Strong normalization of lambda-Sym-Prop and lambda bare-mu-mu tilde*-calculi

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Strong normalization of lambda-Sym-Prop and lambda bare-mu-mu tilde*-calculi

Péter Battyányi, Karim Nour

To cite this version:

Péter Battyányi, Karim Nour. Strong normalization of lambda-Sym-Prop and lambda bare-mu-mu

tilde*-calculi . 2016. �hal-01421201�

Strong normalization of λ ^Sym _Prop - and λµ µ ˜ ^∗ -calculi

Péter Battyányi

Department of Computer Science, Faculty of Informatics, University of Debrecen,

Kassai út 26, 4028 Debrecen, Hungary email : [email protected]

Karim Nour LAMA - Équipe LIMD Université Savoie Mont Blanc

73376 Le Bourget du Lac email : [email protected]

Abstract

In this paper we give an arithmetical proof of the strong normalization of λ

^Sym_Prop

of Berardi and Barbanera [1], which can be considered as a formulae- as-types translation of classical propositional logic in natural deduction style.

Then we give a translation between the λ

^Sym_Prop

-calculus and the λµ˜ µ

^∗

-calculus, which is the implicational part of the λµ µ ˜ -calculus invented by Curien and Herbelin [3] extended with negation. In this paper we adapt the method of [4] for proving strong normalization. The novelty in our proof is the notion of zoom-in sequences of redexes, which leads us directly to the proofs of the main theorems.

1 Introduction

It was revealed by the works of Murthy [8] and Grin [6] that the Curry-Howard isomorphism, which establishes a correspondence between natural deduction style proofs in intuitionistic logic and terms of the typed λ -calculus, can be extended to the case of classical logic, as well. Since their discovery many calculi appeared aiming to give an encoding of proofs formulated either in classical natural deduction or in classical sequent calculus.

The λµ -calculus presented by Parigot in [15] nds its origin in the so called Free Deduction (FD) (cf. [15]). Parigot resolves the deterministic nature of intuitionis- tic natural deduction: unlike in the case of intuitionistic natural deduction, when eliminating an instance of a cut in FD, there can be several choices for picking out the subdeductions to be transformed. By introducing variables of a new kind, the so called µ -variables, Parigot distinguishes formulas that are not active at the mo- ment but the current continuation can be passed over to them. Besides the usual β -reduction, Parigot introduces a new reduction rule called the µ -rule correspond- ing to structural cut eliminations made necessary by the occurrence of new forms of cuts due to the rule in connection with the µ -variables. The result is a calculus, the λµ -calculus (cf. [13]), which is in relation with classical natural deduction. The µ

⁰

-rule is the symmetric counterpart of the µ -rule. It was introduced by Parigot in [14] with the intention of keeping the unicity of representation of data (cf. [11]), the price was, however, that conuence had been lost. In the presence of other simplication rules besides µ and µ

⁰

, even the strong normalization property is lost (cf. [2]).

Historically, the rst calculus reecting the symmetry of classical propositional

logic was the λ

^Sym_Prop

-calculus of Berardi and Barbanera [1] establishing a formulae-

as-types connection with natural deduction in classical logic. The calculus λ

^Sym_Prop

uses an involutive negation which is not dened as A → ⊥ . There are negated and non-negated atomic types, and the main connective is not the arrow but the classical ∧ and ∨ . Berardi and Barbanera make use of the natural symmetry of classical logic expressed by the de Morgan laws in dening negated types. In their paper, Berardi and Barbanera proved that λ

^Sym_Prop

is strongly normalizable with a symmetric version of the Tait-Girard reducibility method (cf. [20]). In this paper, leaning on the combinatorial proof applied by David and Nour in [5], we prove that λ

^Sym_Prop

is strongly normalizing. The novelty in our proof is the application of so-called zoom-in sequences of redexes, which was inspired by the work in [19].

We prove strong normalizability by verifying that it is closed under substitution.

From the assumption that U [x := V ] is strongly normalizing and U , V are strongly normalizing, we can identify a subterm U

⁰

of a reduct of U such that U

⁰

[x := V ] also is strongly normalizing. The reduction sequence leading to U

⁰

is a so-called zoom-in sequence of redexes: each subsequent element is a subterm of the one- step reduct of the preceding one. We prove that zoom-in sequences have useful invariant properties, which makes it relatively easy for us to set the stage for the main theorem. Due to its intrinsic symmetry in dealing with the typing relation, the λ

^Sym_Prop

-calculus also proves to be very close to the calculus named by Nour as classical combinatory logic (CCL). In [12], a calculus of combinators is dened which is equivalent to the full classical propositional logic in natural deduction style. Then a translation is given in both directions between λ

^Sym_Prop

and CCL.

Curien and Herbelin introduced the λµ˜ µ -calculus (cf. [3]), which established a correspondence, via the Curry-Howard isomorphism, between classical Gentzen- style sequent calculus and a logical calculus. The λµ˜ µ -calculus possesses a rather strong symmetry: it has right-hand side and left-hand side terms (also referred to as environments). The strong normalization of the calculus was proved by Polonovski [16], and a proof formalizable in rst order Peano arithmetic was found by David and Nour (cf. [4]).

As to the connection between the λµ and the λµ˜ µ -calculus, Curien and Herbelin (cf. [3]) dened a translation both for the call-by-value and the call-by-name part of the λµ -calculus into the λµ˜ µ -calculus. Rocheteau [17] nished this work by dening simulations between the two calculi in both directions. In this paper we dene the λµ˜ µ

^∗

-calculus, which is the λµ˜ µ -calculus extended with negation, and we describe translations between the λµ˜ µ

^∗

-calculus and the λ

^Sym_Prop

-calculus. As a consequence, we obtain that, if one of the calculi is strongly normalizable, then the other one necessary admits this property.

The proof applied in the paper is an adaptation of that in David and Nour [4]. In [4] arithmetic proofs, that is, proofs formalizable in rst-order Peano arith- metic, are given for the strong normalizability of the λµ˜ µ - and Parigot's symmetric λµ -calculus. It is demonstrated that the set of strongly normalizable terms are closed under substitution. The goal is achieved by applying implicitly an alternat- ing substitution to nd out which part of the substitution would be responsible for being not strongly normalizable provided the basis of the substitution and the terms written in are strongly normalizable. In this paper we reach the same goal by identifying a minimal non strongly normalizing sequence of redexes provided an innite reduction sequence is given. We call this sequence of redexes a minimal zoom-in reduction sequence. The idea of zoom-in sequence was inspired by [19], where perpetual reduction strategies are dened in order to locate the minimal non strongly normalizing subterms of the elements of an innite reduction sequence.

Again, alternating substitutions are dened inductively starting from two sets of

terms, and it is proven that zoom-in reduction sequences do not lead out of these

substitutions. With this in hand, the method in [4] can be applied.

We prove the strong normalization of the λ

^Sym_Prop

-calculus, though our proof works with some slight modication in the case of the λµ˜ µ -calculus, as well (cf.[2]). How- ever, instead of repeating the proof here, we give a translation of the λ

^Sym_Prop

-calculus into the λµ˜ µ -calculus, and vice versa. In fact, to make the connection more visible we dene the λµ˜ µ

^∗

-calculus, which is the λµ˜ µ -calculus extended with terms ex- pressing negated types. Hence, we also obtain a new proof of strong normalization of the λµ˜ µ -calculus.

The paper is organized as follows. In the rst section we introduce the λ

^Sym_Prop

- calculus of Berardi and Barbanera, and, as the rst step towards strong normaliza- tion, prove that the permutation rules can be postponed. In the nest section we show that the β , β

^⊥

, π and π

^⊥

rules together are strongly normalizing. Section 4 introduces the λµ˜ µ -calculus dened by Curien and Herbelin, and we augment the calculus with negation in order to make the comparison of the λ

^Sym_Prop

- and the λµ˜ µ -calculi simpler. Section 5 provides translations between the λ

^Sym_Prop

- and the λµ˜ µ

^∗

-calculi such that the strong normalization of one of the calculi implies that of the other. The last section contains conclusions with regard to the results of the paper.

2 The λ ^Sym _Prop -calculus

The λ

^Sym

-calculus was introduced by Berardi and Barbanera in [1]. It is organized entirely around the duality in classical logic. It has a negation built-in: the nega- tion of A is not dened as A → ⊥ . Each type is rather related to its natural negated type by the notion of duality introduced by negation in classical logic. In fact, Be- rardi and Barbanera dened a calculus equivalent to rst order Peano arithmetic.

However, we only consider here its propositional part, denoted by λ

^Sym_Prop

, since all the other calculi treated by us in this work are concerned with propositional logic.

Denition 2.1 The set of types are built from two sets of base types A = {a, b, . . .}

(atomic types) and A

^⊥

= {a

^⊥

, b

^⊥

, . . .} (negated atomic types).

1. The set of m-types is dened by the following grammar A := α | α

^⊥

| A ∧ A | A ∨ A where α ranges over A and α

^⊥

over A

^⊥

.

2. The set of types is dened by the following grammar C := A | ⊥.

3. We dene the negation of an m-type as follows

(α)

^⊥

= α

^⊥

(α

^⊥

)

^⊥

= α (A ∧ B )

^⊥

= A

^⊥

∨ B

^⊥

(A ∨ B)

^⊥

= A

^⊥

∧ B

^⊥

. In this way we get an involutive negation, i.e. for every m -type A , (A

^⊥

)

^⊥

= A . 4. The complexity of a type is dened inductively as follows.

cxty(A) = 0 , if A ∈ A ∪ A

^⊥

∪ {⊥} .

cxty(A

1

∧ A

2

) = cxty(A

1

∨ A

2

) = cxty(A

1

) + cxty(A

2

) + 1 .

Then, for every m -type A , cxty(A) = cxty(A

^⊥

) .

Denition 2.2 1. We denote by V ar the set of term-variables. The set of terms T of the λ

^Sym_Prop

-calculus together with their typing rules are dened as follows. In the denition below the type of a variable must be an m -type and Γ denotes a context (the set of declarations of variables).

var Γ, x : A ` x : A

h , i Γ ` P

1

: A

1

Γ ` P

2

: A

2

Γ ` hP

1

, P

2

i : A

1

∧ A

2

σ i

Γ ` P i : A i

Γ ` σ i (P i ) : A

1

∨ A

2

i ∈ {1, 2}

λ Γ, x : A ` P : ⊥

Γ ` λxP : A

^⊥

? Γ ` P

1

: A

^⊥

Γ ` P

2

: A Γ ` (P

1

? P

2

) : ⊥

2. As usuel, we denote by F v(M ) , the set of the free vraribles of the term M . 3. The complexity of a term of T is dened as follows.

cxty(x) = 0 ,

cxty(hP

1

, P

2

i) = cxty((P

1

? P

2

)) = cxty(P

1

) + cxty(P

2

) , cxty(λxP ) = cxty(σ i (P)) = cxty(P ) + 1 , for i ∈ {1, 2} . Denition 2.3 1. The reduction rules are enumerated below.

(β) (λxP ? Q) → β P [x := Q]

(β

^⊥

) (Q ? λxP ) → _β

⊥

P [x := Q]

(η) λx(P ? x) → η P if x / ∈ F v(P ) (η

^⊥

) λx(x ? P ) → _η

⊥

P if x / ∈ F v(P ) (π) (hP

1

, P

2

i ? σ i (Q i )) → π (P i ? Q i ) i ∈ {1, 2}

(π

^⊥

) (σ i (Q i ) ? hP

1

, P

2

i) → _π

⊥

(Q i ? P i ) i ∈ {1, 2}

(T riv) E[P ] → T riv P (∗)

(*) If E[−] is a context with type ⊥ and E[−] 6= [−] , P has type ⊥ and E[−]

does not bind any free variables in P .

2. Let us take the union of the above rules. Let → stand for the compatible closure of this union and, as usual, →

^∗

denote the reexive, symmetric and transitive closure of → . The notions of reduction sequence, normal form and normalization are dened with respect to → .

3. Let M, N be terms. The length (i.e. the number of steps) of the reduction M →

^∗

N is denoted by lg(M N) .

We enumerate below some theoretical properties of the λ

^Sym_Prop

-calculus (cf. [1]

and [9]).

Proposition 2.4 (Type-preservation property) If Γ ` P : A and P →

^∗

Q , then Γ ` Q : A .

Proposition 2.5 (Subformula property) If Π is a derivation of Γ ` P : A and P is in normal form, then every type occurring in Π is a subformula of a type occurring in Γ , or a subformula of A .

Theorem 2.6 (Strong normalization) If Γ ` P : A , then P is strongly nor-

malizable, i.e. every reduction sequence starting from P is nite.

Berardi and Barbanera proved Theorem 2.6 for the extension of the λ

^Sym_Prop

- calculus equivalent to rst-order Peano-arithmetic. The proof of this result by Berardi-Barbanera [1] is based on reducibility candidates, but the denition of the interpretation of a type relies on non-arithmetical xed-point constructions.

We present a syntactical and arithmetical proof of the strong normalization of the λ

^Sym_Prop

-calculus in Section 3. The proof was inspired by [4]. First we establish that the permutation rules η , η

^⊥

and T riv can be postponed so that we can restrict our attention uniquely to the rules β , β

^⊥

, π , and π

^⊥

.

2.1 Permutation rules

First of all, we prove that the η - and η

^⊥

-reductions can be postponed w.r.t. β , β

^⊥

, π , and π

^⊥

.

Denition 2.7 1. Let λ _βπ -calculus denote the calculus with only the reduction rules → _β , → _β

⊥

, → _π , and → _π

⊥

.

2. Let → _βπ stand for the union of → _β , → _β

⊥

, → _π , → _π

⊥

and let M → _e N denote the fact that M → η N or M → _η

⊥

N .

3. We denote by → β

₀

(resp. by → _β

⊥

0

) the β -reduction (λxM ? N ) → β M [x := N]

(resp. the β

^⊥

-reduction (N ? λxM ) → _β

⊥

M [x := N ] ), where x occurs at most once in M .

4. We use the standard notation →

⁺

and →

^∗

for the transitive and reexive, transitive closure of a reduction, respectively.

We examine the behaviour of a → e rule followed by a → β and a → β

₀

rule in Lemmas 2.8 and 2.9.

Lemma 2.8 If U → e V → β W , then U → β V

⁰

→

^∗

_e W or U → β

0

V

⁰

→ β W for some V

⁰

.

Proof We assume → e is an η -reduction, the proof of the case of → β is similar.

The proof is by induction on cxty(U ) . The only interesting case is U = (U

1

? U

2

) . We consider only some of the subcases.

1. U

1

= λx(U

3

? x) , with x / ∈ F v(U

3

) , and V = (U

3

? U

2

) → β U

4

[y := U

2

] = W , where U

3

= λyU

4

. In this case U = (λx(U

3

? x) ? U

2

) → β

0

(U

3

? U

2

) → β

U

₄

[y := U

₂

] = W , so → η → β turns into → β

0

→ β .

2. U

₁

= λxU

₃

, U

₃

→ η U

₄

and V = (λxU

₄

? U

₂

) → β U

₄

[x := U

₂

] = W . Then U → _β V

⁰

= U

₃

[x := U

₂

] → _η U

₄

[x := U

₂

] = W .

3. U

₁

= λxU

₃

, U

₂

→ _η U

₄

and V = (λxU

₃

? U

₄

) → _β U

₃

[x := U

₄

] = W . Then U → β V

⁰

= U

3

[x := U

2

] →

^∗

_η U

3

[x := U

4

] .

Lemma 2.9 If U → e V → β

0

W , then U → β

0

W or U → β

0

V

⁰

→ e W or U → β

0

V

⁰

→ _β

₀

W for some V

⁰

.

Proof By induction on cxty(U) . We assume U = (U

1

? U

2

) and we consider some of the more interesting cases.

1. U

1

= λx(U

3

? x) , with x / ∈ F v(U

3

) , and V = (U

3

? U

2

) → β

0

U

4

[y := U

2

] = W , where U

₃

= λyU

₄

. In this case U = (λx(U

₃

? x) ? U

₂

) → β

0

(U

₃

? U

₂

) → β

0

U

₄

[y := U

₂

] = W , thus → η → β

0

turns into → β

0

→ β

0

.

2. U

1

= λxU

3

, U

3

→ η U

4

and V = (λxU

4

? U

2

) → β

₀

U

4

[x := U

2

] = W . Then U → β

₀

V

⁰

= U

3

[x := U

2

] → η U

4

[x := U

2

] = W .

3. U

1

= λxU

3

, U

2

→ η U

4

and V = (λxU

3

? U

4

) → β

₀

U

3

[x := U

4

] = W . Then U → β

₀

V

⁰

= U

3

[x := U

2

] → η U

3

[x := U

4

] provided x occurs in U

3

. Otherwise U → β

₀

U

3

= W .

We obtain easily the following lemma on the behaviour of several → _e rules followed by a → _β or a → _β

₀

rule.

Lemma 2.10 If U →

^∗

_e V → β

₀

W , then U → β

₀+

V

⁰

→

^∗

_e W for some V

⁰

, and lg(U→ β

₀+

V

⁰

→

^∗

_e W ) ≤ lg(U →

^∗

_e V → β

₀

W ) .

Proof By induction on lg(U →

^∗

_e V → β

₀

W ) , using Lemma 2.9.

Lemma 2.11 If U →

^∗

_e V → _β W , then U → _β

⁺

V

⁰

→

^∗

_e W for some V

⁰

.

Proof By induction on lg(U →

^∗

_e V → β W ) . Use Lemmas 2.8, 2.10.

Lemma 2.12 If U →

^∗

_e V → β

^⊥

W , then U → β

^⊥+

V

⁰

→

^∗

_e W for some V

⁰

.

Proof Similar to that of the previous lemma.

We investigate now how a → e rule behaves when followed by a → π or → _π

^⊥

rule.

Lemma 2.13 If U → e V → π W (resp. U → e V → _π

^⊥

W ), then U → π V

⁰

→ e W or U → π W (resp. U → _π

^⊥

V

⁰

→ e W or U → _π

^⊥

W ) for some V

⁰

.

Proof Observe that in case of U → e V → π W the following possibilities can occur: either U = λx(V ? x) and V → π W or U = (hP

1

, P

2

i ? σ(Q)) and V = (hP

₁⁰

, P

₂⁰

i ? σ(Q

⁰

)) , where exactly one of P i → e P _i

⁰

, Q → e Q

⁰

holds, the other two terms are left unchanged. From this, the statement easily follows.

Lemma 2.14 If U →

^∗

_e V → βπ W , then U →

⁺

_βπ V

⁰

→ e W for some V

⁰

.

Proof By Lemmas 2.11, 2.12 and 2.13.

Lemma 2.15 If U →

^∗

_e V →

^∗

_βπ W , then U →

⁺

_βπ V

⁰

→

^∗

_e W for some V

⁰

.

Proof Follows from the previous lemma.

We are now in a position to prove the main result of the section.

Lemma 2.16 The η - and the η

⊥

-reductions are strongly normalizing.

Proof The η - and η

_⊥

-reductions on M reduce the complexity of M . Denition 2.17 1. Let λ βπη -calculus denote the calculus obtained from the

λ βπ -calculus by adding the η - and η

^⊥

-reductions to it.

2. Let → βπη denote the union of → β , → _β

⊥

, → π , → _π

⊥

, → η and → _η

⊥

.

3. Assume M is a term strongly normalizable in the λ _βπ -calculus. Then we denote by η _βπ (M ) the length of the longest reduction sequence →

^∗

_βπ starting from M .

Corollary 2.18 If the λ βπ -calculus is strongly normalizing, then the λ βπη -calculus is also strongly normalizing.

Proof Let M be a term, we prove by induction on η βπ (M ) that M is strongly normalizable in the λ βπη -calculus. Assume S is an innite βπη -reduction sequence starting from M . If S begins with a → βπ , then the induction hypothesis applies.

In the case when S contains only → e -reductions, we are done by Lemma 2.16.

Otherwise there is an initial subsequent M →

⁺

_e M

⁰

→ βπ N . By Lemma 2.14, we

have M →

⁺

_βπ M

⁰⁰

→

^∗

_e N . Thus, we can apply the induction hypothesis to M

⁰⁰

.

What has remained is to augment the calculus treated so far with the rule T riv . For strong normalization, it is enough to show that → T riv can be postponed w.r.t.

→ βπη .

Lemma 2.19 If U →

^∗

_{T riv} V → βπη W , then U →

⁺

_βπη V

⁰

→

^∗

_{T riv} W for some V

⁰

. Proof It is enough to prove that if U → T riv V → βπη W , then U → βπη V

⁰

→ T riv

W for some V

⁰

. Observe that if U = E[V ] → T riv V → βπη W , then V : ⊥ and

W : ⊥ , from which the statement follows.

Lemma 2.20 The reduction → _{T riv} is strongly normalizing.

Proof The reduction → T riv on M reduces the complexity of M . Corollary 2.21 If the λ βπ -calculus is strongly normalizing, then the λ

^Sym

_{P rop} -calculus is also strongly normalizing.

Proof By Corollary 2.18 and Lemmas 2.19 and 2.20.

3 Strong normalization of the λ βπ -calculus

In this section, we give an arithmetical proof for the strong normalization of the λ βπ -calculus. In the sequel we detail the proofs for the β - and π -reductions only, all the proofs below can be extended with the cases of the β

^⊥

- and π

^⊥

-reduction rules in a straightforward way. We intend to examine how substitution behaves with respect to strong normalizability. The rst milestone in this way is Lemma 3.7. Before stating the lemma, we formulate some auxiliary statements.

Denition 3.1 1. Let SN βπ denote the set of strongly normalizable terms of the λ _βπ -calculus.

2. Let M ∈ SN _βπ , then ηc(M ) stands for the pair hη βπ (M ), cxty(M )i .

Lemma 3.2 Let us suppose M ∈ SN _βπ , N ∈ SN _βπ and (M ? N ) ∈ / SN _βπ . Then there are P ∈ SN _βπ , Q ∈ SN _βπ and R / ∈ SN _βπ such that M →

^∗

_βπ P and N →

^∗

_βπ Q and (P ? Q) → βπ R .

Proof By induction on ηc(M ) + ηc(N ) . Assume M ∈ SN βπ , N ∈ SN βπ and (M ? N ) ∈ / SN _βπ . When (M ? N ) → (M

⁰

? N ) or (M ? N ) → (M ? N

⁰

) , then the induction hypothesis applies. Otherwise (M ? N) → P / ∈ SN _βπ , and we have the

result.

Denition 3.3 1. A proper term is a term diering from a variable.

2. For a type A , Σ A denotes the set of simultaneous substitutions of the form [x

₁

:= N

₁

, . . . , x _k := N _k ] where N _i ( 1 ≤ i ≤ n ) is proper and has type A . 3. A simultaneous substitution σ ∈ Σ _A is said to be in SN _βπ , if, for every

x ∈ dom(σ) , σ(x) ∈ SN _βπ holds.

Lemma 3.4 Let M, N be terms such that M →

^∗

_βπ N . 1. If N = λxP , then M = λxP

1

with P

1

→

^∗

_βπ P .

2. If N = hP, Qi , then M = hP

1

, Q

1

i with P

1

→

^∗

_βπ P and Q

1

→

^∗

_βπ Q . 3. If N = σ i (P ) , then M = σ i (P

1

) with P

1

→

^∗

_βπ P , for i ∈ {1, 2} .

Proof Straightforward.

We remark that in the presence of the → η and →

^⊥

_η rules the above lemma would not work. For example, λx(y ? x) → η y .

Lemma 3.5 If M ∈ SN βπ and x ∈ V ar , then (M ? x) ∈ SN βπ (resp. (x ? M ) ∈ SN _βπ ).

Proof Let us suppose M ∈ SN βπ and (M ? x) ∈ / SN βπ . By Lemma 3.2, we must have M →

^∗

_βπ λyM

1

∈ SN βπ such that (M ? x) →

^∗

_βπ (λyM

1

? x) → βπ M

1

[y := x]

and M

1

[y := x] ∈ / SN βπ . Being a subterm of a reduct of M ∈ SN βπ , we also have M

1

∈ SN βπ . Moreover, M

1

[y := x] is obtained from M

1

by α -conversion, hence

M

1

[y := x] ∈ SN βπ , a contradiction.

Denition 3.6 1. Let M, N be terms.

(a) We denote by M ≤ N (resp. M < N ) the fact that M is a sub-term (resp. a strict sub-term) of N .

(b) We denote by M ≺ N the fact that M ≤ P for some P such that N →

^∗

_βπ P and either N →

⁺

_βπ P or M < P . We denote by the reexive closure of ≺ .

(c) Let R be a βπ -redex. We write M → ^R N if N is the term M after the reduction of R .

2. Let R = [R

1

, . . . , R n ] where R i is a βπ -redex (1 ≤ i ≤ n) . Then R is called zoom-in if, for every 1 ≤ i < n , R _i → ^R

ⁱ

R

⁰

_i and R _i+1 ≤ R

⁰

_i . Moreover, R is minimal, if, for each R _i = (P _i ? Q _i ) , we have P _i , Q _i ∈ SN _βπ and (P _i ? Q _i ) ∈ / SN _βπ . We write M →

^R

N , if M → ^R

¹

... → ^R

ⁿ

N .

For the purpose of proving the strong normalization of the calculus, it is enough to show that the set of strongly normalizable terms is closed under substitution. To this end, we show that, if U , S ∈ SN βπ and U [x := S] ∈ / SN βπ , then there is a term W ≤ U of a special form such that W ∈ SN βπ and W [x := S] ∈ / SN βπ . Moreover, we show that the sequence of redexes leading to W is not completely general, this is a zoom-in sequence dened below. Reducing the outermost redexes of a zoom-in sequence preserve some useful properties, which is the statement of Lemma 3.9.

Lemma 3.7 Let U , S ∈ SN _βπ and suppose U [x := S] ∈ / SN _βπ . Then there are terms P, V U and a zoom-in minimal R such that U [x := S] →

^R

V [x := S] , (x ? P ) ≤ V (or (P ? x) ≤ V ), P[x := S] ∈ SN _βπ and (x ? P )[x := S] ∈ / SN _βπ (or (P ? x)[x := P ] ∈ / SN βπ ).

Proof The proof goes by induction on ηc(U ) . If U is other than an application, we can apply the induction hypothesis. Assume U = (U

1

? U

2

) with U i [x := S] ∈ SN βπ

(i ∈ {1, 2}) and U[x := S] ∈ / SN βπ . By Lemma 3.2 and the induction hypothesis we may assume that (U

1

? U

2

)[x := S] → ρ U

⁰

∈ / SN βπ , where ρ ∈ {β, β

^⊥

, π, π

^⊥

} . Let us suppose ρ = β , the other cases can be treated similarly. If U

₁

= λyU

₁⁰

, then the induction hypothesis applies to U

₁⁰

[y := U

₂

] . Otherwise U

₁

= x , and we have

obtained the result.

Next we dene an alternating substitution: we start from two sets of terms of

complementer types and the substitution is dened in a way that we keep track

of which newly added sets of substitutions come from which of the two sets. The

reason for this is that Lemma 3.7 in itself is not enough for proving the strong

normalizability of λ

^Sym_Prop

even if we would consider the β and β

^⊥

rules alone. We

have to show that, if we start from a term (U

1

? U

2

) , where U

1

and U

2

∈ SN βπ and

we assume that U

1

[x := U

2

] ∈ / SN βπ , then there are no deep interactions between

the terms which come from U

₁

and from U

₂

. We can identify a subterm of a reduct

of U

₁

which is the cause for being non SN _βπ , when performing a substitution with

U

₂

.

Denition 3.8 1. A set A of proper terms is called -closed from below if, for all terms U, U

⁰

, if U

⁰

U , U ∈ A and U

⁰

is proper, then U

⁰

∈ A .

2. Let A, B be sets -closed from below and A a type. We dene simultaneously two sets of substitutions

(a) Π A (B) ⊆ Σ A and Θ _A

⊥

(A) ⊆ Σ _A

⊥

as follows.

∅ ∈ Π _A (B) ,

[y

1

:= V

1

τ

1

, . . . , y m := V m τ m ] ∈ Π A (B) if V i ∈ B such that type(V i ) = A and τ i ∈ Θ _A

^⊥

(A) (1 ≤ i ≤ m) .

∅ ∈ Θ _A

⊥

(A) .

[x

1

:= U

1

ρ

1

, . . . , x m := U m ρ m ] ∈ Θ _A

⊥

(A) if U i ∈ A such that type(U i ) = A

^⊥

and ρ i ∈ Π A (B) (1 ≤ i ≤ m) .

(b) Let S A (A, B) = {U ρ | U ∈ A and ρ ∈ Π A (B)} ∪ {V τ | V ∈ B and τ ∈ Θ _A

⊥

(A)} . It is easy to see that, from U ≤ V and V ∈ S A (A, B) , it follows U ∈ S A (A, B) .

Lemma 3.9 Let n be an integer, A a type of length n and R = [R

1

, . . . , R m ] a zoom-in minimal sequence of redexes. Assume the property H if U , V ∈ SN βπ and cxty(type(V )) < n , then U [x := V ] ∈ SN βπ holds. If R

1

∈ S A (A, B) for some sets A and B -closed from below, then R m ∈ S A (A, B) .

Proof The proof goes by induction on m . We prove the induction step from m = 1 to m = 2 , the proof is the same when m ∈ N is arbitrary. We only treat the more interesting cases. Assume R

₁

∈ S A (A, B) .

1. R

1

= (λxQ ? S) → β R

⁰₁

= Q[x := S] and R

2

≤ R

⁰₁

.

(a) Suppose R

1

= U ρ for some U ∈ A and ρ ∈ Π A (B) . Then U = (U

1

? U

2

) with U

1

ρ = λxQ and U

2

ρ = S , and, since ρ ∈ Σ A , U

1

must be proper.

Then we have U

1

= λxU

₁⁰

and U

₁⁰

ρ = Q for some U

₁⁰

. Now, R

⁰₁

= U

₁⁰

[x :=

U

2

]ρ ∈ S A (A, B) , which yields R

2

∈ S A (A, B) .

(b) Assume now R

1

= V τ . Then V = (V

1

?V

2

) with V

1

τ = λxQ and V

2

τ = S , and, since τ ∈ Σ _A

⊥

, V

2

must be proper. If V

1

is proper, then, as before, we obtain the result. Otherwise V

1

τ = U ρ = λxQ . Since U ∈ A is proper, U = λxU

1

and U

1

ρ = Q for some U

1

. Then U

1

ρ

1

∈ S A (A, B) with ρ

1

= ρ + [x := V

2

τ] , since type(V

2

τ) = type(S) = A . This implies R

2

∈ S A (A, B) .

2. R

1

= (hQ

1

, Q

2

i ? σ

1

(S)) → π (Q

1

? S) = R

⁰₁

and R

2

≤ R

⁰₁

.

(a) Assume R

1

= U ρ for some U ∈ A and ρ ∈ Π A (B) . Then U

1

ρ = hQ

1

, Q

2

i and U

2

ρ = σ

1

(S) .

- Let U

1

and U

2

be proper. Then U

1

= hU

₁⁰

, U

₁⁰⁰

i and U

2

= σ

1

(U

₂⁰

) such that U

₁⁰

ρ = Q

1

, U

₁⁰⁰

ρ = Q

2

and U

₂⁰

ρ = S . We have R

⁰₁

= (U

₁⁰

? U

₂⁰

)ρ ∈ S A (A, B) , which yields the result.

- Assume U

₂

∈ V ar . Then V τ = σ(S) , and cxty(type(S)) <

cxty(type(σS)) = n . Then assumption H and the fact that hQ

₁

, Q

₂

i ∈ SN _βπ , together with Lemma 3.5, lead to (Q

₁

? S) ∈ SN _βπ , which is not possible. Since ρ ∈ Σ _A , U

₁

∈ V ar is impossible.

(b) Assume R

₁

= V τ for some V ∈ B and τ ∈ Θ _A

⊥

(A) . Then V

₁

τ = hQ

₁

, Q

₂

i and V

₂

τ = σ

₁

(S) , where V = (V

₁

? V

₂

) .

- Let V

₁

and V

₂

be proper. Then V

₁

= hV

₁⁰

, V

₁⁰⁰

i and V

₂

= σ

₁

(V

₂⁰

) such

that V

₁⁰

τ = Q

₁

, V

₁⁰⁰

τ = Q

₂

and V

₂⁰

τ = S . We have R

⁰₁

= (V

₁⁰

? V

₂⁰

)τ ∈

S _A (A, B) .

- Assume V

1

∈ V ar . Then U ρ = hQ

1

, Q

2

i , where cxty(type(Q

1

)) <

cxty(type(hQ

1

, Q

2

i)) = n . Then assumption H and the fact that S ∈ SN βπ , together with Lemma 3.5, lead to (Q

1

? S) ∈ SN βπ , which is not possible. Since τ ∈ Σ _A

^⊥

, the case of V

2

∈ V ar is impossible.

The next lemma identies the subterm of U being responsible for the non strong normalizability of U [x := V ] .

Lemma 3.10 Let n be an integer and A a type of length n . Assume the property H if U , V ∈ SN βπ and cxty(type(V )) < n , then U [x := V ] ∈ SN βπ holds.

1. Let U be a proper term, σ ∈ Σ A and a / ∈ Im(σ) . If U σ, P ∈ SN βπ and U σ[a := P] ∈ / SN _βπ , then there exists U

⁰

such that (U

⁰

? a) U and σ

⁰

∈ Σ _A such that U

⁰

σ

⁰

∈ SN _βπ and (U

⁰

σ

⁰

? a)[a := P] ∈ / SN _βπ .

2. Let U be a proper term, σ ∈ Σ _A

⊥

and a / ∈ Im(σ) . If U σ, P ∈ SN _βπ and U σ[a := P] ∈ / SN _βπ , then there exists U

⁰

such that (a ? U

⁰

) U and σ

⁰

∈ Σ _A

⊥

such that U

⁰

σ

⁰

∈ SN _βπ and (a ? U

⁰

σ

⁰

)[a := P] ∈ / SN _βπ .

Proof Let us consider only case (1). We identify the reason of U σ[a := P] being non strongly normalizable, we nd a subterm (U

⁰

? a) of a reduct of U such that, for a substituted instance of (U

⁰

? a) , (U

⁰

? a)σ

⁰

∈ SN _βπ and (U

⁰

? a)σ

⁰

[a := P] ∈ / SN _βπ . This will contradict by some minimality assumption concerning U in the next lemma. For this we dene two sets of substitutions as in Denition 3.8 with the sets A and B as below. We note that Property H of the previous lemma implicitly ensures that the type of U and the type of the elements in σ can be of the same lengths.

Let

A = {M | M U and M is proper },

B = {V | V σ(b) for some b ∈ dom(σ) and V is proper }.

Then U σ ∈ S A (A, B) . By Lemma 3.7, there exists a minimal zoom-in R = [R

1

, . . . , R n ] and there are terms U

^∗

and V U σ such that U σ[a := P] →

^R

V [a := P ] and (U

^∗

? a) ≤ V and (U

^∗

? a) ∈ SN βπ and (U

^∗

? a)[a := P ] ∈ / SN βπ

or (a ? U

^∗

) ≤ V and (a ? U

^∗

) ∈ SN βπ and (a ? U

^∗

)[a := P ] ∈ / SN βπ . Assume the former. By Lemma 3.9, (U

^∗

? a) ∈ S A (A, B) . Then (U

^∗

? a) = Sρ for some S ∈ A or (U

^∗

? a) = W τ for some W ∈ B . Since a / ∈ Im(σ) , the latter is impossible. The former case, however, yields S = (U

⁰

? a) with U

⁰

ρ = U

^∗

for some U

⁰

∈ A , which

proves our assertion.

The next lemma states closure of strong normalizability under substitution.

Lemma 3.11 If M, N ∈ SN βπ , then M [x := N ] ∈ SN βπ .

Proof We are going to prove a bit more general statement. Suppose M, N i ∈ SN βπ are proper, type(N i ) = A (1 ≤ i ≤ k) . Let τ i ∈ Σ _A

⊥

are such that τ i ∈ SN βπ

(1 ≤ i ≤ k) and let ρ = [x

1

:= N

1

τ

1

, . . . , x k := N k τ k ] . Then we have M ρ ∈ SN βπ . The proof is by induction on (cxty(A), η βπ (M ), cxty(M ) , Σ i η βπ (N i ), Σ i cxty(N i )) where, in Σ i η βπ (N i ) and Σ i cxty(N i ) , we count each occurrence of the substituted variable. For example if k = 1 and x

1

has n occurrences, then Σ i η βπ (N i ) = n · η βπ (N

1

) .

The only nontrivial case is when M = (M

1

? M

2

) and M ρ / ∈ SN βπ . By the induction hypothesis M _i ρ ∈ SN _βπ (i ∈ {1, 2}) . We select some of the typical cases.

(A) M

₁

ρ → _βπ λzM

⁰

and M

⁰

[z := M

₂

] ∈ / SN _βπ .

1. M

1

is proper, then there is an M

3

such that M

1

= λzM

3

and M

3

ρ → βπ

M

⁰

. In this case (M

3

[z := M

2

])ρ / ∈ SN βπ and since η βπ (M

3

[z := M

2

]) <

η βπ (M ) , the induction hypothesis gives the result.

2. M

1

∈ V ar . Then M

1

= x ∈ dom(ρ) , ρ(x) = N j τ j → βπ λzM

⁰

for some (1 ≤ j ≤ k) . Since N j is proper, there is an N

⁰

such that N j = λzN

⁰

, N

⁰

τ j → βπ M

⁰

. Then N

⁰

τ j [z := M

2

ρ] ∈ / SN βπ and type(z) = type(N j )

^⊥

= type(τ j ) , so, by the previous lemma, we have N

⁰⁰

≺ N

⁰

and τ

⁰

such that (N

⁰⁰

τ

⁰

? M

2

ρ) ∈ / SN βπ . Now we have (N

⁰⁰

τ

⁰

? M

2

ρ) = (y ? M

2

ρ)[y := N

⁰⁰

τ

⁰

] , type(N

⁰⁰

) = type(τ

⁰

)

^⊥

= A and η βπ cxty(N

⁰⁰

) <

ηc(N _j ) , which contradicts the induction hypothesis.

(B) M

₁

ρ → βπ hM

⁰

, M

⁰⁰

i and either (M

⁰

? M

₂

) ∈ / SN _βπ or (M

⁰⁰

? M

₂

) ∈ / SN _βπ . Suppose the former.

1. M

₁

, M

₂

are proper, then there are M

₃

, M

₄

such that M

₁

= hM

₃

, M

₄

i and M

3

ρ → βπ M

⁰

, or M

4

ρ → βπ M

⁰⁰

. Assume the former. Then we have (M

3

? M

2

)ρ / ∈ SN βπ and η βπ ((M

3

? M

2

)) < η βπ (M ) , a contradiction.

2. M

1

= x ∈ dom(ρ) , then ρ(x) = N j τ j → βπ hM

⁰

, M

⁰⁰

i , N j is proper and N j = hU, V i , U τ j → βπ M

⁰

or V τ j → βπ M

⁰⁰

. Now (U τ j ? M

2

) = (y ? M

2

)[y := U τ j ] ∈ / SN βπ , but cxty(type(U )) < cxty(type(N j )) , a contradiction again.

3. M

2

∈ V ar . This is similar to the previous case. By the same argument as in part (A)-2.-(a) of the proof of the previous lemma, M

1

and M

2

cannot be both variables. This completes the proof of the lemma.

Theorem 3.12 The λ βπ -calculus is strongly normalizing.

Proof It is enough to show that, for every term, M , N ∈ SN _βπ implies (M ?N) ∈ SN βπ . Supposing M, N ∈ SN βπ , Lemma 3.5 gives (M ? x) ∈ SN βπ , which yields, by the previous lemma, (M ? N ) = (M ? x)[x := N ] ∈ SN βπ .

4 The λµ µ ˜ - and the λµ˜ µ ^∗ -calculus

In this section we introduce the λµ˜ µ -calculus together with one of its extensions, the λµ˜ µ

^∗

-calculus, by which we establish a translation of the λ

^Sym_Prop

-calculus and thus obtain the strong normalization of the λµ˜ µ

^∗

-calculus as a consequence.

4.1 The λµ˜ µ -calculus

The λµ˜ µ -calculus was introduced by Curien and Herbelin (cf. [10] and [3]). We ex- amine here the calculus dened in [3], which is a simply typed one. The λµ˜ µ -calculus was invented for representing proofs in classical Gentzen-style sequent calculus: un- der the Curry-Howard correspondence a version of Gentzen-style sequent calculus is obtained as a system of simple types for the λµ˜ µ -calculus. Moreover, the system presents a clear duality between call-by-value and call-by-name evaluations.

Denition 4.1 There are three kinds of terms, dened by the following grammar,

and there are two kinds of variables. We assume that we use the same set of variables

in the λµ˜ µ

^∗

-calculus, too. In the literature, dierent authors use dierent termi-

nology. Here, we will call them either c -terms, or l -terms or r -terms. Similarly,

the variables will be called either l -variables (and denoted as x, y, ... ) or r -variables (and denoted as a, b, ... ).

p ::= bt, ec

t ::= x | λxt | µαp e ::= α | (t.e) | µxp ˜

As usual, we denote by F v(u) , the set of the free variables of the term u .

Denition 4.2 The types are built from atomic formulas (or, in other words, atomic types) with the connector → . We assume that the same set of type variables A is used in the λµ˜ µ

^∗

-calculus, also. The typing system is a sequent calculus based on judgements of the following form.

p : (Γ B 4) Γ B t : A | 4 Γ | e : A B 4

where Γ (resp. 4 ) is a set of declarations of the form x : A (resp. a : A ), x (resp.

a ) denoting a l -variable (resp. an r -variable) and A representing a type, such that x (resp. a ) occurs at most once in an expression of Γ (resp. 4 ) of the form x : A (resp. a : A ). We say that Γ an l -context and 4 is an r -context, respectively. The typing rules are as follows

V ar

1

Γ, x : A B x : A | 4 V ar

2

Γ | α : A B α : A, 4

λ Γ, x : A B t : B | 4

Γ B λxt : A → B | 4 (.) Γ B t : A | 4 Γ | e : B B 4 Γ | (t.e) : A → B B 4

b, c Γ B t : A | 4 Γ | e : A B 4 bt, ec : (Γ B 4)

µ p : (Γ B α : A, 4)

Γ B µαp : A | 4 µ ˜ p : (Γ, x : A B 4)

Γ | µxp ˜ : A B 4

Denition 4.3 The cut-elimination procedure (on the logical side) corresponds to the reduction rules (on the terms) given below.

(λ) bλxt, (t

⁰

.e)c , → λ bt

⁰

, µx ˜ bt, ecc (µ) bµαp, ec , → µ p[α := e]

(˜ µ) bt, µxpc ˜ , → µ

˜

p[x := t]

(s l ) µαbt, αc , → s

_l

t if α 6∈ F v(t) (s _r ) µxbx, ec ˜ , → s

r

e if x 6∈ F v(e)

Let us take the union of the above rules. Let , → stand for the compatible closure of this union and, as usual, , →

^∗

denote the reexive, symmetric and transitive closure of , → . The notions of reduction sequence, normal form and normalization are dened with respect to , → .

We present below some theoretical properties of the λµ˜ µ -calculus (cf. [10], [3], [9], [16] and [5]).

Proposition 4.4 (Type-preservation property) If Γ B t : A | 4 (resp. Γ | e : A B 4 , resp. p : (Γ B 4) ) and t , →

^∗

t

⁰

(resp. e , →

^∗

e

⁰

, resp. p , →

^∗

p

⁰

), then Γ B t

⁰

: A | 4 (resp. Γ | e

⁰

: A B 4 , resp. p

⁰

: (Γ B 4) ).

Proposition 4.5 (Subformula property) If Π is a derivation of Γ B t : A | 4

(resp. Γ | e : A B 4 , resp. p : (Γ ` 4) ) and t (resp. e , resp. p ) is in normal

form, then every type occurring in Π is a subformula of a type occurring in Γ ∪ 4 ,

or a subformula of A (only for t and e ).

Theorem 4.6 (Strong normalization property) If Γ B t : A | 4 (resp. Γ | e : A B 4 , resp. p : (Γ B 4) ), then t (resp. e , resp. p ) is strongly normalizable, i.e.

evrey reduction sequence starting from t (resp. e , resp. p ) is nite.

The proof of Theorem 4.6 can be found in [16], as well as in [5], where an arithmetical proof is presented.

4.2 The λµ˜ µ ^∗ -calculus

Since we work in a sequent calculus, where dealing with negation is implicitly built in the rules, the typing rules of the λµ˜ µ -calculus do not handle negation. However, for a full treatment of propositional logic we found it more convenient to introduce rules concerning negation. Since c -terms, which could have been candidates for objects of type ⊥ , are distinctly separated from terms, adding new term- and type- forming operators seems to be the easiest way to dene negation.

Denition 4.7 1. The terms of the λµ˜ µ

^∗

-calculus are dened by the following grammar.

p ::= bt, ec

t ::= x | λx t | µα p | e e ::= α | (t.e) | µx p ˜ | e t

As an abuse of terminology, in the sequel when speaking about the syntactic elements of the λµ˜ µ

^∗

-calculus, we may not distinguish l -, r - and c -terms, we may speak about terms in general. We denote by T the set of terms of the λµ˜ µ

^∗

-calculus.

2. The complexity of a term of T is dened as follows.

cxty(x) = cxty(α) = 0 ,

cxty(λxt) = cxty(e t) = cxty(t) + 1 , cxty(e) = cxty(e) + 1 ,

cxty(µαp) = cxty(˜ µx p) = cxty(p) + 1 , cxty(bt, ec) = cxty((t.e)) = cxty(t) + cxty(e) .

Denition 4.8 The type inference rules are the same as in the λµ˜ µ -calculus with two extra rules added for the types of the complemented terms. Moreover, we in- troduce an equation between types (for all types A , (A

^⊥

)

^⊥

= A ) to ensure that our negation is involutive.

. Γ | e : A B 4

Γ B e : A

^⊥

| 4 e . Γ B t : A | 4 Γ | e t : A

^⊥

B 4 We also dene the complexity of types in the λµ˜ µ

^∗

-calculus.

cxty(A) = 0 for atomic types,

cxty(A → B) = cxty(A) + cxty(B) + 1 , cxty(A

^⊥

) = cxty(A) .

That is, the complexity of a type A provides us with the number of arrows in A .

The presence of negation makes it necessary for us to introduce new rules handling

negation.

Denition 4.9 Besides the reduction rules already present in λµ˜ µ , we endow the calculus with some more new rules to handle the larger set of terms. In what follows cl stands for the name: complementer rule. We shall denote the cl

1,l

- and cl

1,r

-rules with a common notation as the cl

1

-rules.

(cl

1,l

) e t , → cl

_1,l

t (cl

1,r

) e e , → cl

_1,r

e (cl

2

) be, e tc , → cl

₂

bt, ec

In the sequel, we continue to apply the notation , → and , →

^∗

in relation with this new calculus.

Obviously, the statements analogous to Propositions 4.4 and 4.5 are still valid.

5 Relating the λ ^Sym _Prop -calculus to the λµ˜ µ ^∗ -calculus

In [17], Rocheteau dened a translation between the λµ˜ µ -calculus and the λµ - calculus, treating both a call-by-value and a call-by-name aspect of λµ˜ µ . In this subsection, we give a translation (in both directions) between the λ

^Sym_Prop

-calculus and the λµ˜ µ

^∗

-calculus, which is a version of the λµ˜ µ -calculus extended with negation.

The translations are such that strong normalization of one calculus follows from that of the other in both directions. We omit issues of evaluation strategies, however. In the end of the section we give an exact description of the correspondence between the two translations. Preparatory to presenting the translations, let us introduce some denitions and notation below. We assume that the two calculi have the same sets of variables and atomic types. Moreover, as an abuse of notation, if α : A

^⊥

stems form the r -variable α : A in the λµ˜ µ

^∗

-calculus, then we suppose that in the λ

^Sym_Prop

-calculus α denotes a variable with type A

^⊥

.

5.1 A translation of the λµ˜ µ ^∗ -calculus into the λ ^Sym _Prop -calculus

Denition 5.1 1. Let us consider the λ

^Sym_Prop

-calculus. For i ∈ {1, 2} , we write π i (y) = λz(y ? σ i (z)) . Then, we can observe that y : A

1

∧ A

2

` π i (y) : A i , for i ∈ {1, 2} .

2. We dene a translation .

^e

: T −→ T as follows.

p

^e

= (u

^e

? v

^e

) if p = bv, uc.

t

^e

=



 



 



x if t = x,

λy(λx(π

₂

(y) ? u

^e

) ? π

₁

(y)) if t = λxu, λx(e

^e

? t

^e

) if t = ˜ µxbt, ec,

u

^e

if t = u.

e

^e

=



 



 



α if e = α,

ht

^e

, h

^e

i if e = t.h, λα(e

^e

? t

^e

) if e = µαbt, ec, h

^e

if e = e h.

3. The translation .

^e

also applies to types.

• A

^e

= A , where A is an atomic type,

• (A

^⊥

)

^e

= (A

^e

)

^⊥

,

• (A → B)

^e

= (A

^e

)

^⊥

∨ B

^e

.

4. Let Γ , 4 be l - and r -contexts, respectively. Then Γ

^e

= {x : A

^e

| x : A ∈ Γ} and similarly for 4 . Furthermore, for any r -context 4 , let 4

^⊥

= {α : A

^⊥

| α : A ∈ 4} .

Lemma 5.2 1. If Γ B t : A | 4 , then Γ

^e

, (4

^e

)

^⊥

` t

^e

: A

^e

. 2. If Γ | e : A B 4 , then Γ

^e

, (4

^e

)

^⊥

` e

^e

: (A

^e

)

^⊥

.

3. If p : (Γ B 4) , then Γ

^e

, (4

^e

)

^⊥

` p

^e

: ⊥ .

Proof The above statements are proved simultaneously according to the length of the λµ˜ µ

^∗

-deduction. We remark that .

^e

is dened in Denition 5.1 exactly in the way to make the assertions of the lemma true. Let us examine some of the more interesting cases.

1. Suppose

Γ, x : A B u : B | 4 Γ B λxu : A → B | 4 .

Then we have, by the induction hypothesis and Notation 5.1, Γ

^e

, (4

^e

)

^⊥

, x : A

^e

, y : A

^e

∧ (B

^e

)

^⊥

` u

^e

: B

^e

,

Γ

^e

, (4

^e

)

^⊥

, y : A

^e

∧ (B

^e

)

^⊥

` π

1

(y) : A

^e

,

Γ

^e

, (4

^e

)

^⊥

, y : A

^e

∧ (B

^e

)

^⊥

` π

2

(y) : (B

^e

)

^⊥

. Thus we can conclude

Γ

^e

, (4

^e

)

^⊥

, x : A

^e

, y : A

^e

∧ (B

^e

)

^⊥

` (π

₂

(y) ? u

^e

) : ⊥,

Γ

^e

, (4

^e

)

^⊥

, y : A

^e

∧ (B

^e

)

^⊥

` λx(π

₂

(y) ? u

^e

) : (A

^e

)

^⊥

. From which, we obtain

Γ

^e

, (4

^e

)

^⊥

` λy(λx(π

2

(y) ? u

^e

) ? π

1

(y)) : (A

^e

)

^⊥

∨ B

^e

. 2. Assume now

Γ B t : A | 4 Γ | e : B B 4 Γ | t.e : A → B B 4 . Then we have

Γ

^e

, (4

^e

)

^⊥

` t

^e

: A

^e

Γ

^e

, (4

^e

)

^⊥

` e

^e

: (B

^e

)

^⊥

Γ

^e

, (4

^e

)

^⊥

` ht

^e

, e

^e

i : A

^e

∧ (B

^e

)

^⊥

. 3. From

Γ B t : A | 4 Γ | e : A B 4 bt, ec : (Γ B 4) , we obtain

Γ

^e

, (4

^e

)

^⊥

` t

^e

: A

^e

Γ

^e

, (4

^e

)

^⊥

` e

^e

: (A

^e

)

^⊥

Γ

^e

, (4

^e

)

^⊥

` (e

^e

? t

^e

) : ⊥ .

Our next aim is to prove that λµ˜ µ

^∗

can be simulated by the λ

^Sym_Prop

-calculus. To this end we introduce a new notion of equality in the λ

^Sym_Prop

-calculus.

Denition 5.3 We dene an equivalence relation ∼ on T .

• x ∼ x ,

• if M ∼ M

⁰

, then λxM ∼ λxM

⁰

and σ _i (M ) ∼ σ _i (M

⁰

) for i ∈ {1, 2} ,

• if M ∼ M

⁰

and N ∼ N

⁰

, then hM, N i ∼ hM

⁰

, N

⁰

i and (M ? N ) ∼ (M

⁰

? N

⁰

) and (M ? N) ∼ (N

⁰

? M

⁰

) .

We say that M and N are equal up to symmetry provided M ∼ N . Lemma 5.4 Let M, M

⁰

, N, N

⁰

∈ T .

1. If M ∼ M

⁰

and N ∼ N

⁰

, then M [x := N] ∼ M

⁰

[x := N

⁰

] .

2. If M ∼ M

⁰

and M

⁰

→ N , then there is N

⁰

for which M → N

⁰

and N ∼ N

⁰

.

Proof 1. By induction on cxty(M ) . 2. By 1.

Lemma 5.5 Let u, t, e ∈ T . Then (u[x := t])

^e

= u

^e

[x := t

^e

] and (u[a := e])

^e

= u

^e

[a := e

^e

] .

Proof By induction on cxty(u) .

Now we can formulate our assertion about the simulation of the λµ˜ µ

^∗

-calculus by the λ

^Sym_Prop

-calculus.

Theorem 5.6 Let v, w ∈ T .

1. If v , → r w and r ∈ {β , µ , µ , s ˜ l , s r } , then v

^e

→

⁺

w

^e

. 2. If v , → r w and r ∈ {cl

1,l

, cl

1,r

, cl

2

} , then v

^e

∼ w

^e

. Proof

1. Let us only treat the typical cases.

(a) If v = bλxu, (t.e)c , → β bt, µxbu, ecc ˜ = w , then v

^e

= (ht

^e

, e

^e

i?λy(λx(π

2

(y)?

u

^e

)?π

1

(y))) → _β

⊥

(λx(π

2

(ht

^e

, e

^e

i)?u

^e

)?π

1

(ht

^e

, e

^e

i)) →

^∗

(λx(e

^e

?u

^e

)?t

^e

) = w

^e

.

(b) If v = bµap, ec , → µ p[a := e] = w , then, by Lemma 5.5, v

^e

= (e

^e

? λap

^e

) → _β

⊥

p

^e

[a = e

^e

] = w

^e

.

(c) If v = µabw, ac , → s

_l

w , a / ∈ w , then v

^e

= λa(a ? w

^e

) → _η

⊥

w

^e

. 2. (a) If v = u , e → cl

_1,l

u = w , then v

^e

= ( e u)

^e

= u

^e

= w

^e

.

(b) If v = bv, uc e , → cl

₂

bu, vc = w , then v

^e

= bv, uc e

^e

= (u

^e

? v

^e

) ∼ w

^e

. Corollary 5.7 The λµ˜ µ

^∗

-calculus is strongly normalizable.

Proof Let S be an innite reduction sequence in the λµ˜ µ

^∗

-calculus. By Theorem

5.6 and Lemma 5.4, S cannot contain an innite number of β -, µ -, µ ˜ -, s l - and s r -

reductions. Thus, there would exist an innite reduction sequence consisting entirely

of cl

_1,l

-, cl

_1,r

- and cl

₂

-reductions, which is impossible.

5.2 A translation of the λ ^Sym _Prop -calculus into the λµ˜ µ ^∗ -calculus

Now we are going to deal with the converse relation. That is we will present a translation of the λ

^Sym_Prop

-calculus into the λµ˜ µ

^∗

-calculus which faithfully reects the typability relations of one calculus in the other one. Then we prove that our translation is in fact a simulation of the λ

^Sym_Prop

-calculus in the λµ˜ µ

^∗

-calculus.

Denition 5.8 1. The translation .

^f

: T −→ T is dened as follows.

M

^f

=



 

 

 

 

 

 

 

 

x if M = x,

bQ

^f

, P f

^f

c if M = (P ? Q),

˜

µxN

^f

if M = λxN , (P

^f

.f Q

^f

) if M = hP, Qi,

λxµβbN

^f

, xc e if M = σ

1

(N ) , x / ∈ F v(N

^f

) and β / ∈ F v(bN

^f

, xc), e λxN

^f

if M = σ

2

(N ) and x / ∈ F v(N

^f

).

2. The translation .

^f

applies to the types as follows.

• α

^f

= α ,

• (α

^⊥

)

^f

= α

^⊥

,

• (A ∧ B)

^f

= (A

^f

→ (B

^f

)

^⊥

)

^⊥

,

• (A ∨ B)

^f

= (A

^f

)

^⊥

→ B

^f

.

We remark that .

^f

maps the terms of the λ

^Sym_Prop

-calculus with type ⊥ to c -terms of the λµ˜ µ

^∗

-calculus, which have no types. We also have, for all types A , (A

^⊥

)

^f

= (A

^f

)

^⊥

. Therefore the translation .

^f

maps equal types to equal types.

Lemma 5.9 1. If Γ ` M : A and A 6= ⊥ , then Γ

^f

B M

^f

: A

^f

. 2. If Γ ` M : ⊥ , then M

^f

: (Γ

^f

B ) .

Proof The proof proceeds by a simultaneous induction on the length of the derivation in the λ

^Sym_Prop

-calculus. We can observe again that the notion of .

^f

in Denition 5.8 is conceived in a way to make the statements of the lemma true. Let us only examine some of the typical cases of the rst assertion.

1. Suppose

Γ, x : A ` u : ⊥ Γ ` λxu : A

^⊥

. Then, applying the induction hypothesis,

u

^f

: (Γ

^f

, x : A

^f

B ) Γ

^f

| µxu ˜

^f

: A

^f

B Γ

^f

B µxu ˜

^f

: (A

^f

)

^⊥

. 2. If

Γ ` u : A Γ ` σ

1

(u) : A ∨ B , then, we obtain

Γ

^f

, x : (A

^f

)

^⊥

B u

^f

: A

^f

| β : B

^f

Γ

^f

, x : (A

^f

)

^⊥

B x : (A

^f

)

^⊥

| β : B

^f

Γ

^f

, x : (A

^f

)

^⊥

| x e : A

^f

B β : B

^f

bu

^f

, xc e : (Γ

^f

, x : (A

^f

)

^⊥

B β : B

^f

)

Γ

^f

, x : (A

^f

)

^⊥

B µβbu

^f

, xc e : B

^f

Γ

^f

B λxµβbu

^f

, xc e : (A

^f

)

^⊥

→ B

^f

.

3. From

Γ ` u : A

^⊥

Γ ` v : A Γ ` (u ? v) : ⊥ , we obtain

Γ

^f

B u

^f

: (A

^f

)

^⊥

Γ

^f

| u e

^f

: A

^f

B Γ

^f

B v

^f

: A

^f

bv

^f

, u e

^f

c : (Γ

^f

B )

.

Now we turn to the proof of the simulation of the λ

^Sym_Prop

-calculus in the λµ˜ µ

^∗

- calculus.

Lemma 5.10 Let M, N ∈ T . Then (M [x := N])

^f

= M

^f

[x := N

^f

] .

Proof By induction on cxty(M ) .

Theorem 5.11 Let M, N ∈ T . If M → N , then M

^f

, →

⁺

N

^f

. Proof Let us prove some of the more interesting cases.

1. If M = (λxP ? Q) → β P[x := Q] = N , then, applying Lemma 5.10, M

^f

= bQ

^f

, µxP ˜ ]

^f

c , → cl

₁

bQ

^f

, µxP ˜

^f

c , → µ

˜

P

^f

[x := Q

^f

] = N

^f

.

2. If M = (Q ? λxP) → β

_⊥

P [x := Q] = N , then

M

^f

= b˜ µxP

^f

, Q f

^f

c , → cl

₂

bQ

^f

, µxP ˜

^f

c , → µ

˜

P

^f

[x := Q

^f

] = N

^f

. 3. If M = (hP, Qi ? σ

1

(R)) → π (P ? R) = N , then

M

^f

= bλxµbbR

^f

, xc, e ^

(P

^f

.f Q

^f

)c , → cl

₁

bλxµbbR

^f

, e xc, (P

^f

.f Q

^f

, )c , → λ

bP

^f

, µxbµbbR ˜

^f

, xc, e Q f

^f

cc , → µ

˜

bµbbR

^f

, P f

^f

c, Q f

^f

c , → µ bR

^f

, P f

^f

c = N

^f

.

We could as well demonstrate that the λµ˜ µ

^∗

-calculus is strongly normalizable by applying the method presented in Section 3 (cf. [2]). The following result states that in this case the strong normalizability of the λ

^Sym_Prop

-calculus would arise as a direct consequence of that of the λµ˜ µ

^∗

-calculus.

Corollary 5.12 If the λµ˜ µ

^∗

-calculus is strongly normalizable, then the same is true for the λ

^Sym_Prop

-calculus as well.

Proof By Theorem 5.11 .

5.3 The connection between the two translations

In this subsection we examine the connection between the two transformations. We prove that both compositions .

^ef

: T −→ T and .

^fe

: T −→ T are such that we can get back the original terms by performing some steps of reduction on u

^ef

or on M

^fe

, respectively. That is, the following theorems are valid. The case of .

^fe

is the easier one. First we describe the eect of .

^fe

on the typing relations.

Lemma 5.13 If Γ ` M : A , then Γ ` M

^fe

: A .

Proof Combining Lemmas 5.9 and 5.2.