = n+1 nA∗ : each σi ∈ CA satisfies the requirements of the lemma, as each nA∗ +1 τi ∈ CA does. • R is ?W or ?C: the cases are analogous to the previous one. • R is !: π has this form .. .πA∗ .. ⇒?Σ, ?E ∗ , B .π2 ! ⇒ Γ, !E ⇒?Σ, ?E ∗ , !B Cut ⇒ Γ, ?Σ, !B

22

Note that µ can’t have this form .. .. .πA∗ .π2 ⇒ Γ, !E ⇒?Σ, ?E ∗ , B Cut ⇒ Γ, ?Σ, B ! ⇒ Γ, ?Σ, !B because the !-rule would violate the restriction that all the contextual occurrences of the premise must be interrogated, since we do not have any warrant that all the formulas in Γ are interrogated. To solve this problem, we will use the lemmas from 1 to 3 concerning derivations in which exclaimed formulas occur. We can apply lemma 1 to π2 . n2 +1 Then in π2 there are is a set R!A (n2 is the number of applications of the &-rule in π2 ), which is relevant for !A, of subderivations λk of ⇒?Θk , !E (with 1 ≤ k ≤ r, for some r ≤ n2 + 1). The structure of π is: .. .λk ! .. ⇒?Θk , !E . . . ⇒ Λ1 . . . ⇒ Λm .πA∗ .. ⇒?Σ, ?E ∗ , B .λ ! ⇒ Γ, !E ⇒?Σ, ?E ∗ , !B Cut ⇒ Γ, ?Σ, !B We call λ the derivation of ⇒ Γ, !E with hypotheses ⇒ Θk , !E, ⇒ Λ1 , . . . , ⇒ Λm , obtained by cutting off the subderivations of ⇒ Θk , !E from π2 . Note that all the ⇒ Λj , if any at all (that is, if 1 ≤ j ≤ m, for some m ≥ 1), are axioms. The application of lemma 2 to λ warrants that the family of the occurrence of !E in the conclusion is noble (briefly λ(!E)). The structure of λ(!E) is: ⇒?Θk , !E

Hyp

. . . .⇒ Λ1 .. λ . ⇒ Γ, !E

Ax

. . . ⇒ Λm

Ax

So, we can apply lemma 3 to λ(!E) in order to obtain a derivation λ(!B, ?Σ) of ⇒ Γ, !B, ?Σ with hypotheses ⇒?Θk , !B, ?Σ and ⇒ Λj . Note that d(λ(!B, ?Σ)) = d(λ(!E)). The structure of λ(B, ?Σ) is the following: ⇒?Θk , !B, ?Σ . ... ⇒ Λ1 . . . ⇒ Λm .. λ . ⇒ Γ, !B, ?Σ

23

We have a derivation of each hypothesis ⇒?Θk , !B, ?Σ: .. .. .πA∗ .λk ! ⇒?Θk , !E ⇒?Σ, ?E ∗ , B Cut ⇒?Θk , B, ?Σ ! ⇒?Θk , !B, ?Σ So, the proposition warrants the existence of a derivation µ of ⇒ Γ, ?Σ, !B. The structure of the derivation can be sketched as follows: .. .. .πA∗ .λk ! ⇒?Θk , !E ⇒?Σ, ?E ∗ , B Cut ⇒?Θk , B, ?Σ ! ⇒ Λ1 . . . ⇒ Λm ⇒?Θk , !B, ?Σ .. .λ ⇒ Γ, !B, ?Σ We call µk the subderivations of ⇒?Θk , B, ?Σ. Note that d(µk ) ≤ d(π) We can apply the induction hypothesis to each µk , since 1) there is an application of the Cut-rule which is irregular only (so, at most) on the right that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πA∗ is lower than that of π1 : in particular h(π1 ) = h(πA∗ ) + 1 > h(πA∗ ) The induction hypothesis warrants the existence of derivations µ0k such that: d(µ0k ) = d(µk ); if in µk there are no Cuts of degree d, so in µ0k . Each µ0k is regular. So, we can build the derivation π 0 , by substituting in µ the derivations µ0k to all the µk (the proposition warrants the correctness of such operation) µ0k ⇒?Θk , B, ?Σ ! ⇒ Λ1 . . . ⇒ Λm ⇒?Θk , !B, ?Σ .. .λ ⇒ Γ, !B, ?Σ Note that: d(π 0 ) = sup(d(µ0k ), d(λ)) = d(π); if in π there are no Cuts of degree d, so in π 0 , since the same holds for µk and µ0k ; π 0 is regular. 24

• R is Cut: the form of π is .. .. .πH .πH ∗ .. R R ⇒ ∆1 , H, A∗ 1 ⇒ ∆2 , H ∗ 2 .π2 Cut ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ Note that H is the principal occurrence of R1 and H ∗ is the principal occurrence of R2 since, by hypothesis, there is only one irregular Cut in π. We have different sub-cases, depending on the form of the Cutoccurrences H and H ∗ : – H is an atomic formula of form B: then π and π 0 are respectively .. .π2 ⇒ Γ, B

Ax Ax ⇒ B, B ⊥ ⇒ B, B ⊥ Cut ⇒ B, B ⊥ Cut ⇒ Γ, B

.. .π2 Ax Γ, B ⇒ B, B ⊥ Cut ⇒ Γ, B

– H is an atomic formula of form B ⊥ : then π and π 0 are respectively .. .π2 ⇒ Γ, B ⊥

Ax Ax ⇒ B, B ⊥ ⇒ B, B ⊥ Cut ⇒ B, B ⊥ Cut ⇒ Γ, B ⊥

.. .π2 Ax Γ, B ⊥ ⇒ B, B ⊥ Cut ⇒ Γ, B ⊥

– H is B ⊗ C: then π is .. .. πF .πB .πC ∗ ⇒ Λ1 , A , B ⇒ Λ2 , C ∆2 , B ∗ , C ∗ .. F ⊗ ∗ ⇒ ∆1 , A , B ⊗ C ⇒ ∆2 , B ∗ FC ∗ .π2 Cut ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ First we produce a proof µ of the following form: .. .. .πB .π2 .. πF ⇒ Γ, A ⇒ Λ1 , A∗ , B .πC Cut ⇒ ∆2 , B ∗ , C ∗ ⇒ Γ, Λ1 , B ⇒ Λ2 , C F ⊗ ⇒ Γ, ∆1 , B ⊗ C ⇒ ∆2 , B ∗ FC ∗ Cut ⇒ Γ, ∆ We call µ1 the subderivation of ⇒ Γ, Λ1 , B. Note that d(µ1 ) ≤ d(π)

25

We can apply the induction hypothesis to µ1 , since 1) there is one application of the Cut-rule which is irregular at least on the right that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πB is lower than that of π1 : in particular h(π1 ) = sup(sup(h(πB ), h(πC )) + 1, h(πF )) + 1 > h(πB ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left nB +1 irregular, then µ01 is regular; otherwise it contains a set CA (nB is the number of applications of the &-rule in πB ): each nB +1 τi ∈ CA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .. πF .µ1 .πC ⇒ ∆2 , B ∗ , C ∗ ⇒ Γ, Λ1 , B ⇒ Λ2 , C F ⊗ ⇒ Γ, ∆1 , B ⊗ C ⇒ ∆2 , B ∗ FC ∗ Cut ⇒ Γ, ∆ Note that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 , since the same holds for µ1 and µ01 . Furthermore either nB +1 n+1 π 0 is regular or in π 0 there is a set CA = CA , since n = nB : n+1 each σi ∈ CA satisfies the requirements of the lemma, as each nB +1 τi ∈ CA does. – H is BFC: then π is .. .πF .. .π⊗ ⇒ ∆1 , A∗ , B, C .. F ⊗ ∗ ⇒ ∆1 , A , BFC ⇒ ∆2 , B ∗ ⊗ C ∗ .π2 Cut ⇒ ∆, A∗ ⇒ Γ, A Cut ⇒ Γ, ∆ First we produce a proof µ of the following form: .. .. .πF .π2 ⇒ Γ, A ⇒ ∆1 , A∗ , B, C .. Cut .π⊗ ⇒ Γ, ∆1 , B, C ⊗ F ⇒ Γ, ∆1 , BFC ⇒ ∆2 , B ∗ ⊗ C ∗ Cut ⇒ Γ, ∆ 26

We call µ1 the subderivation of ⇒, ∆1 , B, C. Note that d(µ1 ) ≤ d(π) We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule which is irregular at least on the right that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πF is lower than that of π1 : in particular h(π1 ) = sup(h(πF ) + 1, h(π⊗ )) + 1 > h(πF ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left n +1 irregular, then µ01 is regular; otherwise it contains a set CAF (nF is the number of applications of the &-rule in πF ): each n +1 τi ∈ CAF is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .µ1 .. .π⊗ ⇒ Γ, ∆1 , B, C ⊗ F ⇒ Γ, ∆1 , BFC ⇒ ∆2 , B ∗ ⊗ C ∗ Cut ⇒ Γ, ∆ Note that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 , since the same holds for µ1 and µ01 . Furthermore either nF+1 n+1 π 0 is regular or in π 0 there is a set CA = CA , since n ≥ nF : n+1 each σi ∈ CA satisfies the requirements of the lemma, as each n +1 τi ∈ CAF does. – H is B&C: then π is .. .. .. .π⊕ .πB .πC ∗ ∗ ⇒ ∆1 , A , B ⇒ ∆1 , A , C ∆2 , B ∗ .. ⊕ & ∗ ⇒ ∆1 , A , B&C ⇒ ∆2 , B ∗ ⊕ C ∗ .π2 Cut ⇒ ∆, A∗ ⇒ Γ, A Cut ⇒ Γ, ∆ First we produce a proof µ of the following form: . . . . ..π ..π ..π ..π B C 2 2 ∗ ⇒ Γ, A ⇒ ∆1 , A , B ⇒ Γ, A ⇒ ∆1 , A∗ , C Cut Cut ⇒ Γ, ∆1 , B ⇒ Γ, ∆1 , C & ⇒ Γ, ∆1 , B&C ⇒ Γ, ∆

27

.. .π⊕ ⊕ ⇒ ∆2 , B ∗ ⊕ C ∗ Cut

We call µ1 and µ2 the subderivations of (respectively) ⇒ Γ, ∆1 , B and ⇒ Γ, ∆1 , C. Note that d(µ1 ) ≤ d(π) and d(µ2 ) ≤ d(π). We can apply the induction hypothesis to µ1 and µ2 , since 1) there is one application of the Cut-rule which is irregular at least on the right that concludes each subderivation; 2) all other applications of the Cut-rule in both subderivations are regular; 3) the heights of πB and πC are lower than that of π1 : in particular h(π1 ) = sup(sup(h(πB ), h(πC )) + 1, h(π⊕ )) + 1 > h(πB ) h(π1 ) = sup(sup(h(πB ), h(πC )) + 1, h(πF )) + 1 > h(πC ) The induction hypothesis warrants the existence of two derivations µ01 and µ02 such that: d(µ01 ) = d(µ1 ) and d(µ02 ) = d(µ2 ); if in µ1 (µ2 ) there are no Cuts of degree d, so in µ01 (µ02 ). Furthermore, if the irregular Cut in π wasn’t left irregular, then both µ01 and µ02 are regular; otherwise each one contains a set (respectively) nB +1 nC +1 CA and CA (nB and nC are the numbers of applications nB +1 of the &-rule (respectively) in πB and πC ): each τj ∈ CA and nC +1 each τz ∈ CA are such that their conclusive Cut are irregular at most on the left and all their other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivations µ01 and µ02 to (respectively) the subderivations µ1 and µ2 (the proposition warrants the correctness of such operation) .. 0 .. 0 .µ2 .µ1 .. .π⊕ ⇒ Γ, ∆1 , B ⇒ Γ, ∆1 , C ⊕ & ⇒ Γ, ∆1 , B&C ⇒ ∆2 , B ∗ ⊕ C ∗ Cut ⇒ Γ, ∆ Note that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 , since the same holds for µ1 and µ01 and for µ2 and µ02 . Furthermore either π 0 is regular or in π 0 there is a set nB +1 nC +1 n+1 CA = CA ∪ CA n+1 since n ≥ nB + nC + 1; each σi ∈ CA satisfies the requirements nB +1 nC +1 of the lemma, as each τj ∈ CA and each τz ∈ CA do.

– H is B ⊕ C: then π is .. .πB .. ⇒ ∆1 , A∗ , B .π& .. ⊕ & ∗ ⇒ ∆1 , A , B ⊕ C ⇒ ∆2 , B ∗ &C ∗ .π2 Cut ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ 28

First we produce a proof µ of the following form: .. .. .πB .π2 ⇒ Γ, A ⇒ ∆1 , A∗ , B .. Cut ⇒ Γ, ∆1 , B .π& ⊕ & ⇒ Γ, ∆1 , B ⊕ C ⇒ ∆2 , B ∗ &C ∗ Cut ⇒ Γ, ∆ We call µ1 the subderivation of ⇒ Γ, ∆1 , B. Note that d(µ1 ) ≤ d(π) We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule which is irregular at least on the right that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of πB is lower than that of π1 : in particular h(π1 ) = sup(h(πB ) + 1, h(π& )) + 1 > h(πB ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left nB +1 irregular, then µ01 is regular; otherwise it contains a set CA (nB is the number of applications of the &-rule in πB ): each nB +1 τi ∈ CA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .µ1 .. ⇒ Γ, ∆1 , B .π& F & ⇒ Γ, ∆1 , B ⊕ C ⇒ ∆2 , B ∗ &C ∗ Cut ⇒ Γ, ∆ Note that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 , since the same holds for µ1 and µ01 . Furthermore either nB +1 n+1 π 0 is regular or in π 0 there is a set CA = CA , since n ≥ nB : n+1 each σi ∈ CA satisfies the requirements of the lemma, as each nB +1 τi ∈ CA does.

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– H is ?B: then π is10 .. .π? .. ⇒ ∆01 , A∗ .π! .. # ! ⇒ ∆1 , A∗ , ?B ⇒ ∆2 , !B ∗ .π2 Cut ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ First we produce a proof µ of the following form: .. .. .π? .π2 ⇒ Γ, A ⇒ ∆01 , A∗ .. Cut ⇒ Γ, ∆01 .π! # ! ⇒ Γ, ∆1 , ?B ⇒ ∆2 , !B ∗ Cut ⇒ Γ, ∆ We call µ1 the subderivation of ⇒ Γ, ∆01 . Note that d(µ1 ) ≤ d(π). We can apply the induction hypothesis to µ1 , since 1) there is an application of the Cut-rule which is irregular at least on the right that concludes the subderivation; 2) all other applications of the Cut-rule in the derivation are regular; 3) the height of π? is lower than that of π1 : in particular h(π1 ) = sup(h(π? ) + 1, h(π! )) + 1 > h(π? ) The induction hypothesis warrants the existence of a derivation µ01 such that: d(µ01 ) = d(µ1 ); if in µ1 there are no Cuts of degree d, so in µ01 . Furthermore, if the irregular Cut in π wasn’t left n? +1 irregular, then µ01 is regular; otherwise it contains a set CA (n? n? +1 is the number of applications of the &-rule in π? ): each τi ∈ CA is such that its conclusive Cut is irregular at most on the left and all its other applications of the Cut-rule are regular. So, we can build the derivation π 0 , by substituting in µ the derivation µ01 to the subderivation µ1 (the proposition warrants the correctness of such operation) .. 0 .µ1 .. ⇒ Γ, ∆01 .π! # ! ⇒ Γ, ∆1 , ?B ⇒ ∆2 , !B ∗ Cut ⇒ Γ, ∆ 10

We use the symbol # to denote one of the three rule ?; ?W ; ?C (possibly) together with some applications of the Ex-rule. Furthermore, if # =?W then ∆01 = ∆1 ; if # =?C then ∆01 = ∆1 , ?B, ?B; if # =? then ∆01 = ∆1 , B.

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Note that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 , since the same holds for µ1 and µ01 . Furthermore either n? +1 n+1 π 0 is regular or in π 0 there is a set CA = CA , since n ≥ n? : n+1 each σi ∈ CA satisfies the requirements of the lemma, as each n? +1 τi ∈ CA does. – H is !B: then π is11

.. .π2 ⇒ Γ, !E

.. .π! .. ⇒?Σ, ?E ∗ , !B .π? # ! ∗ ⇒?Σ, ?E , !B ⇒ ∆2 , ?B ∗ Cut ⇒ ∆, ?E ∗ Cut ⇒ Γ, ∆

As in the ! case of the h(π1 ) induction, here as well we can’t directly apply the permutation, that is, we can’t expect µ of this form .. .. .π! .π2 ⇒ Γ, !E ⇒?Σ, ?E ∗ , B .. Cut ⇒ Γ, ?Σ, B .π? # ! ⇒ Γ, ?Σ, !B ⇒ ∆2 , ?B ∗ Cut ⇒ Γ, ∆ since the application of the !-rule would violate the requirement that all the formulas in the context must be interrogated: this is not warranted, because there are no bounds on the formulas of Γ. Using the lemmas from 1 to 3, anyway, we will be able to find an adequate µ. We apply lemma 1 to π2 . It warrants that it has the following form .. .λk ! ⇒?Θk , !E . . . .⇒ Λ1 . . . ⇒ Λm .. λ . ⇒ Γ, !E where 1 ≤ k ≤ r for some r ≤ nπ2 +1 (n the number of application of the &-rule in π2 ). We call λ the derivation of ⇒ Γ, !E with hypotheses ⇒?Θi , !E; ⇒ Λ1 , . . . ; ⇒ Λm . Note that As clearly appears from the structure of the derivation, the formulas A and A∗ must be respectively exclaimed and interrogated; all the formulas in ∆1 must be interrogated as well. For this reason we refer to them as !E, ?E ∗ and ?Σ. Furthermore, as in the previous case, we use # to refer to one of the three rule ?; ?W and ?C. 11

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1. The lemma warrants that all the occurrences of !E in the conclusive sequents of the subderivations are principal (that is the subderivations end with an application of the !-rule). 2. In λ there are no applications of the !-rule. 3. All the ⇒ Λj if any at all (that is, if 1 ≤ j ≤ m, for some m ≥ 1), are axioms. 4. By lemma 2 the family of occurrences of !E in the conclusion is noble (in particular each leaf occurrence of the family belongs to one of the ⇒?Θk , ?E hypothesis of λ). Lemma 3 can be applied to λ(!E) in order to obtain a derivation λ(∆) of ⇒ Γ, ∆ with hypotheses ⇒ Θk , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm . Derivations of the hypotheses of λ(∆) can be obtained, and, consequently, by proposition we can get a derivation µ (without hypothesis) of ⇒ Γ, ∆ .. .. .π! .λk ! ⇒?Θk , !E ⇒, ?Σ, ?E ∗ , B .. Cut ⇒?Θk , ?Σ, B .π? # ! ⇒?Θk , ?Σ, !B ⇒ ∆2 , ?B ∗ Cut ⇒?Θk , ∆ . . . ⇒ Λ1 . . . ⇒ Λm .. .. λ ⇒?Θk , ∆

We call µk the derivations of ⇒?Θk , ?Σ, B. Note that d(µk ) ≤ d(π) We can apply the induction hypothesis to each µk , since 1) there is an application of the Cut-rule which is irregular on the right that concludes the subderivations; 2) all other applications of the Cut-rule in the derivation are regular; 3) The height of π! is lower than that of π1 : in particular h(π1 ) = sup(h(π! ) + 1, h(π? )) + 1 > h(π! ) Since the irregular applications of the Cut concluding the subderivations µk were irregular only on the right, then the induction hypothesis warrants the existence of regular derivations µ0k of ⇒ Θk , ?Σ, B, with d(µ0k ) = d(µk ) and such that if in µk there are no Cuts of degree d, so in µ0k . So, we can build the derivation π 0 , by substituting in µ the derivations µ0k to the subderivations µk (the proposition warrants the

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correctness of such operation) .. .µk .. ⇒?Θk , ?Σ, B .π? # ! ⇒?Θk , ?Σ, !B ⇒ ∆2 , ?B ∗ Cut ⇒?Θk , ∆ . . . ⇒ Λ1 . . . ⇒ Λm .. .. λ ⇒?Θk , ∆ Note that: d(π 0 ) = sup(d(µ0k ), d(πc )) = d(π); if in π there are no Cuts of degree d, so in π 0 , since the same holds for µk and µ0k ; π 0 is regular. cπ > 0 We have two cases depending on whether the last rule of π1 is ?C, with the occurrence of ?E in the conclusion of π1 as pseudo-principal occurrence, or not. In the latter case the permutation to be applied is one of those presented above. In the former, π is .. .π? .. ⇒ ∆, ?E ∗ , ?E ∗ .π! ?C ⇒ Γ, !E ⇒ ∆, ?E ∗ Cut ⇒ Γ, ∆ We call r and s the numbers of pseudo-principal occurrences in the families of the ?E ∗ occurrences in the conclusion of π? . Thus cπ = r + s + 1. As in the ! sub-case of the cπ = 0 case, here as well we can’t directly apply the permutation, that is we can’t expect µ of this form .. .. .π? .π! .. ⇒ Γ, !E ⇒ ∆, ?E ∗ , ?E ∗ .π! Cut ⇒ Γ, !E ⇒ Γ, ∆, ?E ∗ Cut ⇒ Γ, Γ, ∆ ?C ⇒ Γ, ∆ since the application(s) of the ?C-rule would violate the requirement that the formulas to be contracted must be interrogated: this is not warranted, because there are no bounds on the formulas of Γ. Using the lemmas from 1 to 3, anyway, we will be able to find an adequate µ. We apply lemma 1 to π! . It warrants that it has the following form .. .λk ! ⇒?Θk , !E . . . .⇒ Λ1 . . . ⇒ Λm .. λ . ⇒ Γ, !E 33

where 1 ≤ k ≤ r, for some r ≤ n! + 1 (n! is the number of application of the &-rule in π! ). We call λ the derivation of ⇒ Γ, !E with hypotheses ⇒?Θk , !E; ⇒ Λ1 ; . . . ; ⇒ Λm . Note that 1. Lemma 1 warrants that all the occurrences of !E in the conclusive sequents of the subderivations are principal (that is, the subderivations end with an application of the !-rule). 2. In λ there are no applications of the !-rule. 3. All the ⇒ Λj , if any at all (that is, if 1 ≤ j ≤ m, for some m ≥ 1), are axioms. 4. By lemma 2 the family of occurrences of !E in the conclusion is noble (in particular each leaf occurrence of the family belongs to one of the ⇒?Θk , ?E hypothesis of λ). Lemma 3 can be applied to λ(!E) in order to obtain a derivation λ(∆) of ⇒ Γ, ∆ with hypotheses ⇒ Θk , ∆; ⇒ Λ1 ; . . . ; ⇒ Λm . Derivations of the hypotheses of λ(∆) can be obtained and, consequently, by proposition we can get a derivation µ (without hypothesis) of ⇒ Γ, ∆ .. .. .π? .λk .. ! ?Θk , !E ⇒, ∆, ?E ∗ , ?E ∗ .λk Cut ! ?Θk , !E ⇒?Θk , ∆, ?E ∗ Cut ⇒?Θk , ?Θk , ∆ ?C ⇒?Θk , ∆ . . . ⇒ Λ1 . . . ⇒ Λm .. .. λ ⇒ Γ, ∆ We call µk the derivations of ⇒?Θk , ∆, ?E ∗ . We can apply the induction hypothesis to each µk , since the number cµk = s < cπ = r + s + 1. The induction hypothesis warrants the existence of regular derivations µ0k Then we can build a derivation µ0 , by substituting in µ each derivation µ0k to each subderivation µk (the proposition warrants the correctness of such operation) .. 0 .. .µk .λk ! ?Θk , !E ⇒?Θk , ∆, ?E ∗ Cut ⇒?Θk , ?Θk , ∆ ?C ⇒?Θk , ∆ . . . ⇒ Λ1 . . . ⇒ Λm .. .. λ ⇒ Γ, ∆

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If ?E ∗ is the principal occurrence of the last rule of each µ0k , then µ00 is regular. Otherwise we call µ00k the subderivations of µ0 of ⇒?Θk , ?Θk , ∆. Note that the condition ♠ warrants that cµ00k ≤ r since the family of the ?E ∗ occurrence in the conclusion of µ0k does not contain more pseudo-principal occurrences than the family of the corresponding occurrence in the conclusion of µk . Since r < r + s + 1 we have that cmu00k < cπ . We can apply the induction hypothesis to µ00k . It warrants the existence of regular derivations 0 00 0 µ000 k that, substituted in µ to the subderivations µk , give the derivation π .  Lemma 5 Given a derivation π of ⇒ Ξ such that: • The last rule applied in π is a Cut which is at most irregular on the left. Its Cut-occurrences are A and A∗ ; • Every other Cut in π is regular; there is a regular derivation π 0 of ⇒ Ξ such that: d(π) = d(π 0 ) and if in π there are no Cuts of degree d, so in π 0 . Furthermore π 0 is such that ♠ The family of each occurrence in the conclusion of π 0 does not contain more pseudo-principal occurrences then the family of the corresponding occurrence in the conclusion of π. Proof (Sketch) The general form of π is the following: .. .. .π1 .π2 R R0 ⇒ Γ, A ⇒ ∆, A∗ Cut ⇒ Γ, ∆ The proof of the lemma follows the lines of the previous one, with one difference: the Cut is regular on the right, that means that A∗ is the principal occurrence of R0 . This implies 1) The major strength of the result; 2) a strong simplification in the proof: in the ! case of the induction on h(π2 ), constituting the proof of the cπ = 0 case, we need not apply the lemmas from 1 to 3. For, the hypothesis of the Cut regularity on the right amounts, in this case, to the warrancy that the last rule of π1 is an application of the !-rule; but this, in turn, implies that all the formulas in ∆ are interrogated12 : .. .πA .. ⇒?Ω, ?E, B .π1 ! ! ⇒?Ω, ?E, !B ⇒?∆, !E ∗ Cut ⇒?Ω, ?Σ, !E ∗ 12

As in the previous lemma, in this case the formulas A and those in ∆ are interrogated, that is, A =?E and ∆ =?Σ. Furthermore here we have Γ =?Ω.

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Thus we can construct a derivation µ of the following form: .. .. .π1 .πA ⇒?Ω, ?E, B ⇒?Σ, !E ∗ Cut ⇒?Ω, ?Σ, B ! ⇒?Ω, ?Σ, !B Applying the induction hypothesis to the subderivation of ⇒?Ω, ?Σ, B, we get a derivation µ1 that substituted in µ gives us π 0 .. .µ1 ⇒?Ω, ?Σ, B ! ⇒?Ω, ?Σ, !B In the same way, also the ! sub-case of the inductive step involving the Cut-rule doesn’t require the use of the lemmas from 1 to 3: π has the following form13 .. .π! .. ⇒?Ω, ?E, B .π? .. # ! ⇒?Ω, ?E, !B ⇒ Γ2 , ?B ∗ .π1 Cut ⇒ Γ, ?E ⇒?Σ, !E ∗ Cut ⇒ Γ, ?Σ We can construct a derivation µ in the following way .. .. .π! .π1 ⇒?Ω, ?E, B ⇒?Σ, !E ∗ .. Cut ⇒?Ω, ?Σ, B .π? # ! ⇒?Ω, ?Σ, !B ⇒ Γ2 , ?B ∗ Cut ⇒ Γ, ?Σ Applying the induction hypothesis to the subderivation of ⇒?Ω, ?Σ, B, we get a derivation µ1 that substituted in µ gives us π 0 .. .µ1 .. ⇒?Ω, ?Σ, B .π? # ! ⇒?Ω, ?Σ, !B ⇒ Γ2 , ?B ∗ Cut ⇒ Γ, ?Σ 13

As before, we have A =?E and ∆ =?Σ; furthermore here we have Γ1 =?Ω.

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Finally, also the cπ > 0 case doesn’t require the use of the lemmas from 1 to 3: π has the following form14 .. .π? .. ⇒ ∆, ?E, ?E .π! ! ?C ⇒ ∆, ?E ⇒?Ω, !E ∗ Cut ⇒?Ω, ∆ We can construct a derivation µ of this form .. .. .π! .π? .. ! ⇒ ∆, ?E, ?E ⇒?Ω, !E ∗ .π! ! Cut ⇒?Ω, ∆, ?E ⇒?Ω, !E ∗ Cut ⇒?Ω, ?Ω, ∆ ?C ⇒?Ω, ∆ We call r and s the numbers of pseudo-principal occurrences in the families of the ?E occurrences in the conclusion of π? . Thus cπ = r + s + 1. We call µ1 the subderivation of ⇒?Ω, ∆, ?E; note that cµ1 = s < cπ . Applying the induction hypothesis to µ1 , we get a regular derivation µ01 that substituted in µ gives us µ00 .. 0 .. .µ1 .π! ! ⇒?Ω, ∆, ?E ⇒?Ω, !E ∗ Cut ⇒?Ω, ?Ω, ∆ ?C ⇒?Ω, ∆ Either µ00 is regular or we call µ001 the subderivation of ⇒?Ω, ?Ω, ∆. The condition ♠ warrants that cµ001 = r < cπ (the reasoning is analogous to the one in the previous lemma). The induction hypothesis warrants the 00 00 existence of a regular derivation µ000 1 which substituted to µ1 in µ gives us the regular derivation π 0 .. 000 .µ ⇒?Ω, ?Ω, ∆ ?C ⇒?Ω, ∆  Theorem 1 Given a derivation π of ⇒ Γ, then there is a regular derivation π 0 of Γ such that d(π 0 ) = d(π) and if in π there are no Cuts of degree d, so in π 0 . 14

As before, we have A =?E; furthermore here we have Γ =?Ω.

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Proof We prove the lemma by induction on π: h(π) = 1 Then π is regular and π 0 = π. h(π) > 1 We have different cases depending on the last rule of π: • R is a unary rule: then π is .. .. π1 ⇒ Γ0 ∗ ⇒Γ Applying the induction hypothesis to π1 , we get a regular derivation π10 of ⇒ Γ0 . By substituting in π the derivation obtained to π1 we get the desired regular derivation π 0 of ⇒ Γ. • R is &; ⊗, or a regular application of the Cut-rule: then π is .. .. .. π1 .. π2 ⇒ Γ1 ⇒ Γ2 ∗ ⇒Γ Applying the induction hypothesis to π1 and π2 , we get two regular derivations π10 and π20 of (respectively) ⇒ Γ1 and ⇒ Γ2 . By substituting in π the derivations obtained to π1 and π2 we get the desired regular derivation π 0 of ⇒ Γ. • R is an application of the Cut-rule which is irregular at most on the left: then π is .. .. .π2 .π1 R R ⇒, Γ1 , A 1 ⇒ Γ2 , A∗ 2 Cut ⇒Γ where A∗ is the principal occurrence of R2 . Applying the induction hypothesis to π1 and π2 , we get two regular derivations π10 and π20 of (respectively) ⇒ Γ1 , A and ⇒ Γ2 A∗ . Note that π20 and π2 end with an application of the same rule. By substituting in π the derivations obtained to π1 and π2 we get a derivation ν such that – The last rule applied in ν is a Cut which is at most irregular on the left. Its Cut-occurrences are A and A∗ ; – Every other Cut in ν is regular; Thus we can apply lemma 5 to ν. We get a regular derivation ν 0 of ⇒ Γ. Then π 0 = ν 0

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• R is an application of the Cut-rule which is irregular at least on the right: then π is .. .. .π2 .π1 R R ⇒, Γ1 , A 1 ⇒ Γ2 , A∗ 2 Cut ⇒Γ Applying the induction hypothesis to π1 and π2 , we get two regular derivations π10 and π20 of (respectively) ⇒ Γ1 , A and ⇒ Γ2 , A∗ . By substituting in π the derivations obtained to π1 and π2 we get a derivation ν. There are three possibilities: 1) ν is regular: then π 0 is ν; 2) all Cuts in ν are regular except the last one, which is irregular only on the left: this possibility reduces to the previous case; 3) ν is such that: – The last rule applied in ν is a Cut, which is at least irregular on the right. Its Cut-occurrences are A and A∗ ; – Every other Cut in ν is regular; Thus we can apply lemma 4 to ν. We get a derivation ν 0 of ⇒ Γ such that: if the Cut is irregular only on the right, then ν 0 is regular. Then π0 = ν 0. Otherwise 1) all the applications of the Cut-rule in ν 0 are regular except at most n2 + 1 (n2 is the number of applications of the &-rule in π20 ) that are irregular at most on the left. 2) their Cut-occurrences are occurrences of A and A∗ . 3) the νi subderivations of ν 0 ending with these applications of the Cut-rule are such that all their other applications of the Cut-rule are regular. Applying the lemma 5 to each νi we get regular derivations νi0 that, substituted to each νi in ν 0 , give us the regular derivation π 0 . 

3

The Cut-elimination

At this point we only Cut-rule from regular work with the Cuts of such that there are no derivations15 .

have to show how to eliminate applications of the derivations. We have two possibilities: either we maximum degree or with the highest Cuts (that is, Cuts in the subderivations of their premises) of the

15

The two possibilities roughly amount to (respectively) the Tait-Girard and the Gentzen-Takeuti Cut-elimination strategies (see [Girard 1987b] and [Gentzen 1935] for the former and the latter).

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We choose the former because, at this point, it leads to a simpler Cutelimination procedure: this is due to the fact that in the process of “regularization”, the number of Cuts in the derivation increases, that is, if a derivation contains ni Cuts of degree i, the regular derivation we obtain from it via theorem 1 has mi Cuts of degree i, where mi ≥ ni (as the reader can realize by analyzing the proof of theorem 1, mi depends on ni and on the number of applications of the &-rule in the given derivation). The latter strategy would start with the proof of the following result. Given a regular derivation π such that: 1. The last rule applied in π is a Cut with Cut-occurrences A and A∗ ; 2. π contains no other Cuts; then there is another regular derivation φ of ⇒ Ξ which is Cut-free. The proof of this lemma is by induction on the degree of A and A∗ . We see, examining only one case, that, because of the increase of the number of Cut due to the application of theorem 1, a nested induction on such a number is required: suppose we have a regular derivation ending with an application of the Cut-rule whose Cut-occurrences are B ⊗ C and B ∗ FC ∗ . .. .. .. .πF .πC .πB ⇒ ∆, B ∗ , C ∗ ⇒ Γ1 , B ⇒ Γ2 , C F ⊗ ⇒ Γ, B ⊗ C ⇒ ∆, B ∗ FC ∗ Cut ⇒ Γ, ∆ First we produce a derivation π 0 of the following form .. .. .πF .πB .. ⇒ Γ1 , B ⇒ ∆, B ∗ , C ∗ .πC Cut ∗ ⇒ Γ1 , ∆, C ⇒ Γ2 , C Cut ⇒ Γ, ∆ Note that the derivation π 0 may be irregular. We call π1 the subderivation of ⇒ Γ, ∆, C ∗ : we have to apply to it theorem 1. We get a regular derivation π10 . Its degree is the same as that of π1 and if in π1 there are no Cuts of degree n so in π10 . So in π10 we have only Cuts with Cut-occurrences B and B ∗ . Nonetheless in π1 we will have a possibly great number (depending on how many application of the &-rule are there in π) of applications of the Cut-rule of this degree. First we try to eliminate all this Cuts, that is we try to get a regular derivation λ of ⇒ Γ1 , ∆, C ∗ . In order to achieve this, we are forced to work by induction on the number n of Cuts in π10 . If n = 1 then we can apply the main induction hypothesis to the subderivation ending with the application of the Cut, since d(B) < d(B ⊗ C).

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We substitute the derivation obtained in π10 . Thus we obtain the Cut-free derivation λ. If n > 1 then there are π1i subderivations of π10 (1 ≤ i ≤ r, for some r ≤ n) such that the last rule applied in each π1i is a Cut with Cut-occurrences C and C ∗ ; each π1i contains no other applications of the Cut-rule. We can apply to each of them the main induction hypothesis, and then we substitute the derivations obtained in π10 . We call π100 the derivation obtained. Note that nπ100 < nπ10 : the induction hypothesis on n warrants the existence of a a Cut-free derivation λ. We substitute λ to π1 in π 0 . We call π 00 the derivation so obtained: .. .. .πC .λ ∗ ⇒ Γ1 , ∆, C ⇒ Γ2 , C Cut ⇒ Γ, ∆ We apply to π 00 theorem 1. We get a derivation π 000 . We must work again by induction on the number of applications of the Cut rule in π 000 . The reader will agree that, even if the ⊗-case is the worst one (the original Cut is substituted by two new ones of lower degree that are “consecutive”), a full detailed proof of this lemma would result rather tedious. Then it appears preferable to work with Cuts of maximum degree. Here as well, anyway, the increase in the number of Cuts due to theorem 1 requires a strengthening in the formulation of lemma 6 with respect to Girard’s corresponding result: we do not just take Cuts of maximum degree, but, among these, only those such that all the Cuts above them has lower degree16 . Taking Cuts that are only of maximum degree would require, in the proof of the lemma, a further induction on the number of Cuts of maximum degree, that is, the complication of the Gentzen-Takeuti procedure that we wanted to escape. Lemma 6 Given a regular derivation π of ⇒ Ξ such that: 1. The last rule applied in π is a Cut with Cut-occurrences A and A∗ ; 2. if π contains other applications of the Cut-rule with Cut-occurrences B, B ∗ , then d(B) < d(A). then there is another regular derivation φ of ⇒ Ξ such that dφ < dπ . Proof The general form of π is .. .. .π2 .π1 R R ⇒ Γ1 , A 1 ⇒ Γ2 , A∗ 2 Cut ⇒Γ 16 In [Girard 1987b], the main lemma (see the introduction) is formulated so that the derivation of ⇒ Γ1 and ⇒ Γ2 have degree ≤ A, where A is the fusion-formula; here, instead, we are requiring that if the degree of the last Cut is d(A), than the degree of the derivations of its premises must be strictly lower than d(A). Nonetheless in chapter 13 of [Girard 1989] the lemma is formulated with our restriction.

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Note that A and A∗ are the principal occurrences of (respectively) R1 and R2 . We prove the lemma by induction on the degree of A and A∗ : d(A) = 1 We have two cases: • A is atomic of form B. Then π is Ax Ax ⇒ B, B ⊥ ⇒ B, B ⊥ Cut ⇒ B, B ⊥ In this case φ is ⇒ B, B ⊥

Ax

that is, φ is Cut-free. • A is atomic of form B ⊥ . The case is analogous to the previous one. d(A) > 1 We have different sub-cases: • A is B ⊗ C: then π is .. .. .. .πF .πB .πC ⇒ ∆, B ∗ , C ∗ ⇒ Γ1 , B ⇒ Γ2 , C F ⊗ ⇒ Γ, B ⊗ C ⇒ ∆, B ∗ FC ∗ Cut ⇒ Γ, ∆ First we produce a derivation π 0 of the following form .. .. .πF .πB .. ⇒ Γ1 , B ⇒ ∆, B ∗ , C ∗ .πC Cut ⇒ Γ1 , ∆, C ∗ ⇒ Γ2 , C Cut ⇒ Γ, ∆ Note that dπ0 < dπ . By applying theorem 1 to π 0 we get the regular derivation φ. Since dφ = dπ0 , dφ < dπ . • A is BFC. This case is the dual of the previous one. • A is B&C: then π is .. .. .. .π⊕ .πB .πC ⇒ ∆, B ∗ ⇒ Γ, B ⇒ Γ, C ⊕ & ⇒ Γ, B&C ⇒ ∆, B ∗ ⊕ C ∗ Cut ⇒ Γ, ∆ First we produce a derivation π 0 of the following form .. .. .π⊕ .πB ⇒ Γ, B ⇒ ∆, B ∗ Cut ⇒ Γ, ∆ 42

Note that dπ0 < dπ . By applying theorem 1 to π 0 we get the regular derivation φ. Since dφ = dπ0 , dφ < dπ . • A is B ⊕ C. This case is the dual of the previous one. • A is !B. We have two sub-cases: 1. The form of π is17 .. .. .π? .π! ⇒ ∆, B ∗ ⇒?Ω, B ? ! ⇒?Ω, !B ⇒ ∆, ?B ∗ Cut ⇒?Ω, ∆ First we produce a derivation π 0 of the following form .. .. .π? .π! ⇒?Ω, B ⇒ ∆, B ∗ Cut ⇒?Ω, ∆ Note that dπ0 < dπ . By applying theorem 1 to π 0 we get the regular derivation φ. Since dφ = dπ0 , dφ < dπ . 2. The form of π is .. .. .π! .π? ⇒?Ω, B ⇒∆ ?W ! ⇒?Ω, !B ⇒ ∆, ?B ∗ Cut ⇒?Ω, ∆ The form of φ is the following .. .π? ⇒∆ ?W ⇒?Ω, ∆ Note that φ is regular and dφ < dπ . • A is ?B. This case is the dual of the previous one.  Theorem 2 If there is a regular derivation π of ⇒ Γ, then there is a Cutfree derivation λ of ⇒ Γ. 17

All the formula in Γ are interrogated, that is, Γ =?Ω.

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Proof By induction on the number nπ of Cuts of maximal degree in π: nπ = 0 Then π is Cut-free. nπ > 0 Then there are πi subderivations of π, 1 ≤ i ≤ r, for some r ≤ nπ , such that for each of them the last rule applied is a Cut with Cutoccurrences A and A∗ ; if πi contains other occurrences of the Cut-rule with Cut-occurrences B, B ∗ , then d(B) < d(A). We can apply to them lemma 6. By substituting the derivations obtained in π we get a derivation π 0 such that: nπ0 = nπ − r < nπ . By applying the induction hypothesis to π 0 we get the Cut-free derivation λ.  Corollary If there is a derivation of ⇒ Γ then there is a Cut-free derivation of ⇒ Γ. Proof The result follows directly from theorems 1 and 2. 

4

Conclusions

We conclude this work with two brief considerations. We didn’t estimate the computational complexity of the procedure, that is how big would be a Cut-free derivation obtained from a derivation of height h. As the authors of [Lincoln et alia 1994] did, we too suggest a hyper-exponential increase in the height; nonetheless we hope to have somehow underscored the relevance of the additive (and of course exponential) connectives in the economy of the whole process. A possible refinement of the proof strategy here presented would be to strengthen theorem 1, in order to get derivations not only regular, but also such that, for each application of the Cut-rule in them, all other applications above it have equal or lower degree. It seems that, in this case, it would be possible to apply the reductions so that the derivations obtained are already in regular form. The procedure would then appear much more elegant, since there wouldn’t be any need of apply theorem 1 before the induction hypothesis during the proof of lemma 6, with the consequence of splitting the permutation and the reduction procedures in a much more stronger sense than here. Finally the author would like to thank Duccio Pianigiani and Aldo Ursini from the Departments of Philosophy and Mathematics of the Siena University for their patience, encouragement and essential contributions to the development of this paper.

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References [Bra¨ uner 1996] T. Bra¨ uner, Introduction to Linear Logic, http://www.brics.dk, 1996

BRICS,

[Bra¨ uner de Paiva 1996] T. Bra¨ uner, V. de Paiva, Cut-Elimination for Full Intuitionistic Linear Logic, BRICS, http://www.brics.dk, 1996 [Buss 1998] S. Buss, Introduction to proof theory, in S. Buss (ed.), Handbook of proof theory, Elsevier Science, 1998 [Gentzen 1935] G. Gentzen, Untersuchungen u ¨ber das logische schliessen, in “Mathematische Zeitschrift”, 39, 1935, eng. trans. in Szabo (ed.), The collected papers of Gerhard Gentzen, North Holland, AmsterdamLondon, 1969, pp.68-103. [Girard 1987a] G. Y. Girard, Linear Logic, Theoretical Computer Science, 50, pp. 1–102, 1987 [Girard 1987b] G. Y. Girard, Proof theory an logical complexity, vol.1, Bibliopolis, Napoli, 1987 [Girard 1989] G. Y. Girard, Proof and Types, Cambridge University Press, Cambridge, 1989 [Lincoln et alia 1994] P. Lincoln, J. Mitchell, A. Scedrov and N. Shankar, Decision Problem for Propositional Linear Logic, 1994

Contacts Address: Luca Tranchini via Stefani, 2 31029 Vittorio Veneto (TV) Italy e-mail: [email protected]

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