Olivier Laurent- A proof of the focalization property of Linear Logic

8/3/2019 Olivier Laurent- A proof of the focalization property of Linear Logic

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A proof of the focalization property of Linear Logic

Olivier LAURENT

Preuves Programmes Systemes

CNRS Universite Paris VII

UMR 7126 Case 7014

2, place Jussieu 75251 Paris Cedex 05 FRANCE

[email protected]

May 5, 2004

The focalization property of linear logic has been discovered by Jean-Marc Andreoli [1] in thebeginning of the 90s. It is one of the main properties of Linear Logic that appeared after theoriginal paper of Jean-Yves Girard [3]. This property is proved in various papers [1, 4, 2] but, asfar as we know never for the usual LL sequent calculus (or with an intricate induction in Andreolisthesis, which is replaced here by a cut elimination property).

The proof we give is a compilation of the previous proofs and proof techniques, our goal is justto give a complete presentation of a proof of this property which appears to be more and moreimportant in the current research in Linear Logic [4, 2, 5, 6].

We decompose the usual focalization property into two technically different steps: weak focal-ization (the decompositions of two positive formulas are not interleaved, see LLfoc) and reversing(negative rules are applied as late as possible from a top-down point of view, see LLFoc).

1 Linear Logic and the focalization property

Formulas of Linear Logic are given by the usual grammar:

A ::= X | A A | A A | 1 | 0 | !A| X | A ` A | A & A | | | ?A

We split this grammar into two sub-classes: positive formulas and negative formulas.

P ::= X | A A | A A | 1 | 0 | !AN ::= X | A ` A | A & A | | | ?A

In the sequel:

A, B, ... denote arbitrary formulas;

P, Q, R denote positive formulas;

N, M, L denote negative formulas;

X, X, ... are called atoms or atomic formulas.

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We consider the usual rules of Linear Logic [3]:

ax A, A

, A , Acut

,

,A,B` , A ` B

, A , B ,, A B

, A , B&

, A & B , A

1 , A B

, B2

, A B

1 1

,

,

?, A!

?, !A , A

?d , ?A

?w

, ?A , ?A, ?A

?c , ?A

The main connectives of positive (resp. negative) formulas are called positive connectives (resp.

negative connectives), that is X, , , 1, 0 and ! (resp. X, `, &, , and ?). The rulesintroducing positive (resp. negative) connectives are called positive rules (resp. negative rules).

A positive (resp. negative) formula is strictly positive (resp. strictly negative) if its main con-nective is not ! (resp. ?). A positive (resp. negative) connective is strictly positive (resp. strictlynegative ) if it is not ! (resp. ?). A positive (resp. negative) rule is strictly positive (resp. strictlynegative ) if it is not introducing a !A formula (resp. ?A formula).

Definition 1 (Main positive tree)IfA is a formula, its main positive tree T+(A) is defined by:

T+(N) =

T+

(X) = XT+(A B) = T+(A) T+(B)

T+(A B) = T+(A) T+(B)

T+(1) = 1

T+(0) = 0

T+(!A) = !

T+(A) is a (possibly empty) tree whose nodes are , (with arity at most 2) and !, 1, 0 or X(with arity 0).

Definition 2 (Weakly +-focalized proof)

A proof in LL is weakly +-focalized if it is cut-free and, for any subproof of with conclusion , A, the only positive rules of between two rules introducing connectives of T+(A) are rulesintroducing connectives of T+(A).

Definition 3 (Strongly +-focalized proof)A proof in LL is strongly +-focalized if it is cut-free and, for any subproof of with conclusion , A, the only rules of between two rules introducing connectives of T+(A) are rules introducingconnectives of T+(A).

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Definition 4 (Reversed proof)A proof in LL is reversed if, for any subproof of with conclusion , N with N a strictlynegative non-atomic formula, the last rule of is a strictly negative rule.

Definition 5 (Focalized proof)

A proof

inLL

is focalized if it is both strongly +-focalized and reversed.Remark: Various systems for classical logic related with focalization constraints have been intro-duced. In particular Girards LC [4] corresponds to weak +-focalization, Danos-Joinet-SchellinxsLK [2] corresponds to strong +-focalization and Quatrini-Tortora de Falcos LK,pol [7] correspondsto focalization since their -constraint is a reversing constraint.

2 Weakly Focalized Linear Logic

2.1 Sequent calculus LLfoc

A sequent of LLfoc has the shape ; where is a multi-set of formulas and is either empty

or contains a unique positive formula.The rules are a linear logic version of the rules of Girards LC [4]:

ax P ; P

; Pfoc

, P ;

; P , P ; p-cut

, ; , P ; , P ;

n-cut , ;

,A,B ; `

, A ` B ; ; P ; Q

, ; P Q

; P ,M; , ; P M

,N ; ; Q , ; N Q

,N ; ,M ; , ; N M

, A ; , B ; &

, A & B ;

; P1

; P B , N ;

1 ; N B

; Q2

; A Q ,M ;

2 ; A M

1 ; 1

;

, ;

, ;

?, A ;!

? ; !A ; P

?d , ?P ;

, N ;?d

, ?N ;

; ?w

, ?A ; , ?A, ?A ;

?c , ?A ;

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2.2 Expansion of axioms

Given a positive formula P, the sequent P ; P is provable in LLfoc by means of the ax rule.Moreover, ifP is not atomic, it is also possible to give a proof based only on ax rules applied tothe sub-formulas ofP (and thus, by an easy induction, to the atomic sub-formulas of P):

P ; P Q ; Q

P, Q ; P Q`

P ` Q ; P Q

N ; N

foc N, N ; Q ; Q

N,Q ; N Q

` N ` Q ; N Q

P ; P

M ; Mfoc

M,M ;

P,M ; P M`

P ` M ; P M

N ; Nfoc

N, N ;

M ; Mfoc

M,M ;

N,M ; N M`

N ` M ; N M

P ; P1

P ; P Q

Q ; Q2

Q ; P Q&

P & Q ; P Q

N ; N

foc N, N ;

1 N ; N Q

Q ; Q2

Q ; NQ&

N & Q ; N Q

P ; P1

P ; P M

M ; Mfoc

M,M ;2

M ; P M&

P & M ; P M

N ; Nfoc

N, N ;1

N ; N M

M ; Mfoc

M,M ;2

M ; N M&

N & M ; N M

1

; 1 ; 1 ; 0

P ; Pfoc

P, P ;?d

?P, P ;!

?P ; !P

N ; N?d

?N,N ;!

?N ; !N

2.3 Embedding of LL (weak +-focalization)

The translation (.) of LL into LLfoc does not modify formulas, translates the sequent as ;

and acts on proofs by adding a lot of cut rules:ax

P, P ax

P ; P

, P , Pcut

,

, P ; , P ;n-cut

, ;

,A,B`

, A ` B

,A,B ;`

, A ` B ;

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, P , Q

,, P Q

, P ;

, Q ;

ax P ; P

ax Q ; Q

P,Q ; P Q

foc P,Q, P Q ;

n-cut , P, P Q ;

n-cut

,, P Q ;

, N , Q

,,N Q

, N ;

, Q ;

ax N ; N

foc N, N ;

ax Q ; Q

N,Q ; N Q

foc N,Q, N Q ;

n-cut , N,N Q ;

n-cut ,,N Q ;

, N ,M

,,N M

,N ;

,M ;

ax N ; N

foc

N, N

;

ax M ; M

foc

M,M

; N,M ; N M

foc N,M,N M;

n-cut ,N, N M ;

n-cut ,,N M;

, A , B&

, A & B

, A ; , B ;&

, A & B ;

, P1 , P B

, P ;

ax P ; P

1

P

; P B foc P, P B ;

n-cut , P B ;

,N1

, N B

,N ;

ax N ; N

foc N, N ;

1 N ; N B

foc N,N B ;

n-cut , N B ;

1 1 1

; 1 foc 1 ;

,

;

, ;

,

, ;

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?, A!

?, !A

?, A ;!

?; !Afoc

?, !A ;

, P

?d , ?P

, P ;

ax P ; P

?d P, ?P ; n-cut

, ?P ;

,N?d

, ?N

, N ;?d

, ?N ;

?w

, ?A

;?w

, ?A ;

, ?A, ?A?c

, ?A

, ?A, ?A ;?c

, ?A ;

2.4 Focalization in LLfoc

If is a proof in LLfoc then is the LL proof obtained by erasing all the ; in the sequents.

Proposition 1 (Cut-free weak +-focalization)If is a cut-free proof of ; in LLfoc then

is a weakly +-focalized proof of , in LL.

Corollary 1.1 (Weak +-focalization)If is provable in LL, is provable with a weakly +-focalized proof.

Proof: Starting from a proof of , we translate it into the proof of ; i n LLfoc. Usingthe cut elimination property given in appendix A (corollary 5.2), this leads to a cut-free proof of ; in LLfoc. By proposition 1, (

) is a weakly +-focalized proof of in LL. 2

3 Focalized Linear Logic

3.1 Sequent calculus LLFoc

In the spirit of [5], we define a sub-system LLFoc of LLfoc in which the strictly negative formulas inthe context of positive rules must be atomic.

There are two kinds of sequents: P,N ; and P, ?,X ; P where X contains only negativeatoms. In order to simplify the notations we will write P,N ; for either a sequent with emptyor a sequent with = P and N = ?,X.

ax X ; X P, ?,X

; P foc P, ?,X, P ;

P,N,A,B ;`

P,N, A ` B ;

P, ?,X ; P P, ?,X ; Q

P,P, ?, ?,X,X ; P Q

P, ?,X ; P P, ?,X,M;

P,P, ?, ?,X,X ; P M

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P, ?,X, N ; P, ?,X ; Q

P,P, ?, ?,X,X ; N Q

P, ?,X,N ; P, ?,X,M ;

P,P, ?, ?,X,X ; N M

P,N, A ; P,N, B ;&

P,N, A & B ;

P, ?,X ; P1

P, ?,X ; P B

P, ?,X, N ;1

P, ?,X ; N B

P, ?,X ; Q2

P, ?,X ; A Q

P, ?,X,M ;2

P, ?,X ; A M

1 ; 1

P,N ;

P,N, ;

P,N, ;

?, A ;!

? ; !A

P, ?,X ; P?d

P, ?,X

, ?P ;

P, ?,X, N ;?d

P, ?,X

, ?N ;

P, ?,X ;?w

P, ?,X, ?A ;

P, ?,X, ?A, ?A ;?c

P, ?,X, ?A ;

3.2 Embedding of LLfoc

We embed proofs of LLfoc of sequents of the shape ; into LLFoc. We proceed in two steps by firstshowing that LLFoc sequents (that is of the shape P,N ; or P, ?,X ; P) provable in LLfoc areprovable in LLFoc enriched with the two rules:

P, ?,X ; P

?w P, ?,X, ?A ; P

P, ?,X, ?A, ?A ; P

?c P, ?,X, ?A ; P

The size of a sequent ; is the sum of the sizes of the formulas of plus one. The size of asequent ; P is the sum of the sizes of the formulas of plus the size of P.

We consider a cut-free proof with expanded axioms (according to section 2.2). We proceed byinduction on the size of the final sequent. We look at the last rule of the proof. The non trivialcases are the foc rules and ?d rules on positive formulas (called a positive ?d rule), the other rulesare immediately valid in LLFoc (in our restricted case where their conclusion is an LLFoc sequent).

We only consider the foc rule (the case of the positive ?d rule is very similar). If the context isof the shape P, ?,X, the rule is valid in LLFoc. Otherwise this context contains a formula withmain connective `, &, or :

If we have a context of the shape , A ` B, using the required proof of section 2.2 withoutthe dashed rule, we build:

, A ` B ; P A,B ; A B

p-cut ,A,B ; P

foc ,A,B,P;

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By induction hypothesis, after cut elimination and expansion of axioms in LLfoc, this proofgives a proof of LLFoc. We conclude with:

,A,B,P;`

, A ` B, P ;

If we have a context of the shape , A & B, in the same way we build:

, A & B ; P A ; A B

p-cut , A ; P

foc ,A,P ;

and

, A & B ; P B ; A

B

p-cut , B ; Pfoc

,B ,P ;

After cut elimination and expansion of axioms, we apply the induction hypothesis and weconclude with:

,A,P ; ,B ,P ;&

, A & B, P ;

If we have a context of the shape ,, we build:

, ; P ; 1

p-cut ; P

foc , P ;

After cut elimination and expansion of axioms, we apply the induction hypothesis and weconclude with:

, P ;

,, P ;

If we have a context of the shape ,, we can immediately use the LLFoc proof:

,, P ;

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We now have to eliminate the additional ?w and ?c rules. We show by induction on the size ofthe proof that if P, ?,X ; P (resp. P,N ; ) is provable in LLFoc with the two additional rulesthen P, ?,X ; P (resp. P,N ; ) is provable in LLFoc with ?

? (where ? ? meansthat for each formula ?A in ? there is at least an occurrence of ?A in ?, or equivalently that wehave an inclusion of the underlying sets).

If the last rule is not one of the two additional rules and neither a foc rule nor a positive ?drule, we translate the rule by itself. Otherwise:

If the last rule is a foc rule or a positive ?d rule, we apply the corresponding rule and weapply the required ?w and ?c rules to move from ? to ?.

In the case of the added ?wrule, by induction hypothesis, P, ?,X ; P is provable in LLFocwith ? ?, thus we have the result since ? ?, ?A.

In the case of the added ?c rule, by induction hypothesis, P, ?,X ; P is provable in LLFocwith ? ?, ?A, ?A, thus we have the result since ? ?, ?A.

This shows that if ; is provable inLL

foc, then it is also provable inLL

Foc.

3.3 Focalization in LLFoc

If is a proof in LLFoc then is the LL proof obtained by erasing all the ; in the sequents.

Proposition 2 (Cut-free focalization)If is a cut-free proof of P,N ; in LLFoc then

is a focalized proof of P,N, in LL.

Corollary 2.1 (Focalization)If is provable in LL, is provable with a focalized proof.

Proof: Starting from a proof of , we translate it into the proof of ; in LLfoc and then

into a cut-free proof of ; i n LLFoc. By proposition 2, is a focalized proof of inLL. 2

Remark: It is possible to define the embedding ofLL proofs into LLFoc directly (without using LLfocas an intermediary step). We have chosen to decompose it into two steps in order to show thatthe key property is weak +-focalization. Strong +-focalization is then obtained through reversingwhich is in general easy to do.

4 Additional remarks

4.1 Decomposition of exponentials

The attentive reader has certainly remarked that an hidden decomposition of the exponentialconnectives underlies the whole text (as suggested by Girard [5]):

!A = A ?A = A

with A negative and A positive, and and are used as the same connectives of LLpol (seeappendix A.2.1).

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The corresponding rules in LLfoc would be:

?, A ; ()

?, A ; ()

; P

; P

,N ;

; N

, A ; ; A

; P , P ;

, N ; , N ;

However it is difficult to give a meaning to A (resp. A) under any other connective than (resp. ).

4.2 Quantifiers

Our method can perfectly be extended to quantification (of any order) by adding A (resp. A)in positive (resp. negative) formulas and the following rules in LLfoc:

; P

;

P

,N ;

;

N

, A ;

,

A ;

with free neither in nor in in the rule.

References

[1] Jean-Marc Andreoli. Proposition pour une synthese des paradigmes de la programmation logiqueet de la programmation par objets. These de doctorat, Universite Paris VI, June 1990.

[2] Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. A new deconstructive logic: linearlogic. Journal of Symbolic Logic, 62(3):755807, September 1997.

[3] Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1102, 1987.

[4] Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Com-puter Science, 1(3):255296, 1991.

[5] Jean-Yves Girard. Locus solum: From the rules of logic to the logic of rules. MathematicalStructures in Computer Science, 11(3):301506, June 2001.

[6] Olivier Laurent. Etude de la polarisation en logique. These de doctorat, Universite Aix-Marseille II, March 2002.

[7] Myriam Quatrini and Lorenzo Tortora de Falco. Polarisation des preuves classiques et renverse-ment. Compte-Rendu de lAcademie des Sciences de Paris, 323:113116, 1996.

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A Cut elimination in LLfoc

A.1 Cut elimination steps

Key steps

ax P ; P , P ; p-cut

, P ; , P ;

; Pax

P ; Pp-cut

; P ; P

; Pfoc

, P ; , P ;n-cut

,, P ;

; P , P ;p-cut

,, P ;

; P ; Q , ; P Q

, P, Q ; ` , P ` Q ; p-cut

,, ;

; P

; Q , P, Q ; p-cut

,, P ; p-cut

,, ;

; P ,M ;

, ; P M

, P,M ; `

, P ` M ; p-cut

,, ;

; P

,M ; , P,M ; n-cut ,, P ;

p-cut ,, ;

,N ; ,M ;

, ; N M

,N,M ; `

,N ` M ; p-cut

,, ;

,N ;

,M ; , N,M ; n-cut

,, N ; n-cut

,, ;

; P 1 ; P B

, P

; , B

; & , P & B ;

p-cut , ;

; P , P ;

p-cut , ;

, N ;1

; N B

,N ; , B ; &

, N & B ; p-cut

, ;

,N ; ,N ;

n-cut , ;

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1 ; 1

;

, ; p-cut

;

;

?,N ;!

? ; !N

; N?d

, ?N

; p-cut ?, ;

?, N ; ; N

p-cut ?, ;

?, P ;!

? ; !P

, P ;?d

, ?P ;p-cut

?, ;

?, P ; , P ;

n-cut ?, ;

?, A ;!

? ; !A

; ?w

, ?A ; p-cut

?, ;

; ?w

?, ;

?, A ;!

? ; !A

, ?A, ?A ; ?c

, ?A ; p-cut

?, ;

?, A ;!

? ; !A

?, A ;! ? ; !A , ?A, ?A ;

p-cut ?,, ?A ;

p-cut ?, ?, ;

?c ?, ;

Left commutative p-steps

,A,B ; P`

, A ` B ; P , P ; p-cut

,, A ` B ;

,A,B ; P , P ; p-cut

,,A,B ; `

,, A ` B ;

, A ; P , B ; P&

, A & B ; P , P ; p-cut

,, A & B ;

, A ; P , P ; p-cut

,, A ; , B ; P , P ;

p-cut ,, B ;

& ,, A & B ;

; P

, ; P , P ; p-cut

,, ;

; P , P ; p-cut

, ;

,, ;

, ; P , P ; p-cut

,, ;

,, ;

; P?w

, ?A ; P , P ; p-cut

,, ?A ;

; P , P ; p-cut

, ; ?w

,, ?A ;

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, ?A, ?A ; P?c

, ?A ; P , P ; p-cut

,, ?A ;

, ?A, ?A ; P , P ; p-cut

,, ?A, ?A ; ?c

,, ?A ;

Right commutative p-steps

; P

, P ; Pfoc

, P, P ;p-cut

,, P ;

; P , P ; Pp-cut

, ; Pfoc

,, P ;

; P

,A,B,P ; `

, A ` B, P ; p-cut

,, A ` B ;

; P ,A,B,P ; p-cut

,,A,B ; `

,, A ` B ;

; P

, P ; P ; Q

,, P ; P Qp-cut

,, ; P Q

; P , P ; Pp-cut

, ; P ; Q

,, ; P

Q

; P

, P,N ; ; Q

,, P ; N Qp-cut

,, ; N Q

; P , P,N ;p-cut

,, N ; ; Q

,, ; N Q

; P

, N ; , P ; Q

,, P ; N Qp-cut

,, ; N Q

,N ; ; P , P ; Q

p-cut , ; Q

,, ; N Q

; P

, P,N ; ,M ;

,, P ; N M p-cut ,, ; N M

; P , P, N ;

p-cut ,,N ; ,M ; ,, ; N M

; P

, P, A ; , P, B ; &

, P, A &B ; p-cut

,, A & B ;

; P , P, A ; p-cut

,, A ; ; P , P, B ;

p-cut ,, B ;

& ,, A & B ;

; P

, P ; P

1 , P ; P Bp-cut

, ; P B

; P , P ; P

p-cut , ; P1

, ; P B

; P

, P, N ;1

, P ; N Bp-cut

, ; N B

; P , P, N ;p-cut

,,N ;1

, ; N B

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; P

, P ;

, P, ; p-cut

,, ;

; P , P ; p-cut

, ;

,, ;

; P

,, P ;

p-cut ,, ;

,, ;

? ; !A

?, ?A, B ;!

?, ?A ; !Bp-cut

?, ? ; !B

? ; !A ?, ?A, B ;p-cut

?, ?, B ;!

?, ?; !B

; P

, P ; P?d

, P, ?P ;p-cut

,, ?P ;

; P , P ; Pp-cut

, ; P?d

,, ?P ;

; P

, P, N ;

?d , P, ?N ;p-cut

,, ?N ;

; P , P, N ;p-cut ,,N ;

?d ,, ?N ;

; P

, P ; ?w

, P, ?A ; p-cut

,, ?A ;

; P , P ; p-cut

, ; ?w

,, ?A ;

; P

, P, ?A, ?A ; ?c

, P, ?A ; p-cut

,, ?A ;

; P , P, ?A, ?A ; p-cut

,, ?A, ?A ; ?c

,, ?A ;

Commutative n-steps

, P ; Pfoc

,P ,P ; , P ;n-cut

,, P ;

, P ; P , P ;n-cut

, ; Pfoc

,, P ;

,A,B,P; `

, A ` B, P ; , P ;n-cut

,, A ` B ;

,A,B,P; , P ;n-cut

,,A,B ; `

,, A ` B ;

, P ; P ; Q

,, P ; P Q , P ;n-cut

,, ; P Q

, P ; P , P ;n-cut , ; P ; Q

,, ; P Q

,P ,N; ; Q

,, P ; N Q , P ;n-cut

,, ; N Q

,P ,N; , P ;n-cut

,,N ; ; Q

,, ; N Q

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,N ; , P ; Q

,, P ; N Q , P ;n-cut

,, ; N Q

,N ; , P ; Q , P ;

n-cut , ; Q

,, ; N Q

,P ,N; ,M ;

,, P ; N M , P

; n-cut ,, ; N M

,P ,N; , P ;n-cut

,,N ; ,M ;

,, ; N M

,P ,A ; ,P ,B ; &

,P ,A & B ; , P ;n-cut

,, A & B ;

,P ,A ; , P ;n-cut

,, A ; ,P ,B ; , P ;

n-cut ,, B ;

& ,, A & B ;

, P ; P1

, P ; P B , P ;n-cut

, ; P B

, P ; P , P ;n-cut

, ; P1

, ; P B

,P ,N;1

, P ; N B , P ;n-cut

, ; N B

,P ,N; , P ;n-cut

,, N ;1

, ; N B

, P ;

, P, ; , P

; n-cut ,, ;

, P ; , P ;n-cut

, ; ,, ;

, P, ; , P ;

n-cut ,, ;

,, ;

, P ; P?d

, P, ?P ; , P ;n-cut

,, ?P ;

, P ; P , P ;n-cut

, ; P?d

,, ?P ;

,P ,N;?d

, P, ?N ; , P

; n-cut ,, ?N ;

,P ,N; , P ;n-cut

,,N ; ?d ,, ?N ;

, P ; ?w

, P, ?A ; , P ;n-cut

,, ?A ;

, P ; , P ;n-cut

, ; ?w

,, ?A ;

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, P, ?A, ?A ; ?c

, P, ?A ; , P ;n-cut

,, ?A ;

, P, ?A, ?A ; , P ;n-cut

,, ?A, ?A ; ?c

,, ?A ;

A.2 Cut elimination property

A.2.1 LLpol

The system LLpol

is obtained from LL by restricting formulas to the following grammar:

P ::= X | P P | P P | 1 | 0 | !N | NN ::= X | N` N | N& N | | | ?P | P

with the following rules for the and connectives:

N,N

N, N

, P

, P

where N is a multi-set of negative formulas.

A.2.2 LLP

The system LLP is obtained from LL by restricting formulas to the following grammar:

P ::= X | P P | P P | 1 | 0 | !NN ::= X | N` N | N& N | | | ?P

with the following generalizations of the exponential rules:

N,N!

N, !N , P

?d , ?P

?w

,N ,N,N

?c , N

where N is a multi-set of negative formulas.

Proposition 3 (Strong normalization)There is no infinite sequence of reductions in LLP if we forbid commutations of cuts.

Proof: Such a sequence of reductions can only contain finitely many steps between two steps thatcorrespond to a reduction step in proof-nets. Thus by strong normalization for proof-nets [6]we can conclude. 2

Corollary 3.1 (Cut elimination)If is provable in LLP, then is provable without the cut rule.

A.2.3 Simulations

The translation (.)! of LLpol into LLP is obtained by replacing N by !N and P by ?P and the twolifting rules by promotion and dereliction.

Lemma 1 (Polarized formulas)IfA is a positive (resp. negative) formula in LLpol thenA

! is a positive (resp. negative) formula in

LLP.

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Proposition 4 (One-to-one simulation)If reduces to in LL

polby one step of reduction then ! reduces to ! in LLP by one step of

reduction.

Corollary 4.1 (Strong normalization)

There is no infinite sequence of reductions inLL

pol if we forbid commutations of cuts.Corollary 4.2 (Cut elimination)If is provable in LLpol, then is provable without the cut rule.

The translation (.) of LLfoc into LLpol

is obtained by adding to formulas exactly the required

liftings to get a polarized formula, and by translating the sequent P,N ; by P,N,.In particular (!P` 1) = !P ` 1.

Proposition 5 (Strict simulation)If reduces to in LLfoc by one step of reduction then

reduces to in LLpol by at least onestep of reduction.

Corollary 5.1 (Strong normalization)There is no infinite sequence of reductions in LLfoc if we forbid commutations of cuts.

Corollary 5.2 (Cut elimination)If ; is provable in LLfoc, then ; is provable without the cut rules.

Proof: We consider a cut rule without any cut above it. We look at the two different cases:

If it is a n-cut, we look at the rule above the premise , P ; . If the rule above itintroduces P, it is either a foc rule or a rule and we apply the corresponding key step(and the n-cut becomes a p-cut) or commutative n-step. Otherwise this rule cannot bean ax rule, a 1 rule or a ! rule and we can apply the corresponding commutative n-step.

If it is a p-cut, we first look at the premise ; P. IfP is not a main formula, we canapply a left commutative p-step. IfP is a main formula and P is not, we can applythe corresponding right commutative p-step (notice that the rule above P cannot be a1 rule). We just have to verify that we can apply the right commutative p-step in thecase of a ! rule above P: since P is main, the rule above it is either an ax rule or a! rule and we can apply the reduction step. If both P and P are main, we apply thecorresponding key step.

So that, either the proof is cut-free or a reduction step can be applied, and we conclude bystrong normalization. 2

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