3.1. Multimodalities in linear logic and the focused system $\mathsf{LNS}_\mathsf{FMLL}$

Similar to modal connectives, exponentials in $\mathsf{LL}$ are not canonical (Danos et al. Reference Danos, Joinet and Schellinx1993), in the sense that, even having the same scheme for introduction rules, marking the exponentials with different labels does not preserve equivalence: if $i\not= j$ then .

Let $\mathcal{I}$ be a set of labels, organized in a preorder $\preceq$ . We will call , subexponentials, which can be seen as substructural multimodalities.

There are, however, two main differences between multimodalities in normal modal logics and subexponentials in linear logic:

(i). Substructural behavior . Subexponentials carry the possibility of being unbounded (or classical) if weakening and contraction are allowed or bounded otherwise (thus having a linear behavior);

(ii). Nature of modalities . Normal modal logics start from the weakest version, assuming only the axiom $\mathsf{K}$ . Then, extensions are considered, by adding other axioms. Exponentials in linear logic, on the other side “take for granted” the behaviors expressed by axioms $\mathsf{K}$ , $\mathsf{T}$ , and $\mathsf{4}$ (Martini and Masini Reference Martini and Masini1994) – and this is inherited by subexponentials.

We observe that (i) opened a venue for proposing different multimodal substructural logical systems, that encountered a number of different applications e.g. in the specification and verification of concurrent systems (Nigam et al. Reference Nigam, Olarte and Pimentel2017), biological systems (Olarte et al. Reference Olarte, Chiarugi, Falaschi and Hermith2016), bigraphs (Chaudhuri and Reis Reference Chaudhuri and Reis2015), applications in linguistics (Kanovich et al. Reference Kanovich, Kuznetsov, Nigam and Scedrov2019), and the specification of systems with multiple contexts, which may be represented by sets or multisets of formulas (Nigam et al. Reference Nigam, Pimentel and Reis2016). Regarding (ii), Guerrini et al. (Reference Guerrini, Martini and Masini1998) unified the modal and $\mathsf{LL}$ approaches, with the exponentials assuming only the linear version of $\mathsf{K}$ , with the possibility of adding modal extensions to it.

In Lellmann et al. (Reference Lellmann, Olarte and Pimentel2017), we brought this discussion to the multimodal case, extending the concept of simply dependent multimodal logics to the substructural setting, where subexponentials consider not only the structural axioms for contraction and weakening:

but also the subexponential version of axioms $\{\mathsf{K},\mathsf{4},\mathsf{D},\mathsf{T}\}$ :

This means that can behave classically ( $\mathsf{U}$ ) or not, but also with exponential behaviors different from those in $\mathsf{LL}$ . Hence, by assigning different modal axioms one obtains, in a modular way, a class of different substructural modal logics. For instance, subexponentials assuming $\mathsf{T}$ allow for dereliction, those assuming $\mathsf{4}$ are persistent, and those assuming only $\mathsf{K}$ are functorial (Girard Reference Girard1998). In particular, substructural $\mathsf{K}\mathsf{D}$ can be seen as a fragment of elementary linear logic $\mathsf{ELL}$ (Guerrini et al. Reference Guerrini, Martini and Masini1998).

The main goal of the present work is to show how this new class of subexponentials can be applied to the problem of characterizing cut admissibility of object-level logical systems. This line of work of using $\mathsf{LNS}$ systems at the meta-level started in Olarte et al. (Reference Olarte, Pimentel and Xavier2020), where we have considered a finite class of unbounded subexponentials for specifying classical modal logics. In this work, we extend the meta-logic to support an infinite set of subexponentials, which can be bounded or unbounded, for specifying classical and linear $\mathsf{LNS}$ systems for multimodal logics.

Before presenting the formal definition, let us start with some intuitions. First of all, observe that systems at meta- and object levels are presented in the linear nested formulation. This allows not only broadening the range of specified systems, but also a more granular control of modalities. However, this comes with a price: the meta-level role of exponentials becomes highly overloaded. Indeed, they should not only handle the internal structural properties of formulas, but also the external structural properties of modalities. For example, how a classical, nonpersistent OL box modality $\Box$ should be encoded at the meta-level? The natural answer would be: via an unbounded, non persistent subexponential . Note that the object rule $\Box_L$ suppresses the modality of the boxed formula when moving between components (reading the rule in Figure 4 bottom-up). If the subexponential modality is also removed at the meta-level (when moving to the next component), this would impose a linear nature on formulas. On the other hand, keeping the subexponential while moving between components implies that the box will be persistent (a mismatch with respect to the OL behavior).

We solve this problem by introducing the unbounded, confined subexponentials , matching the exact same behavior of the $\mathsf{LL}$ exponentials in sequent systems. That is, promotion is restricted to a component, and it can be applied only when the linear context is empty (see rule in Figure 5). Although it seems that we are sacrificing locality for the sake of modularity and uniformity, this is not the case: banged formulas with label $\texttt{c}$ will never appear in the encodings – the confined classical behavior is encoded using the question-mark exponential. Hence, the nonlocal promotion rule will never be applied. It is part of the systems, however, for completeness of the cut-elimination procedure (see Section 3.2).

Figure 5. End-active focused system $\mathsf{LNS}_\mathsf{FMLL}$ . $\Theta^u$ (resp. $\Theta^l$ ) contains only unbounded (resp. linear) subexponentials. In $\mathsf{l_{s}}$ and $\mathsf{I_l}$ , A is atomic. In $\forall$ , y is fresh. In $\mathsf{store}$ , S is a literal or a positive formula. In $\mathsf{\mathsf{R_n}}$ , N is a negative formula. In $\mathsf{D_l}$ , P is positive, and in $\mathsf{D^u_s}, \mathsf{D^l_s}$ , $P_a$ is not atomic. In $\mathsf{D^u_s}, \mathsf{D^l_s}$ and $\mathsf{l_{s}}$ , $\mathsf{T} \in \mathcal{U}(i)$ . In all question-marked rules $i\preceq j$ . Moreover, $i\neq \texttt{c}$ in ; $\mathsf{D}\in \mathcal{U}(i)$ in $\mathsf{D_d}$ ; $\mathsf{4} \in \mathcal{U}(j)$ in $?^i_\mathsf{4}$ ; $\{\mathsf{4},\mathsf{C},\mathsf{W}\} \cap \mathcal{U}(j) = \emptyset$ in $?^i_\mathsf{kl}$ ; $\mathsf{4} \not\in \mathcal{U}(i)$ and $\mathsf{U} \subseteq \mathcal{U}(i)$ in $?^i_\mathsf{ku}$ and in $\mathsf{D^u_s}$ .

• – $\mathcal{I}$ is a set of labels;

• – $\mathcal{U}(i)$ represents the set of axioms within $\{\mathsf{C},\mathsf{W}, \mathsf{K},\mathsf{4},\mathsf{D},\mathsf{T}\}$ that the subexponential $i\in\mathcal{I}$ assumes, with $\mathsf{K}\in\mathcal{U}(i)$ for all $i\in \mathcal{I}$ ;

• – $\preceq$ is a preorder among the elements of $\mathcal{I}$ that is upwardly closed with respect to $\mathcal{U}(i)$ , i.e., if an axiom $\mathsf{A} \in \mathcal{U}(i)$ and $i \preceq j$ , then $\mathsf{A} \in \mathcal{U}(j)$ .

We assume $\texttt{c} \in \mathcal{I}$ to be a distinguished element s.t. $\mathcal{U}(\texttt{c})=\{\mathsf{C},\mathsf{W},\mathsf{K},\mathsf{T}\}$ and $\texttt{c}$ is only $\preceq$ -related with itself. Finally, for every subexponential $i\in \mathcal{I}$ , we assume a subexponential $i+\in \mathcal{I}$ s.t. $\mathcal{U}(i+\!)= \mathcal{U}(i)\cup \{\mathsf{T}\}$ , $i \preceq i+$ , and if $i\preceq j$ then $i+\preceq j+$ .

As usual, the upwardly condition is needed for cut elimination (Danos et al. Reference Danos, Joinet and Schellinx1993). Since we are considering extensions of normal modal logics, all the subexponentials are assumed to have the axiom $\mathsf{K}$ . Finally, the subexponential $i+$ , that adds $\mathsf{T}$ to $\mathcal{U}(i)$ , will be used to give a more deterministic rule for exponentials containing the axiom $\mathsf{4}$ (more on this below).

The focused proof system for $\mathsf{MMLL}$ is presented in Figure 5 and explained in the following. Focusing (Andreoli Reference Andreoli1992) is a discipline on proofs aiming at reducing nondeterminism during proof search. Focusing in $\mathsf{LL}$ -based systems is grounded on two kinds of separations: (a) classical/linear behavior of formulas and (b) invertible/non invertible introduction rules.

For (a), observe that it is possible to incorporate the structural rules of contraction and weakening for formulas of the shape $? F$ into the $\mathsf{LL}$ introduction rules. This is reflected into the syntax in the so called dyadic sequents where the context is split into two, a classical (set of formulas $\Theta$ ) and a linear (multiset of formulas $\Gamma$ ). The dyadic sequent $\vdash \Theta;\ \Gamma$ is then interpreted as the linear logic sequent $\vdash ? \Theta, \Gamma$ where $? \Theta = \{? F \mid F \in\Theta\}$ . This idea can be generalized to the case of the subexponentials. The subexponential context is a multiset of the form $\Theta=\{i:F \mid i\in\mathcal{I}\}$ , and the dyadic sequent $\vdash \Theta;\ \Gamma$ is interpreted as the (subexponential) linear logic sequent $\vdash \{?^i F \mid i:F \in \Theta\}, \Gamma$ . That is, $i:F\in\Theta$ represents that F is stored in the (subexponential) context in $\Theta$ labeled by i. We extend the indexed notation to contexts as follows: given a linear context $\Gamma=\{F_1,\ldots,F_n\}$ , we will write $i:\Gamma$ for the subexponential context $\{i:F_1,\ldots,i:F_n\}$ . Finally, we shall use the superindex $\Theta^u$ (resp. $\Theta^l$ ) to denote a context where all the subexponentials are unbounded (resp. linear). When we simply write $\Theta$ , the context may contain both, linear and unbounded subexponentials.

For (b), it turns out that proofs can be organized in two alternating phases: negative containing only invertible rules, and positive containing only noninvertible rules. The connectives have invertible introduction rules and are thus classified as negative. The remaining connectives are positive. Formulas inherit their polarity from their main connective, e.g., $F \otimes G$ is positive and is negative. Although the bias assigned to atoms does not interfere with provability (Andreoli Reference Andreoli1992; Miller and Saurin Reference Miller and Saurin2007), here we follow Andreoli’s convention of classifying atomic formulas as negative, thus negated atoms as positive.

The inference rules of the focused system $\mathsf{LNS}_\mathsf{FMLL}$ in Figure 5 involve two kinds of sequents

\begin{equation*} \Theta;\ \Gamma\Uparrow L\quad\hbox{and}\quad \Theta;\ \Gamma\Downarrow F\end{equation*}

where L is a (possibly empty) list of formulas. The formula occurrence F in a $\Downarrow$ -sequent is called the focus of that sequent. Moreover, linear nested sequents in $\mathsf{LNS}_\mathsf{FMLL}$ have the shape

\begin{equation*}\vdash\mathcal{ G} \mathbin{\!/\mkern-5mu/^{i}\!} {Seq}\end{equation*}

where $\mathcal{ G}$ is either an unfocused sequent or it is empty (in this case //ⁱ is not present). The superscript in “//ⁱ” indicates that the component was created by a subexponential labeled with i. $\textit{Seq}$ , above, is a focused or unfocused $\mathsf{LNS}_\mathsf{FMLL}$ sequent.

The different $\mathsf{LNS}$ above reflect not only the negative/positive proof phases but also the behavior of the promotion rule:

• – $\vdash \Theta;\ \Gamma \Uparrow L$ belongs to the negative phase. During this phase, all negative formulas in the list L are introduced and all positive formulas and literals are moved to the linear context $\Gamma$ ;

• – $\vdash \Theta;\ \Gamma\Downarrow \, F$ belongs to the positive phase, where all positive connectives at the root of F are introduced; and

• – belongs to the exponential/modal phase. During this phase, only applications of the rules for $?^{i}$ are allowed, ending with an application of $\mathsf{R_{\mathsf{r}}}$ (release rule). The dynamics of the different phases is detailed below.

Negative Phase. Negative rules can be applied eagerly, in any order. This process includes storing formulas. Note that the rule $\mathsf{store}$ moves, to the linear zone $\Gamma$ , the literal or positive formula S since it cannot be decomposed during the negative phase; $\mathsf{store}_\mathsf{s}$ stores formulas marked with into the correspondent subexponential context.

The negative phase ends when the list L in unfocused sequents is empty. A decision rule is then applied. There are four possibilities.

• $\mathsf{D_l}$ , $\mathsf{D_{s}^{u}}$ , $\mathsf{D_s^l}$ : a positive phase starts by focusing on a non-atomic formula F such that

  • – $\mathsf{D_l}$ : F is taken from the linear context (and thus erased from it);

  • – $\mathsf{D_{s}^{u}}$ : a copy of F is taken from the subexponential context i, thus making an implicit contraction ( $\mathsf{U} \subseteq\mathcal{U}(i)$ ) and a dereliction ( $\mathsf{T}\in\mathcal{U}(i)$ );

  • – $\mathsf{D_s^l}$ : analogous to $\mathsf{D_{s}^{u}}$ , with the difference that $\mathsf{U} \cap \mathcal{U}(i) = \emptyset$ and the formula is not contracted.

• $\mathsf{D_d}$ : starts an exponential phase.

Positive Phase. Once we focus on a formula, the proof follows by applying positive rules, where the focus persists on the decomposed subformulas until either: the proof ends with an instance of an axiom; a negative formula is reached, and the positive phase ends with the application of $\mathsf{R_n}$ ; or a banged formula is derived, which either moves the proof to the negative case (if the subexponential is ) or triggers an exponential phase execution (otherwise). In the latter, only the rules for , moving formulas between components, are allowed. When this moving is over, the exponential phase ends with an application of the rule $\mathsf{R_{\mathsf{r}}}$ , starting again a negative phase.

There are two initial rules. In $\mathsf{I_l}$ , the atom A is in the linear zone and the subexponential context must be unbounded (being implicitly weakened); in $\mathsf{I_s}$ , the atom is placed in a subexponential context i with $\mathsf{T} \in \mathcal{U}(i)$ . The need of the axiom $\mathsf{T}$ comes from the fact that it embodies the capacity of using the formula in the same component, corresponding to dereliction in linear logic.

Exponential Phase. The exponential phase starts with the application of a rule, with $i\not= \texttt{c}$ , or the rule $\mathsf{D}_\mathsf{d}$ . The distinguished subexponential behaves as the $\mathsf{LL}$ bang exponential. Hence, its introduction rule simply ends the positive phase and assumes that the subexponential context is unbounded. For any other subexponential in $\mathsf{MMLL}$ , the rule creates a new component, moving the active formula to it. Once the new component is created, the rules dictate the movement of formulas between components, as described in the following.

Usually, nested-based systems for modal/linear logics present one core box left/question mark rule, and other modal rules with different behaviors are added on the top of it, depending on the modal axioms of the underlying modality/subexponential. In Olarte et al. (Reference Olarte, Pimentel and Xavier2020), we have adopted such approach, lifting to the meta-level the OL modal rules. Although resulting into adequate specifications, in the sense of successfully capturing the object-level behavior, this approach is not satisfactory for the meta-level focused system, since it adds a great deal of nondeterminism during proof search.

In this work, we propose more efficient modal rules from the proof theoretical point of view using all the subexponential features present in $\mathsf{MMLL}$ . First of all, when $\mathsf{4} \in \mathcal{U}(j)$ , the rules and cannot be applied. Dually, if $\mathsf{4} \not\in \mathcal{U}(j)$ , then the rule is not enabled and the use of is only possible if $\mathsf{U} \not\subset \mathcal{U}(j)$ (resp. $\mathsf{U} \subseteq \mathcal{U}(j)$ ). Second, the proposed rules have a better control of contraction, thus avoiding the need of guessing the number of times a formula must be copied to the next component. Note that the rule moves the formula F stored in the context j to the context $j+$ . This has two immediate effects: the formula F can be copied to yet another component (once it is created) reflecting the behavior of the modal rule $\mathsf{4}$ (persistence); moreover, since $\mathsf{T} \in \mathcal{U}(j+)$ , the formula F can be also used in the last component by applying the decision rule. Finally, when $\mathsf{U} \subseteq \mathcal{U}(j)$ , $j+$ is also unbounded and decision/dereliction on the same formula can be performed several times. In other words, the rule $?^i_\mathsf{4}$ embeds both the behavior of $\mathsf{K}$ (moving formulas between components) and also $\mathsf{4}$ (by keeping the modality of the formula). On the other hand, the behavior of $\mathsf{K}$ , without $\mathsf{4}$ , is specified by the rules and . In the first case, j is linear and then F is not contracted. In the second case, F is placed in the context $\texttt{c}$ . Since $\mathsf{T}\in \mathcal{U}(\texttt{c})$ , F can be used as many times as needed in the last component (by using the decision rule during the positive phase). Also, since $\texttt{c}$ is not related to any subexponential, the formula F cannot be moved to other components and thus confined to the last component.

All this is illustrated in the next example.

The first derivation involves three components (two of them created using the rule). The information $A^\perp$ is passed from the 1st to the 2nd and 3rd components through the subexponential $4+$ via rule . Since this subexponential is linear, the context $4+$ disappear after the application of $\mathsf{D^l_s}$ . The second derivation illustrates the application of the rule in the unbounded subexp. k. Note that the formula $A \otimes B^\perp$ is copied to the context $\texttt{c}$ . All the other axioms in Figure 4 can be proved similarly, using a subexponential featuring the needed modal behaviors.

Finally, it should be noted that the proposed focused system is complete w.r.t. the unfocused version. For the plain linear logic rules, completeness is guaranteed by Andreoli’s result in Andreoli (Reference Andreoli1992). For the modal rules, it is straightforward to simulate the $\mathsf{LNS}$ rules from the focused ones, following the same lines as the process of obtaining sequent rules from the $\mathsf{LNS}$ ones (see the example for implication right rule for $\mathsf{mLJ}$ in Section 2).

3.2 Cut Admissibility for $\mathsf{LNS}_\mathsf{FMLL}$

In this section, we present a cut-admissibility result for the system $\mathsf{LNS}_\mathsf{FMLL}$ . It should be noted that this is, for the best of our knowledge, the first formally verified proof of cut admissibility for focused $\mathsf{LNS}$ . ^{Footnote 2} This is not a simple enhancement w.r.t. other formalizations: usually cut-admissibility of focused systems is done in two steps, first translating it into the unfocused version, then proving the completeness of focusing. Here, we prove the cut admissibility of $\mathsf{LNS}_\mathsf{FMLL}$ directly, observing that the multimodal flavor of the system makes the task even less trivial.

We proceed by double induction on the complexity of the cut formula and the sum of the premises’ heights of the cut rule. Since we are proving the theorem directly on a focused system, we require to eliminate, simultaneously, different rules dealing with the different phases of the proof (see e.g., Liang and Miller Reference Liang and Miller2009). The cut rules are in Figure 6.

Figure 6. Cut-rules for the system $\mathsf{LNS}_\mathsf{FMLL}$ . $\Lambda$ is unbounded and $\Theta=(\Theta_1,\Theta_2)$ is linear.

In the following, we show the most interesting proof reductions needed to establish the main result (Theorem 3.5). In the repository (Xavier and Olarte Reference Xavier and Olarte2021), the reader can find the formalization in Coq of all the results of this section, together with a companion PDF with extra details about the proof.

Auxiliary results. We start with a series of auxiliary results for the system $\mathsf{LNS}_\mathsf{FMLL}$ . They are not only needed for the proof of the main theorem, but they will also help on understanding better the dynamics of the system.

Notation 5. We use $\vdash \Theta;\ \Gamma \Updownarrow X$ to represent a sequent that can belong to the negative phase – and X is a list of formulas, or to the positive phase – and X is a formula. We use $\vdash_n \mathcal{S}$ to indicate that the linear nested sequent $\mathcal{S}$ has a derivation of height at most n. If $\Theta=\{i_1:F_1,...,i_n:F_n\}$ is a subexponential context, then $\Theta+$ denotes the subexponential context $\{i_1+:F_1,...,i_n+:F_n\}$ and $\mathcal{F}(\Theta)$ denotes the list of formulas $F_1,\ldots,F_n$ . For easing the notation, we will write $i:\Theta$ instead of $i:\mathcal{F}(\Theta)=\{i:F_1,\ldots,i:F_n\}$ . Finally, $\Theta = (\Theta_1,\ldots,\Theta_n)$ denotes that $\Theta_1,\ldots,\Theta_n$ is a partition of $\Theta$ . In this section, $\Omega$ will be used to denote a context assuming axiom $\mathsf{4}$ (called $\mathsf{4}$ -context), and $\Psi$ a context without $\mathsf{4}$ . Recall that the axiom $\mathsf{T}$ allows for dereliction. The following result shows some particular behaviors of $\mathsf{T}$ -contexts.

1. ( T 1) If $\{\mathsf{T},\mathsf{C},\mathsf{W}\}\subseteq\mathcal{U}(i)$ and $\vdash_n \Theta,i:F,\texttt{c}:F;\ \Gamma \Updownarrow X$ , then $\vdash_n \Theta,i:F;\ \Gamma \Updownarrow X$ .

2. ( T 2) If $\{\mathsf{T},\mathsf{C},\mathsf{W}\}\subseteq\mathcal{U}(i)$ and $\vdash_n \Theta,i:F;\ \Gamma,F \Updownarrow X$ , then $\vdash_n \Theta,i:F;\ \Gamma \Updownarrow X$ .

3. ( T 2) If $\mathsf{T}\in\mathcal{U}(i)$ and $\vdash_n \Theta;\ \Gamma,F \Updownarrow X$ , then $\vdash_n \Theta,i:F;\ \Gamma \Updownarrow X$ .

4. ( $\mathcal{L}A_1$ ) If $\{\mathsf{T},\mathsf{C},\mathsf{W}\}\subseteq\mathcal{U}(i)$ and $\vdash \Theta,i:F;\Gamma \Uparrow L,F$ then $\vdash \Theta,i:F;\Gamma \Uparrow L$ .

5. ( $\mathcal{L}A_2$ ) If $\mathsf{T}\in\mathcal{U}(i)$ and $\vdash \Theta;\ \Gamma \Uparrow L,F$ then $\vdash \Theta,i:F;\Gamma \Uparrow L$ .

All these results can be easily generalized to contexts featuring the corresponding axioms. For example, in the case of the absorption lemma ( $\mathcal{L}A_2$ ), if $\vdash \Theta;\Delta \Uparrow L,\Gamma$ , then $\vdash \Theta,i:\Gamma;\Delta \Uparrow L$ . Moreover, (T1) states that if a formula is stored in a context i where $\mathsf{T}$ holds, then it is “redundant” to place that formula also in the local context $\texttt{c}$ . The other results allow us for moving formulas in the different zones of a sequent where $\mathsf{T}$ holds.

The next theorem highlights the difference between multisets and lists in contexts. Namely, while permuting a list preserves provability, it may not preserve the height of a derivation. On the other hand, weakening and contraction are height-preserving admissible for unbounded contexts.

1. ( $\equiv_P$ ) If $L_1$ is a permutation of $L_2$ and $\vdash \Theta;\ \Gamma \Uparrow L_1$ then $\vdash \Theta;\ \Gamma \Uparrow L_2$

2. ( $\mathsf{W}$ ) If $\mathsf{U}\subseteq\mathcal{U}(i)$ and $\vdash_n \Theta;\ \Gamma \Updownarrow X$ then $\vdash_n \Theta,i:F;\ \Gamma \Updownarrow X$ .

3. ( $\mathsf{C}$ ) If $\mathsf{U}\subseteq\mathcal{U}(i)$ and $\vdash_n \Theta,i:F,i:F;\ \Gamma \Updownarrow X$ then $\vdash_n \Theta,i:F;\ \Gamma \Updownarrow X$ .

As usual in focused systems, the shape of phases can be totally characterized. In the subexponential phase, since the only action allowed is moving formulas between components until the release rule ( $\mathsf{R_{\mathsf{r}}}$ ) is applied, every derivation has the shape

where $\Xi$ contains all the unbounded subexponentials not related to i, $\Psi = (\Psi_1^u, \Psi_2^l)$ and $i \preceq j$ for all $j:F$ in $\Omega\cup\Psi_1$ . Observe that $\Xi$ is weakened when the release rule is applied. The formulas in $\Omega$ gain the axiom $\mathsf{T}$ , the unbounded formulas in $\Psi$ are stored in $\texttt{c}$ and the linear ones are placed in the linear context ( $\mathcal{F}(\Psi_2)$ ). On the left, the subexponential phase is triggered by the promotion rule and $L=P$ . On the right, the rule $\mathsf{D_d}$ is applied and L is empty.

( S )

where $n=|\Omega,\Psi|$ .

This result can be further specialized depending whether the index i in the subexponential phase ( $\cdot\mathbin{\!/\mkern-5mu/^{i}\!}\cdot$ ) is unbounded or $\mathsf{4}\in \mathcal{U}(i)$ . Remember the upwardly closeness condition: if $i\preceq j$ and some axiom $\mathsf{A}\in \mathcal{U}(i)$ then, $\mathsf{A}\in \mathcal{U}(j)$ . Hence, if i is unbounded, there is no linear subexponentials in $\Omega,\Psi,\Xi$ and, thus, $\Psi_2$ and $\Omega_2$ are necessarily empty in $\textbf{(S)}$ . Moreover, if $\mathsf{4} \in \mathcal{U}(i)$ , $\Psi$ (containing the indices without $\mathsf{4}$ ) must be necessarily unbounded and weakened in the application of $\mathsf{R_{\mathsf{r}}}$ . More precisely:

1. ( $\textbf{S}\mathsf{U}$ ) $\vdash_{m}\Omega_{1+}, \texttt{c}:\Psi_1 ; \cdot\Uparrow L\quad\mbox{iff}\quad\vdash_{m+k+1}\Omega_1,\Psi_1,\Xi; \cdot\Uparrow \cdot \mathbin{\!/\mkern-5mu/^{i}\!} \cdot ; \cdot\Uparrow L$

2. ( $\textbf{S}\mathsf{4}$ ) $\vdash_{m}\Omega+; \cdot\Uparrow L \quad\mbox{iff}\quad\vdash_{m+r+1}\Omega,\Xi ; \cdot\Uparrow \cdot \mathbin{\!/\mkern-5mu/^{i}\!} \cdot ; \cdot\Uparrow L$

where $k=|\Omega_1,\Psi_1|$ and $r=|\Omega|$ .

It is also interesting to revisit the two different promotion rules: and . The following theorem shows that they coincide when $\{\mathsf{C},\mathsf{W},\mathsf{T}\}\subseteq\mathcal{U}(i)$ .

Cut-rules and reductions. The five cut-rules admissible in $\mathsf{LNS}_\mathsf{FMLL}$ are presented in Figure 6. In all cases, $\Lambda$ contains only unbounded subexponentials whereas $\Theta=(\Theta_1,\Theta_2)$ is a linear context. Below we explain the interdependencies between these rules during the cut-elimination procedure. For the sake of readability, in most of the derivations, we omit the unbounded context $\Lambda$ .

Elimination of $\Uparrow$ C. Since the cut formula is the first element in the list of formulas in the left premise, it must be necessarily principal. Take for instance the following case:

When the cut formula has the shape (resp. a positive formula or a literal), a shorter cut using $\Uparrow$ CC (resp. $\Uparrow$ LC*) is used. For example:

When the cut formula is an atom, the right premise of the cut must necessarily finish with the initial rule. We show below the case where A is in the (linear) subexponential context i (where, necessarily, $\mathsf{T} \in \mathcal{U}(i)$ ) and we use the admissible rule (T3), which is is justified by the respective statement in Theorem 3.1.

In the following, we will adopt the same procedure of reading the results in Theorem 3.2 as rules.

Elimination of $\Uparrow$ CC. The elimination of this rule is easy when the list L is not empty: it suffices to permute down the application of the introduction rule.

When L is empty, the left premise necessarily starts with a decision rule or an application of ${\mathsf{D_d}}$ . We show below the case where Q is taken from the linear subexponential context a and, necessarily, $\mathsf{T}\in \mathcal{U}(a)$ . We proceed with a (smaller) application of $\Downarrow$ CC:

Consider now the case where the left premise starts with an application of ${\mathsf{D_d}}$ . Assume that $\mathsf{D} \in \mathcal{U}(a)$ , $a \preceq i$ and $4 \in \mathcal{U}(i)$ . Moreover, let $\Psi=(\Psi_1^u,\Psi_2^l)$ and $\Omega=(\Omega_1^u,\Omega_2^l,\Omega_3^l)$ . Hence

The leaves in the above derivation are obtained from Theorem 3.3. The notation $\Omega_1^{\succeq a}$ represents the subset of formulas in $\Omega_1$ whose indices are above a and that were not weakened in the end of the subexponential phase. Note that, due to the upwardly closeness condition, the linear context $\Psi_2$ must necessarily be moved to the left premise (since $\mathsf{4}$ is not present in $\Psi$ ). Also, $\Psi_1$ is necessarily weakened in the right premise after applying the release rule. The reduction consists in cutting with instead of as follows:

In the above derivation, $\Omega'$ contains all the formulas in both $\Omega_1^{\succeq a}$ and $\Omega_1^{\succeq i}$ , i.e., the formulas from $\Omega_1$ that were not weakened in neither of the derivations. Note that $a \preceq i$ and then, the application of $\mathsf{D_d}$ can preserve all the formulas in $\Omega_1^{\succeq i}$ (similarly for $\Omega_3$ ). Derivations $\pi_i^w$ for $i\in\{1,2\}$ are obtained from $\pi_i$ by weakening the formulas in $\Omega_1'$ not needed in each derivation.

Elimination of $\Downarrow$ CC. When the focused formula on the left premise is principal and focus cannot be lost, an invertibility lemma is used. Below a representative case using the invertibility lemma $(\mathcal{L}\oplus):$ If $\vdash \Theta;\ \Delta \Uparrow L,F$ then $\vdash \Theta;\ \Delta,F\oplus G \Uparrow L$ .

When the left premise starts with the promotion rule, we also have several cases depending on the axioms of the introduced subexponential. Consider the case where i is unbounded and $\mathsf{4} \notin \mathcal{U}(i)$ . Let $\Psi=(\Psi_1^u,\Psi_2^l)$ and $\Omega=(\Omega_1^u,\Omega_2^l)$ :

Note that H is stored in the context $\texttt{c}$ . The reduction considers a cut with (instead of ):

$\Psi_1'$ contains all the formulas in both $\Psi_1^{\succeq a}$ and $\Psi_1^{\succeq i}$ (the same with $\Omega'$ ). Since i is unbounded, note that the application of above is valid.

Another interesting case, similar to the previous one, is when $ \{\mathsf{C},\mathsf{W},\mathsf{4}\}\subseteq\mathcal{U}(i)$ . Hence, H is moved to the context $i+$ and the reduction considers a (smaller) cut with $i+$ (instead of i).

Elimination of $\Uparrow$ C*. When the list $L_1$ is not empty, the first formula in $L_1$ is necessarily principal in the left premise. Hence, we simply permute down the application of that rule.

When $L_1$ is empty, we consider two cases: either the cut formula is stored in the left premise, or it is necessarily stored in the right premise. The reductions for these cases are as follows:

Elimination of $\Uparrow$ LC* Similarly to the previous case, when the list $L_1$ is not empty the reduction is easy. When this is not the case, the left premise must start with a decision rule. When the focus is on the cut formula, the reduction is as follows:

When the proof continues by focusing on a formula different from the cut formula, an invertibility result is needed. Here, the case of $\oplus$ when F is a positive formula and $F\oplus G$ was taken from the unbounded context $\Lambda$ (omitted in the derivation):

where $\mathcal{L}U$ is the following lemma: if F is a positive formula and $\vdash \Theta;\ \Delta,F \Uparrow L$ then $\vdash \Theta;\ \Delta \Uparrow L,F$ .

$\mathcal{L'}\oplus$ is a tailored version of the invertibility result we have already seen ( $\mathcal{L}\oplus$ ) when $F\oplus G$ is stored in a unbounded context.

5.1 Cut rules, cut coherence, and admissibility of cut

The (multiplicative) OL-cut rule can be specified as the bipole $\mathsf{cut} = \exists F. (\lfloor F \rfloor \otimes \lceil F \rceil)$ . In fact, focusing on that formula mimics exactly the behavior of the cut rule at the object level:

We shall use $\mathsf{cut}_n$ to denote the rule $\mathsf{cut}$ applied to (OL-) formulas of size strictly smaller than n. For instance, if $G = G_1 \star G_2$ , a valid application of $\mathsf{cut}_{|G|}$ can instantiate the existentially quantified variable F in $\mathsf{cut}$ with either $G_1$ or $G_2$ (but not with G).

Using the formulas $\mathsf{cut}$ and $\mathsf{init}$ , we can prove that $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ are duals in the sense that the $\mathsf{LNS}_\mathsf{FMLL}$ sequent $\vdash \omega:\{\mathsf{cut},\mathsf{init}\}; \Uparrow \lfloor F \rfloor \equiv \lceil F \rceil^\perp$ is provable. More interestingly, this duality can be generalized to the right and left bodies of OL rules as well. More precisely,

Definition 7. (Cut coherence). Let $\mathcal{T}_\mathcal{L}$ be the canonical theory of the OL $\mathcal{L}$ . We say that $\mathcal{T}_\mathcal{L}$ is cut-coherent if, for each connective $\star \in \mathcal{C}$ , and $F=\star(F_1,...,F_n)$ , the sequent below is provable

Note that $\mathsf{cut}$ is instantiated with F (a subformula of $\Box F$ ).

We now state the main result of the section. Consider two cut-free proof derivations from the object-level point of view (i.e., using only $\mathcal{T}_\mathcal{L}$ ) introducing the cut formula F. It is possible to prove the same sequent using the rule $\mathsf{cut}$ (at the object-level) with strict subformulas of F.

1. (a) (when both $\mathsf{pos}_i$ and $\mathsf{neg}_\texttt{c}$ are present)

2. (b) (when only $\mathsf{pos}_i$ is present)

We will write $\mathsf{cutC}_n, \mathsf{cutN}_n$ to denote the rules $\mathsf{cutC}, \mathsf{cutN}$ applied to (OL) formulas of size strictly smaller than n. Observe that, in the presence of the corresponding structural rules, $\mathsf{cutC}$ and $\mathsf{cutN}$ are equivalent to $\mathsf{cut}$ (remember that with $\mathsf{pos}_i$ , it is possible to prove the equivalences and ).

Assume that the OL features both, $\mathsf{pos}_i$ and $\mathsf{neg}_\texttt{c}$ . Hence, we consider the rules $\mathsf{cut}$ and $\mathsf{cutC}$ and we show, simultaneously, both:

1. (1) if $\vdash \omega:\mathcal{T}_\mathcal{L},\Lambda,\Theta_1;\Omega,\lfloor F \rfloor \Uparrow \cdot$ and $\vdash \omega:\mathcal{T}_\mathcal{L},\Lambda,\Theta_2;\Psi,\lceil F \rceil \Uparrow \cdot$ are both provable then the sequent $\vdash \omega:\{\mathcal{T}_\mathcal{L},\mathsf{cut}_{|F|},\mathsf{cutC}_{|F|}\},\Lambda,\Theta_1,\Theta_2;\Omega,\Psi \Uparrow \cdot$ is also provable; and

2. (2) if $\vdash \omega:\mathcal{T}_\mathcal{L},\Lambda,\Theta_1,i:\lfloor F \rfloor;\Omega \Uparrow \cdot$ and $\vdash \omega:\mathcal{T}_\mathcal{L},\Lambda,\Theta_2,\texttt{c}:\lceil F \rceil;\Psi \Uparrow \cdot$ are both provable then the sequent $\vdash \omega:\{\mathcal{T}_\mathcal{L},\mathsf{cut}_{|F|},\mathsf{cutC}_{|F|}\},\Lambda,\Theta_1,\Theta_2;\Omega,\Psi \Uparrow \cdot$ is also provable.

In the proof of (1), if the structural rules are used on $\lfloor F \rfloor$ or $\lceil F \rceil$ , we produce a shorter derivation with (2), i.e., with $\mathsf{cutC}$ .

For the proof of (2), consider the case where $\lfloor F \rfloor$ and $\lceil F \rceil$ are principal in both premises thus using, respectively, the left and right introduction rules for the same connective:

Here, means that the existential quantifiers of the clause were instantiated with the formula F. As we already know, the bodies and produce well controlled forms of derivations. They can either end the proof (e.g., with the clause $\texttt{f}_{L}$ ) or produce derivations with one or two premises.

The resulting generated premises, after focusing on the body of the clause, can only add new atoms to the linear or classical context before ending the negative phase. For simplicity, consider that both $\pi$ and $\pi'$ produce one premise. We then have

• – $\Pi$ produces a sequent of the form $\vdash \omega:\mathcal{T}_\mathcal{L}, \Lambda',\Theta_1', i:\lfloor F \rfloor;\Omega' \Uparrow \cdot$ .

• – $\Pi'$ produces a sequent of the form $\vdash \omega:\mathcal{T}_\mathcal{L}, \Lambda'',\Theta_2', \texttt{c}:\lceil F \rceil;\Psi' \Uparrow \cdot$ .

The primed contexts are extensions of the original ones with atoms of the form $\lfloor \cdot \rfloor,\lceil \cdot \rceil$ . By induction, the sequents:

• – $\vdash \omega:\{\mathcal{T}_\mathcal{L},\mathsf{cut}_{|F|},\mathsf{cutC}_{|F|}\},\Lambda',\Theta_1';\Omega' \Uparrow \cdot$ ; and

• – $\vdash \omega:\{\mathcal{T}_\mathcal{L},\mathsf{cut}_{|F|},\mathsf{cutC}_{|F|}\}, \Lambda'',\Theta_2';\Psi' \Uparrow \cdot$

are both provable. Hence, the sequents below are also provable:

• – ; and

• – .

At this point, we rely on cut coherence (see Definition 7) and cut admissibility in $\mathsf{LNS}_\mathsf{FMLL}$ :

where $\mathcal{T} = \mathcal{T}_{\mathcal{L}} \cup \{\mathsf{cut}_{|F|}, \mathsf{cutC}_{|F|}\}$ .

Now let us illustrate a nonprincipal case involving modal behaviors. Consider a derivation where the left and right premises introduce, respectively, the modal formulas $G_n$ and $F_m$

where $\Gamma_F,\Delta_f$ (resp. $\Gamma_G,\Delta_G$ ) represent the atomic subformulas appearing in the decomposition of $\lceil F_m \rceil$ (resp. $\lceil G_n \rceil$ ). In each case, the subindices n and m denote the underlying modalities of these formulas. Note that n and m cannot be $\texttt{c}$ since modalities of the OL are not mapped to this index. Also, $i\neq \texttt{c}$ , otherwise, $\lfloor F_m \rfloor$ must be necessarily weakened in the left derivation (and the end sequent is provable by using $[G_n]$ ). We then have $n \preceq i$ and, by Requirement 5.2, $i \preceq n$ , $i\preceq m$ and hence, $n \preceq m$ . Since $\mathsf{T}\in \mathcal{U}(i)$ , such axiom is also present in n and m and Theorem 3.1 applies on the context $\Lambda,\Theta_i$ . We conclude that $\Lambda,\Theta_i\supseteq \{\Lambda,\Theta_i\}^{\succeq n} \supseteq \{\Lambda,\Theta_i\}^{\succeq m}$ and the resulting reduction is:

$\Pi$ is obtained by Theorem 3.1 and $\Psi$ is the same as the derivation introducing $F_m$ but weakening the formulas in $\Gamma_G$ and $\Delta_G$ in the release rule.

Since for every OL formula F, $|F| > 0$ , by induction we conclude the following.

The mechanization of Theorem 5.1 is unfortunately not feasible due to the impossibility of reasoning about derivations (see discussion in the end of Section 3). For instance, it is not possible to specify, in a general way, the number of premises resulting after the decomposition of a bipole. The current version of the library offers definitions, in the form of class types, for cut coherence of constants, connectives, and OL binders. Hence, the clauses encoding OL rules must be instances of such classes (thus satisfying cut coherence). The proof of OL cut admissibility, for each specific case, follows the proof shown above: the principal cases rely on cut coherence and cut-admissibility of $\mathsf{MMLL}$ . The current version of the repository exemplifies this procedure for classical and intuitionistic logics, as well as for the logic $\mathsf{S4}$ .