Rooted Hypersequent Calculus for Modal Logic S5

TL;DR

This paper introduces a rooted hypersequent calculus for modal logic S5, demonstrating its invertibility, admissibility of key rules, and establishing soundness and completeness.

Contribution

It presents a novel calculus for S5 with invertible rules and proves its soundness and completeness, advancing proof theory for modal logic.

Findings

All rules are invertible.

Weakening, contraction, and cut are admissible.

Soundness and completeness are established.

Abstract

We present a rooted hypersequent calculus for modal propositional logic S5. We show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well.

Equations

$$A:=\perp |\top |p|\neg A|A\wedge A|A\vee A|A\to A|◊A|\square A,$$

$$\text{(Dual)}\square A\leftrightarrow \neg ◊\neg A,$$

$$\text{(K)}\square (A\to B)\to (\square A\to \square B),$$

$$\text{(T)}\square A\to A,$$

$$\text{(5)}◊A\to \square ◊A.$$

$$\text{(4)}\square A\to \square \square A,$$

$$\text{(B)}A\to \square ◊A.$$

$$\Gamma \Rightarrow \Delta ||P_1\Rightarrow Q_1|P_2\Rightarrow Q_2|\dots |P_n\Rightarrow Q_n,$$

$$\text{If}\mathcal{M},w\models \bigwedge M,\text{then}\mathcal{M},w\models \bigvee N\vee A\vee \square (\bigwedge P\to \bigvee Q)\vee \square (\bigwedge P_1\to \bigvee Q_1).$$

$$\mathcal{M},w^{\prime }\models \bigwedge M\wedge \bigwedge P$$

$$\mathcal{M},w^{\prime }⊭\bigvee Q\vee \bigvee N\vee \square A\vee \square (\bigwedge P_1\to \bigvee Q_1).$$

$$\text{If}\mathcal{M},w\models \bigwedge M\wedge \bigwedge P_i,\text{then}$$

$$\mathcal{M},w\models \bigvee Q_i\vee \bigvee N\vee \square (\bigwedge P_1\to \bigvee Q_1)\vee \square (\bigwedge P\to \bigvee Q)\vee \square (\bigwedge P_2\to \bigvee Q_2).$$

$$\mathcal{M},w^{\prime }\models \bigwedge M\wedge \bigwedge P$$

$$\mathcal{M},w^{\prime }⊭\bigvee Q\vee \bigvee N\vee \square (\bigwedge P_1\to \bigvee Q_1)\vee \square (\bigwedge P_i\to \bigvee Q_i)\vee \square (\bigwedge P_2\to \bigvee Q_2)$$

$$\mathcal{M},w\models \bigvee N\vee \square (\bigwedge P_1\to \bigvee Q_1)\vee \square (\bigwedge P\to \bigvee Q)\vee \square (\bigwedge P_2\to \bigvee Q_2).$$

$$\underset{i=1}{\overset{k}{\bigwedge }}(\bigvee P_i\vee \bigvee \neg Q_i\vee \bigvee M_i\vee \bigvee \neg N_i),$$

$$\underset{i=1}{\overset{l}{\bigvee }}(\bigwedge P_i\vee \bigwedge \neg Q_i\vee \bigwedge M_i\vee \bigwedge \neg N_i),$$

$${⊢}_{n+1}\Gamma \Rightarrow \Delta ,\square A||\mathcal{H}$$

$${⊢}_n{M}_i,P_i,M\Rightarrow N,N_i,Q_i||\mathcal{H}|P\Rightarrow Q,$$

$$⊢M_i,M,P\Rightarrow Q,N,N_i||\mathcal{H}|P_i\Rightarrow Q_i,$$

$$M,P\Rightarrow Q,N||\mathcal{H}|P_i\Rightarrow Q_i|\mathcal{G}.$$

$$\begin{array}[]{cc}\ignorespaces\ignorespaces\ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$p,p,\Gamma\Rightarrow\Delta\,||\,{\cal H}$}\hskip 4.0pt}\vskip-1.21666pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to77.92331pt{\xleaders\hrule\hfill}\lower 3.21666pt\hbox{\hskip 3.0pt\sf LC}}\vskip-1.21666pt\nointerlineskip\hbox{\hskip 4.73784pt\hbox{\hbox{\hskip 4.0pt\hbox{$p,\Gamma\Rightarrow\Delta\,||\,{\cal H}$}\hskip 4.0pt}}}}}\ignorespaces&\ignorespaces\ignorespaces\ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta,q,q\,||\,{\cal H}$}\hskip 4.0pt}\vskip-2.18889pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to77.50662pt{\xleaders\hrule\hfill}\lower 2.24443pt\hbox{\hskip 3.0pt\sf RC,}}\vskip-2.18887pt\nointerlineskip\hbox{\hskip 4.63367pt\hbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta,q\,||\,{\cal H}$}\hskip 4.0pt}}}}}\ignorespaces\\[14.22636pt] \ignorespaces\ignorespaces\ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,p,p,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$}\hskip 4.0pt}\vskip-1.81944pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to139.23215pt{\xleaders\hrule\hfill}\lower 3.81943pt\hbox{\hskip 3.0pt${\sf LC}^{\sf c}$}}\vskip-1.81943pt\nointerlineskip\hbox{\hskip 4.73784pt\hbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,p,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$}\hskip 4.0pt}}}}}\ignorespaces&\ignorespaces\ignorespaces\ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},q,q\,|\,{\cal G}$}\hskip 4.0pt}\vskip-1.81944pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to138.81546pt{\xleaders\hrule\hfill}\lower 3.81943pt\hbox{\hskip 3.0pt${\sf RC}^{\sf c}$}}\vskip-1.81943pt\nointerlineskip\hbox{\hskip 4.63367pt\hbox{\hbox{\hskip 4.0pt\hbox{$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},q\,|\,{\cal G}$}\hskip 4.0pt}}}}}\ignorespaces\end{array}$$

$$⊢\Gamma \Rightarrow \Delta ,B,B||\mathcal{H}$$

$$M;P;H\Rightarrow G;Q;N,A$$

$$\Gamma ;H\Rightarrow G;\Delta ,◊A,A$$

$$\bigwedge \Gamma \to \bigvee \Delta \vee \underset{i=1}{\overset{n}{\bigvee }}\square (\bigwedge P_i\to \bigvee Q_i).$$

$$(\Gamma \Rightarrow \Delta ,[{\mathcal{N}}_1],\dots ,[{\mathcal{N}}_n]{)}^{tr}:=\bigwedge \Gamma \to \bigvee \Delta \vee \square ({\mathcal{N}}_1{)}^{tr}\vee \dots \vee \square ({\mathcal{N}}_n{)}^{tr},$$

$$\Gamma \Rightarrow \Delta \parallel P_1\Rightarrow Q_1|\cdots |P_n\Rightarrow Q_n,$$

Full text

1 Introduction

The propositional modal logic S5 is one of the peculiar modal logics in several respects. Most notably from the proof-theoretical point of view, S5 has so far resisted all efforts to provide it with a acceptable cut-free sequent calculus. Whereas the framework of sequent calculi has proven quite successful in providing analytic calculi for a number of normal modal logics such as K, KT or S4 [28]. For some formats of rules it can even be shown that no such calculus can exist [15]. Perhaps, the easiest way of demonstrating this resistance is Euclideanness axiom: ${\sf(5)}\,\,\Diamond A\rightarrow\Box\Diamond A$. Sequent calculus systems for S5 have been widely studied for a long time. Several authors have introduced many sequent calculi for S5, including Ohnishi and Matsumoto [20], Mints [17], Sato [25], Fitting [9], Wansing [28] and Braüner [6]. The efforts to develop sequent calculus to accommodate cut-free systems for S5 leading to introduce a variety of new sequent framework. Notably, labelled sequent calculus (see e.g. [18]), double sequent calculus (see e.g. [10]), display calculus (see e.g. [4, 29]), deep inference system (see e.g. [26]), nested sequent (see [7, 22]), hypersequent calculus, which was introduced independently in [1, 16, 23] and finally, grafted hypersequents ([12]), which combines the formalism of nested sequents with that of hypersequents. Hypersequent calculus provided numerous cut-free formulations for the logic S5, including Pottinger [23], Avron [1], Restall [24], Poggiolesi [21], Lahav [13], Kurokawa [11], Bednarska et al [2], and Lellmann [14].

The aim of this paper is to introduce a new sequent-style calculus for S5 by suggesting a framework of rooted hypersequents. A rooted hypersequent is of the form $\Gamma\Rightarrow\Delta\,||\,P_{1}\Rightarrow Q_{1}\,|\cdots|\,P_{n}\Rightarrow Q_{n}$, where $\Gamma$ and $\Delta$ are multisets of arbitrary formulas and $P_{i}$ and $Q_{i}$ are multisets of atomic formulas. Precisely, a rooted hypersequent is given by a sequent $\Gamma\Rightarrow\Delta$, called its root, together with a hypersequent $\cal H$, called its crown, where all formulas in the components of crown are atomic formulas. The sequents in the crown work as storage for atomic formulas that they might be used to get axioms. This sequent is inspired by the grafted hypersequent in [12]. A difference is that in the grafted hypersequent, all formulas in the crown can be arbitrary formula. Thus, the notion of our calculus is very close to the notion of grafted hypersequent. The main idea for constructing rooted hypersequent is to take an ordinary sequent $\Gamma\Rightarrow\Delta$ as a root and add sequences of multisets of atomic formulas to it.

Our calculus has the subformula property, and we show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. It is worth pointing out that in order to prove admissibility of cut rule, we make use of a normal form, called Quasi Normal Form, which is based on using modal and negation of modal formulas as literals. Soundness and completeness are established as well.

We proceed as follows. In the next section we recall the modal logic S5. In Section 3, we present rooted hypersequent calculus ${\cal R}_{\sf S5}$. In Section 4, we prove soundness of the system with respect to Kripke models. In Section 5, we prove the admissibility of weakening and contraction rules, and some other properties of ${\cal R}_{\sf S5}$. In Section 6, we prove admissibility of cut rule, and completeness of the system. Finally, we conclude the paper in Section 7.

2 Modal logic S5

In this section, we recall the axiomatic formulation of the modal logic S5.

The language of modal logic S5 is obtained by adding to the language of propositional logic the two modal operators $\Box$ and $\Diamond$. Atomic formulas are denoted by $p,q,r,$ and so on. Formulas, denoted by $A,B,C,\ldots$, are defined by the following grammar:

[TABLE]

where $\bot$ is a constant for falsity, and $\top$ is a constant for truth.

Modal logic S5 has the following axiom schemes:

[TABLE]

Equivalently, instead of (5) we can use:

[TABLE]

The proof rules are Modus Ponens and Necessitation:

$A$         $A\rightarrow B$

MP,

$B$

   

$A$

N.

$\Box A$

Rule Necessitation can be applied only to theorems (i.e. to formulas derivable from no premise), for a detailed exposition see [3, 5]. If $A$ is derivable in S5 from assumption $\Gamma$, we write $\Gamma\vdash_{\text{S5}}A$.

3 Rooted Hypersequent $\cal{R}_{\sf S5}$

Our calculus is based on finite multisets, i.e. on sets counting multiplicities of elements. We use certain categories of letters, possibly with subscripts or primed, as metavariables for certain syntactical categories (locally different conventions may be introduced): {tasks}(2) \task[$\bullet$] $p$ and $q$ for atomic formulas, \task[$\bullet$] $P$ and $Q$ for multisets of atomic formulas, \task[$\bullet$] $M$ and $N$ for multisets of modal formulas, \task[$\bullet$] $\Gamma$ and $\Delta$ for multisets of arbitrary formulas.

In addition, we use the following notations.

• •

The union of multisets $\Gamma$ and $\Delta$ is indicated simply by $\Gamma,\Delta$. The union of a multiset $\Gamma$ with a singleton multiset $\{A\}$ is written $\Gamma,A$.

• •

We use $\neg\Gamma$ for multiset of formulas $\neg A$ such that $A\in\Gamma$.

Definition 3.1**.**

A sequent is a pair of multisets $\Gamma$ and $\Delta$, written as $\Gamma\Rightarrow\Delta$. A hypersequent is a multiset of sequents, written $\Gamma_{1}\Rightarrow\Delta_{1}\,|\,\cdots,|\Gamma_{n}\Rightarrow\Delta_{n}$, where each $\Gamma_{i}\Rightarrow\Delta_{i}$ is called a component. A rooted hypersequent is given by a sequent $\Gamma\Rightarrow\Delta$, called its root, together with a hypersequent $\cal H$, called its crown, where all formulas in the components of crown are atomic formulas, and is written as $\Gamma\Rightarrow\Delta\,||\,{\cal H}$. If the crown is the empty hypersequent, the double-line separator can be omitted: a rooted hypersequent $\Gamma\Rightarrow\Delta$ is understood as $\Gamma\Rightarrow\Delta\,||\,\emptyset$. Formulas occurring on the left-hand side of the sequent arrow in the root or a component of the crown are called antecedent formulas; those occurring on the right-hand side succedent formulas.

Therefore, the notion of a rooted hypersequent

[TABLE]

can be seen as a restriction of the notion of grafted hypersequent as in [12].

Definition 3.2**.**

Let $\Gamma\Rightarrow\Delta\,||\,P_{1}\Rightarrow Q_{1}\,|\,P_{2}\Rightarrow Q_{2}\,|\,\ldots\,|\,P_{n}\Rightarrow Q_{n}$ be a rooted hypersequent. Its formula interpretation is the formula $\bigwedge\Gamma\rightarrow\bigvee\Delta\vee\bigvee\limits_{i=1}^{n}\Box(\bigwedge P_{i}\rightarrow\bigvee Q_{i})$.

The axioms and rules of $\cal{R}_{\sf S5}$ are given in the following:

Initial sequents:

[TABLE]

Propositional Rules:

[TABLE]

Modal Rules:

[TABLE]

Structural Rule:

[TABLE]

Let us make some remarks on the L$\Diamond$, R$\Box$ and Exch (abbreviates for exchange). All formulas in the conclusions of these rules are atomic or modal formulas. In a backward proof search by applying these rules, atomic formulas in the root of the conclusions move to the crown of the premises as a new sequent. Suppose, $P$ in the antecedent and $Q$ in the succedent of the root sequent of conclusion move to the crown of premise, the formulas in $P$ and $Q$ are saved in the crowns, until applications of the rule Exch in a derivation. In other words, The sequents in the crown work as storage for atomic formulas that they might be used to get axioms. By applying the rule Exch, the multisets $P_{i}$ and $Q_{i}$ come out from the crown while $P$ and $Q$ move to the crown.

Example 3.3**.**

The following sequents are derivable in $\cal{R}_{\sf S5}$.

$\Rightarrow(r\wedge p)\rightarrow(q\rightarrow\Box(\Diamond(p\wedge q)\wedge\Diamond r))$ 2. 2.

$\Rightarrow\Box(\Box\neg p\vee p)\rightarrow\Box(\neg p\vee\Box p)$

Proof*.*

Ax

$p,q,r\Rightarrow\Diamond(p\wedge q),p$

     

Ax

$r,p,q\Rightarrow\Diamond(p\wedge q),q$

R$\wedge$

$r,p,q\Rightarrow\Diamond(p\wedge q),p\wedge q$

R$\Diamond$

$r,p,q\Rightarrow\Diamond(p\wedge q)$

Exch

$\Rightarrow\Diamond(p\wedge q)\,||\,r,p,q\Rightarrow$

     

Ax

$r,p,q\Rightarrow\Diamond r,r$

R$\Diamond$

$r,p,q\Rightarrow\Diamond r$

Exch

$\Rightarrow\Diamond r\,||\,r,p,q\Rightarrow$

R$\wedge$

$\Rightarrow\Diamond(p\wedge q)\wedge\Diamond r\,||\,r,p,q\Rightarrow$

R$\Box$

$r,p,q\Rightarrow\Box(\Diamond(p\wedge q)\wedge\Diamond r)$

L$\wedge$

$r\wedge p,q\Rightarrow\Box(\Diamond(p\wedge q)\wedge\Diamond r)$

R$\rightarrow$

$r\wedge p\Rightarrow q\rightarrow\Box(\Diamond(p\wedge q)\wedge\Diamond r)$

R$\rightarrow$

$\Rightarrow(r\wedge p)\rightarrow(q\rightarrow\Box(\Diamond(p\wedge q)\wedge\Diamond r))$

Ax

$\Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow p\,||\,\Rightarrow p$

L$\neg$

$\neg p,\Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow\,||\,\Rightarrow p$

L$\Box$

$\Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow\,||\,\Rightarrow p$

Exch

$\Box\neg p,\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow$

     

Ax

$p,\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow$

L$\vee$

$\Box\neg p\vee p,\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow$

L$\Box$,

$\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow$

R$\Box$

$\Box(\Box\neg p\vee p),p\Rightarrow\Box p$

R$\neg$

$\Box(\Box\neg p\vee p)\Rightarrow\neg p,\Box p$

R$\vee$

$\Box(\Box\neg p\vee p)\Rightarrow\neg p\vee\Box p$

R$\Box$

$\Box(\Box\neg p\vee p)\Rightarrow\Box(\neg p\vee\Box p)$

R$\rightarrow$

$\Rightarrow\Box(\Box\neg p\vee p)\rightarrow\Box(\neg p\vee\Box p)$

∎

4 Soundness

In this section we prove soundness of the rules with respect to Kripke models.

A Kripke model $\mathcal{M}$ for S5 is a triple $\mathcal{M}=(W,R,V)$ where $W$ is a set of states, $R$ is an equivalence relation on $W$ and $V:\varPhi\rightarrow\mathcal{P}(W)$ is a valuation function, where $\varPhi$ is the set of propositional variables. Suppose that $w\in W$. We inductively define the notion of a formula $A$ being satisfied in $\mathcal{M}$ at state $w$ as follows:

• •

$\mathcal{M},w\vDash p\quad\text{iff}\quad w\in V(p),\,\text{where}\,\,p\in\varPhi$,

• •

$\mathcal{M},w\vDash\neg A\quad\text{iff}\quad\mathcal{M},w\nvDash A,$

• •

$\mathcal{M},w\vDash A\vee B\quad\text{iff}\quad\mathcal{M},w\vDash A\,\,\text{or}\,\,\mathcal{M},w\vDash B,$

• •

$\mathcal{M},w\Rightarrow A\wedge B\quad\text{iff}\quad\mathcal{M},w\vDash A\,\,\text{and}\,\,\mathcal{M},w\vDash B,$

• •

$\mathcal{M},w\vDash A\rightarrow B\quad\text{iff}\quad\mathcal{M},w\nvDash A\,\text{or}\,\mathcal{M},w\vDash B,$

• •

$\mathcal{M},w\vDash\Diamond A\quad\text{iff}\quad\,\,\mathcal{M},v\Rightarrow A\,\,\text{for some}\,\,v\in W\,\,\text{such that}\,\,R(w,v),$

• •

$\mathcal{M},w\vDash\Box A\quad\text{iff}\quad\mathcal{M},v\Rightarrow A\,\,\text{for all}\,\,v\in W\,\text{such that}\,R(w,v).$

We extend semantical notions to sequents in the following way:

• •

$\mathcal{M},w\vDash\Gamma\Rightarrow\Delta\,||\,P_{1}\Rightarrow Q_{1}\,|\cdots P_{n}\Rightarrow Q_{n}$ iff $\mathcal{M},w\vDash\bigwedge\Gamma\rightarrow\bigvee\Delta\vee\bigvee_{i=1}^{n}\Box(\bigwedge P_{i}\rightarrow\bigvee Q_{i})$.

• •

$\mathcal{M}\vDash\Gamma\Rightarrow\Delta\,||\,{\cal H}$ iff $\mathcal{M},w\Rightarrow\Gamma\Rightarrow\Delta\,||\,{\cal H}$, for all $w$ in the domain of $\mathcal{M}$.

• •

$\vDash\Gamma\Rightarrow\Delta\,||\,{\cal H}$ iff $\mathcal{M}\vDash\Gamma\Rightarrow\Delta\,||\,{\cal H}$, for all S5 models $\mathcal{M}$.

• •

The sequent $\Gamma\Rightarrow\Delta\,||\,{\cal H}$ is called S5-valid if $\vDash\Gamma\Rightarrow\Delta\,||\,{\cal H}$.

Lemma 4.1**.**

Let $\mathcal{M}=(W,R,V)$ be a Kripke model for ${\sf S5}$.

• (1)

$\mathcal{M},w\Rightarrow\Box A\quad\text{iff}\quad\mathcal{M},w^{\prime}\Rightarrow\Box A$, for all $w^{\prime}\in W$, where $wRw^{\prime}$.

• (2)

$\mathcal{M},w\Rightarrow\Diamond A\quad\text{iff}\quad\mathcal{M},w^{\prime}\Rightarrow\Diamond A$, for all $w^{\prime}\in W$, where $wRw^{\prime}$.

• (3)

If $\mathcal{M},w\Rightarrow A$, then $\mathcal{M},w^{\prime}\Rightarrow\Diamond A$, for all $w^{\prime}\in W$, where $wRw^{\prime}$.

Proof*.*

The proof clearly follows from the definition of satisfiability and the fact that $R$ is an equivalence relation. ∎

Theorem 4.2** (Soundness).**

If $\Gamma\Rightarrow\Delta\,||\,{\cal H}$ is provable in $\cal{R}_{\sf S5}$, then it is S5-valid.

Proof*.*

The proof is by induction on the height of the derivation of $\Gamma\Rightarrow\Delta\,||\,{\cal H}$. Initial sequents are obviously valid in every Kripke model for S5. We only check the induction step for rules R$\Box$ and Exch. The rule L$\Diamond$ can be verified similarly.

• •

Rule R$\Box$: Suppose that the sequent $\Gamma\Rightarrow\Delta\,||\,{\cal H}$ is $M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$, the conclusion of rule R$\Box$, with the premise $M\Rightarrow N,A\,||\,P\Rightarrow Q\,|\,{\cal H}$. For convenience, let the hypersequent ${\cal H}$ be a sequent $P_{1}\Rightarrow Q_{1}$. Suppose, by induction hypothesis, that the premise is valid, i.e., for every Kripke model $\cal M$ we have

[TABLE]

Assume the conclusion is not S5-valid i.e., there is a model $\mathcal{M}=(W,R,V)$ and $w^{\prime}\in W$ such that

[TABLE]

Thus, $\mathcal{M},w^{\prime}\nvDash\bigvee N\vee\Box(\bigwedge P\rightarrow\bigvee Q)\vee\Box(\bigwedge P_{1}\rightarrow\bigvee Q_{1})$. Suppose $w^{\prime}Rw$, it follows from 2 and Lemma 4.1 that $\mathcal{M},w\vDash\bigwedge M$ and $\mathcal{M},w\nvDash\bigvee N\vee\Box(\bigwedge P\rightarrow\bigvee Q)\vee\Box(\bigwedge P_{1}\rightarrow\bigvee Q_{1})$. Therefore, by (1) we have $\mathcal{M},w\vDash A$, and so $\mathcal{M},w^{\prime}\vDash\Box A$. This leads to a contradiction with (3).

• •

Rule Exch: Suppose that the sequent $\Gamma\Rightarrow\Delta\,||\,{\cal H}$ is the conclusion of the rule Exch. For convenience, let the hypersequent $\cal{H}$ be a sequent $P_{1}\Rightarrow Q_{1}$ and let $\cal{G}$ be a sequent $P_{2}\Rightarrow Q_{2}$. Suppose, by induction hypothesis, that the premise is valid i.e., for every model $\cal M$ we have:

[TABLE]

Assume the conclusion is not valid i.e., there is a model $\mathcal{M}=(W,R,V)$ and $w^{\prime}\in W$ such that

[TABLE]

By 6, we have $\mathcal{M},w\nvDash\bigwedge P_{i}\rightarrow\bigvee Q_{i}$ for some $w$ such that $w^{\prime}Rw$. Thus, $\mathcal{M},w\vDash\bigwedge P_{i}$, and $\mathcal{M},w\nvDash\bigvee Q_{i}$. In addition, because $M$ is a multiset of modal formula, using Lemma 4.1 and (5) we can conclude that $\mathcal{M},w\vDash\bigwedge M$. That means, $\mathcal{M},w\vDash\bigwedge M\wedge\bigwedge P_{i}$ and $\mathcal{M},w\nvDash\bigvee Q_{i}$ for some $w$ such that $w^{\prime}Rw$. It follows from (• ‣ 4) that

[TABLE]

This, Using Lemma 4.1, leads to a contradiction with (6).

∎

5 Structural properties

In this section, we prove the admissibility of weakening and contraction rules, and also some properties of $\cal{R}_{\sf S5}$, which are used to prove the admissibility of cut rule. Some (parts of) proofs are omitted because they are easy or similar to the proofs in [8, 19, 27].

The height of a derivation is the greatest number of successive applications of rules in it, where initial sequents have height [math]. The notation $\vdash_{n}\Gamma\Rightarrow\Delta\,||\,\cal{H}$ means that $\Gamma\Rightarrow\Delta\,||\,\cal{H}$ is derivable with a height of derivation at most $n$ in the system $\cal{R}_{\sf S5}$. A rule of $\cal{R}_{\sf S5}$ is said to be (height-preserving) admissible if whenever an instance of its premise(s) is (are) derivable in $\cal{R}_{\sf S5}$ (with at most height $n$), then so is the corresponding instance of its conclusion. A rule of $\cal{R}_{\sf S5}$ is said to be invertible if whenever an instance of its conclusion is derivable in $\cal{R}_{\sf S5}$, then so is the corresponding instance of its premise(s).

The following lemma, shows that the propositional rules are height-preserving invertible. The proof is by induction on the height of derivations.

Lemma 5.1**.**

All propositional rules are height-preserving invertible in $\cal{R}_{\sf S5}$.

In order to prove the admissibility of cut rule, we introduce the following structural rules, called Merge and ${\sf Merge}^{\sf c}$. Rule ${\sf Merge}^{\sf c}$ allows us to merge two crown components in the crown, and Rule Merge allows us to merge a component in the crown with the root component.

Lemma 5.2**.**

The following rules are height-preserving admissible

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}$

${\sf Merge}^{\sf c}$

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i},P_{j}\Rightarrow Q_{i},Q_{j}\,|\,{\cal G}\,|\,{\cal I}$

**    **

**

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

Merge

$\Gamma,P_{i}\Rightarrow Q_{i},\Delta\,||\,{\cal H}\,|\,{\cal G}$

---

Proof*.*

Both rules are proved simultaneously by induction on the height of the derivations of the premises. In both rules, if the premise is an initial sequent, then the conclusion is an initial sequent too. For the induction step, we consider only cases where the last rule is L$\Diamond$ or Exch; since the rule R$\Box$ is treated symmetrically and for the remaining rules it suffices to apply the induction hypothesis to the premise and then use the same rule to obtain deduction of the conclusion.

For the rule Merge we consider the following cases.

Case 1. Let $\Gamma=\Diamond A,M,P$ and $\Delta=Q,N$ and let $\Diamond A$ be the principal formula:

$\mathcal{D}$

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P\Rightarrow Q$

L$\Diamond$,

$\vdash_{n+1}\Diamond A,M,P\Rightarrow Q,N\,||\,{\cal H}\,|P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

By induction hypothesis (IH), we have:

$\mathcal{D}$

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P\Rightarrow Q$

IH (${\sf Merge}^{\sf c}$)

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|\,{\cal G}\,|\,P,P_{i}\Rightarrow Q_{i},Q$

L$\Diamond$.

$\vdash_{n+1}\Diamond A,M,P,P_{i}\Rightarrow Q_{i},Q,N\,||\,{\cal H}\,|\,{\cal G}$

Case 2. Let $\Gamma=M,P$ and $\Delta=Q,N$, and let the last rule be as:

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

Exch.

$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

In this case, by induction hypothesis, the sequent $\vdash_{n}M,P,P_{i}\Rightarrow Q_{i},Q,N\,||\,{\cal H}\,|\,{\cal G}$ is obtained.

For the rule ${\sf Merge}^{\sf c}$ we consider the following cases.

Case 1. The premise is derived by Exch. If this rule does not apply to $P_{i}\Rightarrow Q_{i}$ and $P_{j}\Rightarrow Q_{j}$, then the conclusion is obtained by applying induction hypothesis and then applying the same rule Exch. Therefore, let the last rule be as follows in which the rule Exch apply to the component $P_{i}\Rightarrow Q_{i}$:

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}$

Exch,

$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}$

where $\Gamma=M,P$ and $\Delta=Q,N$. Then the conclusion is derived as follows

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}$

Merge (IH)

$\vdash_{n}M,P_{i},P_{j}\Rightarrow Q_{j},Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal I}$

Exch.

$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i},P_{j}\Rightarrow Q_{j},Q_{i}\,|\,{\cal G}\,|\,{\cal I}$

Case 2. The premise is derived by L$\Diamond$.

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}\,|\,P\Rightarrow Q$

L$\Diamond$,

$\vdash_{n+1}\Diamond A,M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}$

where $\Gamma=\Diamond A,M,P$ and $\Delta=Q,N$. Then the conclusion is obtained as follows:

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P_{j}\Rightarrow Q_{j}\,|\,{\cal I}\,|\,P\Rightarrow Q$

IH

$\vdash_{n}A,M\Rightarrow N\,||\,{\cal H}\,|\,P_{i},P_{j}\Rightarrow Q_{i},Q_{j}\,|\,{\cal G}\,|\,{\cal I}\,|\,P\Rightarrow Q$

L$\Diamond$.

$\vdash_{n+1}\Diamond A,M,P\Rightarrow Q,N\,||{\cal H}\,|\,P_{i},P_{j}\Rightarrow Q_{i},Q_{j}\,|\,{\cal G}\,|\,{\cal I}$

∎

5.1 Admissibility of weakening

In this subsection, we prove the admissibility of structural rules of external weakening ${\sf EW}$, crown weakening ${\sf W}^{\sf c}$ , left rooted weakening ${\sf LW}$, right rooted weakening ${\sf RW}$.

Lemma 5.3**.**

The rule of external weakening:

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,\cal{G}$

$\sf EW$

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,\cal{G}$

---

is height-preserving admissible in ${\cal R}_{\sf S5}$.

Proof*.*

By straightforward induction on the height of the derivation of the premise. ∎

Lemma 5.4**.**

The rule of crown weakening:

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

${\sf W}^{\sf c}$

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P,P_{i}\Rightarrow Q_{i},Q\,|\,{\cal G}$

---

is height-preserving admissible in ${\cal R}_{\sf S5}$.

Proof*.*

It follows by the height-preserving admissibility of the two rules $\sf EW$ and ${\sf Merge}^{\sf c}$.

∎

Lemma 5.5**.**

The rules of left and right weakening:

$\Gamma\Rightarrow\Delta\,||\,{\cal H}$

LW

$A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

**       **

**

$\Gamma\Rightarrow\Delta\,||\,{\cal H}$

RW,

$\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$

---

are admissible, where $A$ is an arbitrary formula.

Proof*.*

Both rules are proved simultaneously by induction on the complexity of $A$ with subinduction on the height of the derivations. The admissibility of the rules for atomic formula follows by admissibility of the two rules EW and Merge. The admissibility of the rules for modal formula are straightforward by induction on the height of the derivation. The other cases are proved easily.

∎

5.2 Invertibility

In this subsection, first we introduce a normal form called Quasi Normal Form, which are used to prove the admissibility of the contraction and cut rules. Then we show that the structural and modal rules are invertible.

Definition 5.6** (Quasi-literal).**

A quasi-literal is an atomic formula, a negation of atomic formula, a modal formula, or a negation of modal formula.

Definition 5.7** (Quasi-clause and quasi-phrase).**

A quasi-clause (quasi-phrase) is either a single quasi-literal or a conjunction (disjunction) of quasi-literals.

Definition 5.8**.**

A formula is in conjunctive (disjunctive) quasi-normal form, abbreviated CQNF (DQNF), iff it is a conjunction (disjunction) of quasi-clauses.

Example 5.9**.**

The formula $(\neg\Box(A\rightarrow B)\vee p\vee\Diamond C)\wedge(\neg q)\wedge(\Box\Diamond A\vee\neg\Diamond(A\wedge B)\vee\neg r)$ is in CQNF, which $(\neg\Box(A\rightarrow B)\vee p\vee\Diamond C)$, $(\neg q)$, and $(\Box\Diamond A\vee\neg\Diamond(A\wedge B)\vee\neg r)$ are quasi-clauses.

Without loss of generality we can assume that every formula in CQNF is as follows

[TABLE]

and every formula in DQNF is as follows

[TABLE]

where $P_{i}$ and $Q_{i}$ are multisets of atomic formulae and $M_{i}$ and $N_{i}$ are multisets of modal formulae.

Similar to the propositional logic, Every formula can be transformed into an equivalent formula in CQNF, and an equivalent formula in DQNF.

For these quasi-normal forms we have the following lemmas.

Lemma 5.10**.**

Let $A$ be an arbitrary formula, and let $\bigwedge_{i=1}^{k}(\bigvee\neg P_{i}\vee\bigvee Q_{i}\vee\bigvee\neg M_{i}\vee\bigvee N_{i})$ and $\bigvee_{i=1}^{l}(\bigwedge P^{\prime}_{i}\vee\bigwedge\neg Q^{\prime}_{i}\vee\bigwedge M^{\prime}_{i}\vee\bigwedge\neg N^{\prime}_{i})$ be equivalent formulae in CQNF and DQNF, respectively, for $A$. Then

• (i)

$\vdash\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$* iff $\vdash M_{i},P_{i},\Gamma\Rightarrow\Delta,Q_{i},N_{i}\,||\,{\cal H}$, for every $i=1,\ldots,k$.*

• (ii)

$\vdash A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$* iff $\vdash M^{\prime}_{i},P^{\prime}_{i},\Gamma\Rightarrow\Delta,Q^{\prime}_{i},N^{\prime}_{i}\,||\,{\cal H}$, for every $i=1,\ldots,l$.*

Proof*.*

We consider the first assertion; the second is treated symmetrically.

$(\Leftarrow)$. Suppose that $M_{i},P_{i},\Gamma\Rightarrow\Delta,Q_{i},N_{i}\,||\,{\cal H}$ is derivable for every $i=1,\ldots,k$. By applying propositional rules, where non-modal subformulae of $A$ are principal formulae in a backward proof search with $\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$ in the bottom. We obtain a derivation, in which the topsequents are $M_{i},P_{i},\Gamma\Rightarrow\Delta,Q_{i},N_{i}\,||\,{\cal H}$, $i=1,\ldots,k$.

$(\Rightarrow)$. Suppose $\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$ is derivable. By applying the invertibility of the propositional rules, we reach a derivation for $M_{i},P_{i},\Gamma\Rightarrow\Delta,Q_{i},N_{i}\,||\,{\cal H}$, for every $i=1,\ldots,k$. ∎

Corollary 5.11**.**

Both rules in the above lemma from left to right are height-preserving admissible. In other words:

If $\vdash_{n}\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$, then $\vdash_{n}M_{i},P_{i},\Gamma\Rightarrow\Delta,Q_{i},N_{i}\,||\,{\cal H}$, for every $i=1,\ldots,k$.

If $\vdash_{n}A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$, then $\vdash_{n}M^{\prime}_{i},P^{\prime}_{i},\Gamma\Rightarrow\Delta,Q^{\prime}_{i},N^{\prime}_{i}\,||\,{\cal H}$, for every $i=1,\ldots,l$.

Lemma 5.12**.**

Let $A$ be an arbitrary formula, and let $\bigwedge_{i=1}^{k}(\bigvee\neg P_{i}\vee\bigvee Q_{i}\vee\bigvee\neg M_{i}\vee\bigvee N_{i})$ and $\bigvee_{i=1}^{l}(\bigwedge P^{\prime}_{i}\vee\bigwedge\neg Q^{\prime}_{i}\vee\bigwedge M^{\prime}_{i}\vee\bigwedge\neg N^{\prime}_{i})$ be equivalent formulae in CQNF and DQNF, respectively, for $A$. Then

• (i)

$\vdash\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$* iff $\vdash M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$, for every $i=1,\ldots,k$.*

• (ii)

$\vdash\Diamond A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$* iff $\vdash M^{\prime}_{i},\Gamma\Rightarrow\Delta,N^{\prime}_{i}\,||\,{\cal H}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}$, for every $i=1,\ldots,l$.*

Proof*.*

We prove the first assertion; the second is proved by a similar argument. For the direction from right to left observe that by Lemma 5.10, $\vdash M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$ implies $\vdash M_{i},M,P\Rightarrow N,Q,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$, where $M,P$ and $N,Q$ are multisets of atomic and modal formulae corresponding to formulae in $\Gamma$ and $\Delta$. Then by applying rule Exch we have:

$\vdash M_{i},M,P\Rightarrow N,Q,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$

Exch.

$\vdash M_{i},M,P_{i}\Rightarrow N,Q_{i},N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q$

Therefore, $\vdash M_{i},M,P_{i}\Rightarrow N,Q_{i},N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q$, for $i=1,\ldots,k$, and then by Lemma 5.10 we have

$\vdash M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$. Hence, by applying rule R$\Box$ we have:

$\vdash M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$\vdash M,P\Rightarrow N,Q,\Box A\,||\,{\cal H}$

Again by applying Lemma 5.10 we have $\vdash\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$.

For the opposite direction, by induction on $n$, we prove that if $\vdash_{n}\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$, then

$\vdash_{n}M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$, for $i=1,\ldots,k$. For $n=0$, if $\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$ is an initial sequent, then $\Box A$ is not principal, and so $M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$ is an initial sequent too, for $i=1,\ldots,k$. Assume height-preserving admissible up to height n, and let

[TABLE]

If $\Box A$ is not principal, we apply induction hypothesis to the premise(s) and then use the same rule to obtain deductions of $M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$, for $i=1,\ldots,k$. If on the other hand $\Box A$ is principal, the derivation ends with

$\vdash_{n}M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$,

$\vdash_{n+1}M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

where $\Gamma=M,P$ and $\Delta=Q,N$. Therefore, by applying Corollary 5.11, we have

[TABLE]

for $i=1,\ldots,k$. Hence, by applying rule Exch the conclusion is obtained as follows:

$\vdash_{n}M_{i},P_{i},M\Rightarrow N,N_{i},Q_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q$

Exch.

$\vdash_{n+1}M_{i},P,M\Rightarrow N,N_{i},Q\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$

∎

Corollary 5.13**.**

Both rules in the above lemma from left to right are height-preserving admissible. In other words:

• (i)

If $\vdash_{n}\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$, then $\vdash_{n}M_{i},\Gamma\Rightarrow\Delta,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$, for every $i=1,\ldots,k$.

• (ii)

If $\vdash_{n}\Diamond A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$, then $\vdash_{n}M^{\prime}_{i},\Gamma\Rightarrow\Delta,N^{\prime}_{i}\,||\,{\cal H}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}$, for every $i=1,\ldots,l$.

Lemma 5.14**.**

The structural and modal rules are invertible in $\cal{R}_{\sf S5}$.

Proof*.*

Rules L$\Box$ and R$\Diamond$ have repetition of the principle formula in the premises; so that we can obtain a derivation of premises of these rules by weakening their conclusions.

We prove that the rules R$\Box$ and Exch are invertible; the rule L$\Diamond$ is treated similarly.

Let $\vdash M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$, and let $\bigwedge_{i=1}^{n}(\bigvee P_{i}\vee\bigvee\neg Q_{i}\vee\bigvee M_{i}\vee\bigvee\neg N_{i})$ be an equivalent formula in CQNF for $A$. Then, by applying Lemma 5.12 we have

[TABLE]

for $i=1,\ldots,n$. Thus, by applying rule Exch we have

$M_{i},M,P\Rightarrow Q,N,N_{i}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}$

Exch

$M_{i},M,P_{i}\Rightarrow Q_{i},N,N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q$

Therefore, $\vdash M_{i},M,P_{i}\Rightarrow Q_{i},N,N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q$, for $i=1,\ldots,n$, and then again by applying Lemma 5.10 we have $\vdash M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$.

Finally, we prove that the rule Exch is invertible. Let $\mathcal{D}$ be a derivation of

[TABLE]

If the last rule applied in $\mathcal{D}$ is propositional rules or Exch, where the rule Exch does not apply to $P_{i}\Rightarrow Q_{i}$, apply induction hypothesis to the premise and then apply the last rule with the same principal formula to obtain deduction of $M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$. Therefore, let the last rule be R$\Box$ as follows:

$M\Rightarrow N^{\prime},A\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P\Rightarrow Q$

R$\Box$,

$M,P\Rightarrow Q,N^{\prime},\Box A\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

where $N=N^{\prime},\Box A$, we have

$M\Rightarrow N^{\prime},A\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}\,|\,P\Rightarrow Q$

R$\Box$.

$M,P_{i}\Rightarrow Q_{i},N^{\prime},\Box A\,||\,{\cal H}\,|\,{\cal G}\,|\,P\Rightarrow Q$

If the last rule is L$\Diamond$, the proof is similarly. If the last rule is Exch which is applied to $P_{i}\Rightarrow Q_{i}$, the conclusion is obtained. ∎

5.3 Admissibility of contraction

In order to prove the admissibility of the contraction rule, we need the following lemma.

Lemma 5.15**.**

The following rules are admissible.

$\Diamond A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

(@slowromancapi@)

$A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

**       **

**

$\Gamma\Rightarrow\Delta,\Box A\,||\,{\cal H}$

(@slowromancapii@)

$\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$

---

Proof*.*

We only consider the part ${\rm(\@slowromancap i@)}$; the other is proved similarly. Let $\vdash\Diamond A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$, and let $\bigvee_{i=1}^{k}(\bigwedge P^{\prime}_{i}\vee\bigwedge\neg Q^{\prime}_{i}\vee\bigwedge M^{\prime}_{i}\vee\bigwedge\neg N^{\prime}_{i})$ be an equivalent formula in DQNF for $A$. Then we have

$\vdash\Diamond A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

Lemma 5.12

$\vdash M^{\prime}_{i},\Gamma\Rightarrow\Delta,N^{\prime}_{i}\,||\,{\cal H}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}$

Merge

$\vdash M^{\prime}_{i},\Gamma,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},\Delta,N^{\prime}_{i}\,||\,{\cal H}$

Lemma 5.10

$\vdash A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

∎

Corollary 5.16**.**

The following rule is admissible.

$\Diamond A,\Gamma\Rightarrow\Delta,\Box B\,||\,{\cal H}$

$A,\Gamma\Rightarrow\Delta,B\,||\,{\cal H}$

---

We now prove the admissibility of rules left rooted contraction LC, right rooted contraction RC, left crown contraction ${\sf LC}^{\sf c}$, right crown contraction ${\sf RC}^{\sf c}$ and finally, external contraction EC, which are required for the proof of the admissibility of cut rule. First we consider the rules LC and RC for atomic formula, these rules and rules ${\sf LC}^{\sf c}$ and ${\sf RC}^{\sf c}$ are proved simultaneously.

Lemma 5.17**.**

The following rules are height-preserving admissible,

[TABLE]

where $p$ and $q$ are atomic formulae.

Proof*.*

All rules are proved simultaneously by induction on the height of derivation of the premises. In all cases, if the premise is an initial sequent, then the conclusion is an initial sequent too. We consider some cases; the other cases are proved by a similar argument.

For the rule ${\sf LC}^{\sf c}$, let $\Gamma=M,P$ and $\Delta=Q,N$, and let the last rule be

$\mathcal{D}$

$\vdash_{n}M,p,p,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

Exch,

$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,p,p,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

then we have

$\mathcal{D}$

$\vdash_{n}M,p,p,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

IH (LC)

$\vdash_{n}M,p,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

Exch.

$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,p,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

For the other last rules $\sf R$, use induction hypothesis on the premise, and then apply the rule $\sf R$.

For the rule $\sf LC$, let $\Gamma=M,P$ and $\Delta=Q,N$, and let the last rule be as follows

$\mathcal{D}$

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,p,p,P\Rightarrow Q\,|\,{\cal G}$

Exch,

$\vdash_{n+1}p,p,M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

then we have

$\mathcal{D}$

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,p,p,P\Rightarrow Q\,|\,{\cal G}$

IH (${\sf LC}^{\sf c}$)

$\vdash_{n}M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal H}\,|\,p,P\Rightarrow Q\,|\,{\cal G}$

Exch.

$\vdash_{n+1}M,p,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

Let the last rule be L$\Diamond$ as:

$\mathcal{D}$

$\vdash_{n}B,M\Rightarrow N\,||\,{\cal H}\,|\,p,p,P\Rightarrow Q$

L$\Diamond$,

$\vdash_{n+1}p,p,\Diamond B,M,P\Rightarrow Q,N\,||\,{\cal H}$

where $\Gamma=\Diamond B,M,P$ and $\Delta=Q,N$. Then we have

$\mathcal{D}$

$\vdash_{n}B,M\Rightarrow N\,||\,{\cal H}\,|\,p,p,P\Rightarrow Q$

IH (${\sf LC}^{\sf c}$)

$\vdash_{n}B,M\Rightarrow N\,||\,{\cal H}\,|\,p,P\Rightarrow Q$

L$\Diamond$,

$\vdash_{n+1}p,P,\Diamond B,M\Rightarrow N,Q\,||\,{\cal H}$

The rule R$\Box$ is treated similarly; for the other rules use induction hypothesis and then use the same rule.

Lemma 5.18**.**

The rules of contraction,

$A,A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

LC

$A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

**       **

**

$\Gamma\Rightarrow\Delta,A,A\,||\,{\cal H}$

RC,

$\Gamma\Rightarrow\Delta,A\,||\,{\cal H}$

---

are admissible.

Proof*.*

Both rules are proved simultaneously by induction on the complexity of $A$ with subinduction on the height of the derivations. The lemma holds for atomic formula $A$, by Lemma 5.17. For the other cases, if $A$ is not principal in the last rule (either modal or propositional), apply inductive hypothesis to the premises and then apply the last rule. Here we only consider the rule ${\sf LC}$; the admissibility of the rule RC is proved similarly. Suppose $\vdash A,A,\Gamma\Rightarrow\Delta\,||\,{\cal H}$, we consider some cases, in which $A$ is principal formula in the last rule: ∎

Case 1. $A=\Diamond B$:

$\mathcal{D}$

$\vdash B,\Diamond B,M\Rightarrow N\,||\,{\cal H}\,|\,P\Rightarrow Q$

L$\Diamond$,

$\vdash\Diamond B,\Diamond B,M,P\Rightarrow N,Q\,||\,{\cal H}$

where $\Gamma=M,P$ and $\Delta=N,Q$. Then we have

$\mathcal{D}$

$\vdash B,\Diamond B,M\Rightarrow N\,||\,{\cal H}\,|\,P\Rightarrow Q$

Lemma 5.15

$\vdash B,B,M\Rightarrow N\,||\,{\cal H}\,|\,P\Rightarrow Q$

IH

$\vdash B,M\Rightarrow N\,||\,{\cal H}\,|\,P\Rightarrow Q$

L$\Diamond$.

$\vdash\Diamond B,M,P\Rightarrow N,Q\,||\,{\cal H}$

Case 2. $A=\Box B$:

$\mathcal{D}$

$\vdash B,\Box B,\Box B,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

L$\Box$,

$\vdash\Box B,\Box B,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

then

$\mathcal{D}$

$\vdash B,\Box B,\Box B,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

IH

$\vdash B,\Box B,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

L$\Box$.

$\vdash\Box B,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

Case 3. Let $A=B\rightarrow C$ and let the last rule as

$\mathcal{D}_{1}$

$\vdash B\rightarrow C,\Gamma\Rightarrow\Delta,B\,||\,{\cal H}$

     

$\mathcal{D}_{2}$

$\vdash C,B\rightarrow C,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

L$\rightarrow$,

$\vdash B\rightarrow C,B\rightarrow C,\Gamma\Rightarrow\Delta\,||\,{\cal H}$

By applying the invertibility of the rule L$\rightarrow$, see Lemma 5.1, to the first premise, we get

[TABLE]

and applying to the second premise, we get $\vdash C,C,\Gamma\Rightarrow\Delta\,||\,{\cal H}$. We then use the induction hypothesis and obtain $\vdash\Gamma\Rightarrow\Delta,B\,||\,{\cal H}$ and $\vdash C,\Gamma\Rightarrow\Delta\,||\,{\cal H}$. Thus by L$\rightarrow$, we get $\vdash B\rightarrow C,\Gamma\Rightarrow\Delta\,||\,{\cal H}$.

∎

Lemma 5.19**.**

The rule of external contraction,

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

$\sf EC$

$\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

---

is height-preserving admissible.

Proof*.*

If the premise is derivable, then the conclusion is derived as follows

$\vdash_{n}\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

${\sf Merge}^{\sf c}$

$\vdash_{n}\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i},P_{i}\Rightarrow Q_{i},Q_{i}\,|\,{\cal G}$

${\sf RC}^{\sf c}$, ${\sf LC}^{\sf c}$

$\vdash_{n}\Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal G}$

this rule is height-preserving admissible because the rules ${\sf Merge}^{\sf c}$, ${\sf RC}^{\sf c}$, and ${\sf LC}^{\sf c}$ are height-preserving admissible. ∎

6 Admissibility of cut

In this section, we prove the admissibility of cut rule and completeness theorem.

In cut rule,

$\Gamma\Rightarrow\Delta,D\,||\,{\cal H}$         $D,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H^{\prime}}$

Cut,

$\Gamma,\Gamma^{\prime}\Rightarrow\Delta,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H^{\prime}}$

the crown of the conclusion is the union of crowns of the premises.

If cut formula $D$ is atomic, then the admissibility of the cut rule is proved simultaneously with the following rule which is called crown cut:

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$         $M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Note: The following exchanges between root and crown parts should be noted in the crown cut rule ${\sf Cut}^{\sf c}$. {tasks}(4) \task[$\bullet$] $P$ and $P_{i}$ \task[$\bullet$] $Q$ and $Q_{i}$ \task[$\bullet$] $P^{\prime}$ and $P^{\prime}_{i}$ \task[$\bullet$] $Q^{\prime}$ and $Q^{\prime}_{i}$

Lemma 6.1**.**

The following rules are admissible, where $p$ is an atomic formula.

$\Gamma\Rightarrow\Delta,p\,||\,{\cal H}$         $p,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H^{\prime}}$

Cut,

$\Gamma,\Gamma^{\prime}\Rightarrow\Delta,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H^{\prime}}$

---

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$         $M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

---

Proof*.*

Both rules are proved simultaneously by induction on the sum of heights of derivations of the two premises, which is called cut-height. First, we consider the rule ${\sf Cut}$. If both of the premises are initial sequents, then the conclusion is an initial sequent too, and if only one of the premises is an initial sequent, then the conclusion is obtained by weakening.

If one of the last rules in the derivations of the premises is not L$\Diamond$, R$\Box$, or Exch, then the cut rule can be transformed into cut(s) with lower cut-height as usual. Thus we consider cases which the last rules are L$\Diamond$, R$\Box$, and Exch.

Case 1. The left premise is derived by R$\Box$. Let $\Gamma=M,P$ and $\Delta=Q,N,\Box A$, and let $\Box A$ be the principal formula. We have three subcases according to the last rule in the derivation of the right premise.

Subcase 1.1 The right premise is derived by L$\Diamond$. Let $\Gamma^{\prime}=\Diamond B,M^{\prime},P^{\prime}$ and $\Delta^{\prime}=Q^{\prime},N^{\prime}$, and let the last rules be as follows

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q,p$

R$\Box$

$M,P\Rightarrow Q,N,\Box A,p\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}$

L$\Diamond$

$p,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}$

Cut.

$M,P,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q,N,\Box A,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

For this cut, let $\bigwedge_{i=1}^{k}(\bigvee\neg P_{i}\vee\bigvee Q_{i}\vee\bigvee\neg M_{i}\vee\bigvee N_{i})$ be an equivalent formula in CQNF of $A$. Then, by Corollary 5.11, which is height-preserving admissible, we get the following derivation from $\mathcal{D}$

$\mathcal{D}_{1}$

$M_{i},P_{i},M\Rightarrow N,Q_{i},N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q,p,$

for every clause $(\bigvee\neg P_{i}\vee\bigvee Q_{i}\vee\bigvee\neg M_{i}\vee\bigvee N_{i})$. Then by applying crown cut rule ${\sf Cut}^{\sf c}$, we get the following derivation for every clauses in CQNF of $A$

$\mathcal{D}_{1}$

$M_{i},P_{i},M\Rightarrow N,Q_{i},N_{i}\,||\,{\cal H}\,|\,P\Rightarrow Q,p$

     

$\mathcal{D}^{\prime}$

$B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}$

L$\Diamond$

$\Diamond B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}$

${\sf Cut}^{\sf c}$

$M_{i},P,M,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q,N,N_{i},Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}$

and then using Lemma 5.12, the conclusion is obtained.

Subcase 1.2. The right premise is derived by R$\Box$. Similar to Case 1.1.

Subcase 1.3. The right premise is derived by Exch. Let ${\cal H}^{\prime}={\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$, and let the last rules be as follows:

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q,p$

R$\Box$

$M,P\Rightarrow Q,N,\Box A,p\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$p,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Cut.

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,\Box A,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

This cut is transformed into

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q,p$

R$\Box$

$M\Rightarrow N,\Box A\,||\,{\cal H}\,|\,P\Rightarrow Q,p$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

${\sf Cut}^{\sf c}$.

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,\Box A,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Case 2. The left premise is derived by L$\Diamond$. Similar to the case 1.

Case 3. The left premise is derived by Exch. Let the hypersequent ${\cal H}$ be as ${\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}$. According to the last rule applied in the right premise, we have the following subcases:

Subcase 3.1. The right premise is derived by R$\Box$ or L$\Diamond$. Similar to the case 1.2.

Subcase 3.2. The right premise is derived by Exch. Let the hypersequent ${\cal H}^{\prime}$ be as ${\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$ and let the last rules be as

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal G}\,|\,P\Rightarrow Q,p\,|\,{\cal I}$

Exch

$M,P\Rightarrow Q,N,p\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$p,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Cut,

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

where $\Gamma=M,P$, $\,\Delta=Q,N$, $\,\Gamma^{\prime}=M^{\prime},P^{\prime}$, and $\Delta^{\prime}=Q^{\prime},N^{\prime}\,$. This cut is transformed into

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},N\,||\,{\cal G}\,|\,P\Rightarrow Q,p\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,p,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

${\sf Cut}^{\sf c}$.

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Case 4. The right premise is derived by L$\Diamond$, R$\Box$, or Exch. Similar to the above cases for the left premise.

Now we prove the admissibility of the crown cut rule ${\sf Cut}^{\sf c}$. If the left premise is an instance of initial sequent ${\sf L\bot}$ or ${\sf R\top}$, then so is the conclusion. If the left premise is an instance of initial sequent Ax, we have two cases as follows. If $M$ and $N$ contain a common atomic formula, then so is the conclusion. If $P$ and $Q$ contain a common atomic formula, then apply the rule Exch and then apply the rules weakening to obtain deduction of the conclusion. If the right premise is an initial sequent or both of the premises are initial sequent, the conclusion is obtained by the same argument.

The last rule applied in the premises of the crown cute rule can only be modal rules since all formulae in this rule are modal or atomic. Similar to the proof of the cut rule, we only consider the rules L$\Diamond$, R$\Box$, and Exch.

Case 1. The left premise is derived by L$\Diamond$. Let $M=\Diamond B,M_{1}$ and the derivation be as follows

$\mathcal{D}$

$B,M_{1}\Rightarrow N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}\,|\,P\Rightarrow Q$

L$\Diamond$

$\Diamond B,M_{1},P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$,

$\Diamond B,M_{1},P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

For this case, first we transform $B$ into $\bigvee_{j=1}^{m}(\bigwedge P^{\prime\prime}_{j}\vee\bigwedge\neg Q^{\prime\prime}_{j}\vee\bigwedge M^{\prime\prime}_{j}\vee\bigwedge\neg N^{\prime\prime}_{j})$, an equivalent formula in DQNF for $B$. Then by Corollary 5.11 we get $\mathcal{D}_{1}$ from $\mathcal{D}$ as:

$\mathcal{D}_{1}$

$P^{\prime\prime}_{j},M^{\prime\prime}_{j},M_{1}\Rightarrow N,Q^{\prime\prime}_{j},N^{\prime\prime}_{j}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}\,|\,P\Rightarrow Q$

for every phrases in DQNF. Thus since the rules in Corollary 5.11 are height-preserving admissible by induction hypothesis we have

$\mathcal{D}_{1}$

$P^{\prime\prime}_{j},M^{\prime\prime}_{j},M_{1}\Rightarrow N,Q^{\prime\prime}_{j},N^{\prime\prime}_{j}\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}\,|\,P\Rightarrow Q$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$.

$M^{\prime\prime}_{j},M_{1},P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow N,Q_{i},N^{\prime\prime}_{j},Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P^{\prime\prime}_{j}\Rightarrow Q^{\prime\prime}_{j}\,|\,{\cal G}\,|\,P\Rightarrow Q\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Therefore, by applying Lemma 5.12, the conclusion is obtained.

Case 2. The left premise is derived by R$\Box$. Similar to Case 1.

Case 3. The left premise is derived by Exch. We have two subcases according to the principal formula; the cut formula $p$ is principal formula or not. If the cut formula is not principal, then the derivation is transformed into a derivation with lower cut-height as follows:

Let the hypersequent ${\cal G}$ be as $P^{\prime\prime}\Rightarrow Q^{\prime\prime}\,|\,{\cal I}$ and the last rule Exch be as follows:

$\mathcal{D}$

$M,P^{\prime\prime}\Rightarrow Q^{\prime\prime},N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,P\Rightarrow Q\,|\,{\cal I}$

Exch

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,P^{\prime\prime}\Rightarrow Q^{\prime\prime}\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$.

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,P^{\prime\prime}\Rightarrow Q^{\prime\prime}\,|\,{\cal I}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

This cut is transformed into

$\mathcal{D}$

$M,P^{\prime\prime}\Rightarrow Q^{\prime\prime},N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,P\Rightarrow Q\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P^{\prime\prime}\Rightarrow Q^{\prime\prime}\,|\,P\Rightarrow Q\,|\,{\cal I}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Therefore, let the left premise is derive by Exch where the cut formula $p$ is principal. We have three subcases according to the last rule applied in the right premise.

Subcase 3.1. The right premise is derived by Exch.

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},p,N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

Exch

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Exch

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$.

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

This cut is transformed into the first cut as follows:

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},p,N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

     

$\mathcal{D}^{\prime}$

$M^{\prime},p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Cut

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

Subcase 3.2. The right premise is derived by R$\Box$. Let $N^{\prime}=N^{\prime\prime},\Box A$ and $\Box A$ be the principal formula:

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},p,N\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}$

Exch

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$

     

$\mathcal{D}^{\prime}$

$M^{\prime}\Rightarrow N^{\prime\prime},A\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

R$\Box$

$M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime\prime},\Box A\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}$

${\sf Cut}^{\sf c}$.

$M,P_{i},M^{\prime},P^{\prime}_{i}\Rightarrow Q_{i},N,Q^{\prime}_{i},N^{\prime\prime},\Box A\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal G}^{\prime}$

For this cut, let $\bigwedge_{j=1}^{k}(\bigvee\neg P^{\prime\prime}_{j}\vee\bigvee Q^{\prime\prime}_{j}\vee\bigvee\neg M^{\prime\prime}_{j}\vee\bigvee N^{\prime\prime}_{j})$ be an equivalent formula in CQNF of $A$. Then, by Corollary 5.11, which is height-preserving admissible, we get the following derivation from $\mathcal{D}^{\prime}$

$\mathcal{D}^{\prime}_{1}$

$M^{\prime\prime}_{j},P^{\prime\prime}_{j},M^{\prime}\Rightarrow N^{\prime\prime},Q^{\prime\prime}_{j},N^{\prime\prime}_{j}\,||\,{\cal H}^{\prime}\,|\,p,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime},$

for every clause $(\bigvee\neg P^{\prime\prime}_{j}\vee\bigvee Q^{\prime\prime}_{j}\vee\bigvee\neg M^{\prime\prime}_{j}\vee\bigvee N^{\prime\prime}_{j})$. Then by applying the rule ${\sf Cut}^{\sf c}$, we get the following derivation for every clauses in CQNF of $A$

$M,P\Rightarrow Q,N\,||\,{\cal H}\,|\,P_{i}\Rightarrow Q_{i},p\,|\,{\cal G}$         $\mathcal{D}^{\prime}_{1}$

${\sf Cut}^{\sf c}$

$M,P_{i},M^{\prime\prime}_{j},P^{\prime}_{i},M^{\prime}\Rightarrow Q_{i},N,N^{\prime\prime},Q^{\prime}_{i},N^{\prime\prime}_{j}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}\,|\,{\cal H}^{\prime}\,|\,P^{\prime\prime}_{j}\Rightarrow Q^{\prime\prime}_{j}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

Then, by applying Lemma 5.12, the conclusion is obtained.

Subcase 3.3. The right premise is derived by L$\Diamond$. Similar to Subcase 3.2. ∎

Now, we prove the admissibility of the cut rule for arbitrary formula.

Theorem 6.2**.**

The rule of cut

$\Gamma\Rightarrow\Delta,D\,||\,{\cal H}$         $D,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H^{\prime}}$

Cut,

$\Gamma,\Gamma^{\prime}\Rightarrow\Delta,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H^{\prime}}$

---

where $D$ is an arbitrary formula, is admissible in ${\cal R}_{\sf S5}$.

Proof*.*

The proof proceeds by induction on the structure of the cut formula $D$ with subinduction on the cut-height i.e., the sum of the heights of the derivations of the premises. The admissibility of the rule for atomic cut formula follows from Lemma 6.1, therefore we consider cases where $D$ is not atomic formula. If the cut formula is of the form $\neg A$, $A\wedge B$, $A\vee B$, or $A\rightarrow B$, then using invertibility of the propositional rules, 5.1, the cut rule can be transformed into cut rules where cut formula is reduced, i.e, cut formula is $A$ or $B$. Thus it remains to consider cases, where cut formula is of the form $\Diamond A$ or $\Box A$. We only consider the case where the cut formula is of the form $\Box A$; the other case is proved similarly. We distinguish the following cases:

1. Cut formula $\Box A$ is principal in the left premise only. we consider the last rule applied to the right premise of cut. If the last rule applied is a propositional rule, then the derivation is transformed into a derivation of lower cut-height as usual. Thus we will consider modal rules.

Subcase 1.1. The right premise is derived by R$\Box$. Let $\Gamma^{\prime}=M^{\prime},P^{\prime}$ and $\Delta^{\prime}=Q^{\prime},N^{\prime},\Box B$, and let cut rule be as follows

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Box A,M^{\prime}\Rightarrow N^{\prime},B\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

R$\Box$

$\Box A,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime},\Box B\,||\,{\cal H}^{\prime}$

Cut,

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime},\Box B||\,{\cal H}\,|\,{\cal H}^{\prime}$

where $\Gamma=M,P$ and $\Delta=Q,N$. This cut is transformed into

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$M\Rightarrow N,\Box A\,||\,{\cal H}\,|\,P\Rightarrow Q$

     

$\mathcal{D}^{\prime}$

$\Box A,M^{\prime}\Rightarrow N^{\prime},B\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

Cut

$M,M^{\prime}\Rightarrow N,N^{\prime},B\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

${\sf Merge}^{\sf c}$

$M,M^{\prime}\Rightarrow N,N^{\prime},B\,||\,{\cal H}\,|\,{\cal H}^{\prime}\,|\,P,P^{\prime}\Rightarrow Q,Q^{\prime}$

R$\Box$.

$M,M^{\prime},P,P^{\prime}\Rightarrow Q,Q^{\prime},N,N^{\prime},\Box B\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

Case 1.2. The right premise is derived by L$\Diamond$. This case is treated similar to the above case.

Case 1.2. The right premise is derived by R$\Diamond$. Let $\Delta^{\prime}=\Delta^{\prime\prime},\Diamond B$ and cut rule be as follows

$\mathcal{D}$

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime\prime},\Diamond B,B\,||\,{\cal H}^{\prime}$

R$\Diamond$

$\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime\prime},\Diamond B\,||\,{\cal H}^{\prime}$

Cut,

$M,P,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime\prime},\Diamond B\,\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

where $\Gamma=M,P$ and $\Delta=Q,N$. This cut is transformed into

$\mathcal{D}$

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime\prime},\Diamond B,B\,||\,{\cal H}^{\prime}$

Cut

$M,P,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime\prime},\Diamond B,B\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

R$\Diamond$.

$M,P,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime\prime},\Diamond B\,\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

Case 1.3. The right premise is derived by L$\Box$. This case is treated similar to the above case.

Case 1.4. The right premise is derived by Exch. Let $\Gamma^{\prime}=M^{\prime},P^{\prime}$ and $\Delta^{\prime}=Q^{\prime},N^{\prime}$, and let the hypersequent ${\cal H}^{\prime}$ be as ${\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$:

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Box A,M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$\Box A,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Cut,

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

where $\Gamma=M,P$ and $\Delta=Q,N$. This cut is transformed into

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$M\Rightarrow N,\Box A\,||\,{\cal H}\,|\,P\Rightarrow Q$

     

$\mathcal{D}^{\prime}$

$\Box A,M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Cut

$M,M^{\prime},P^{\prime}_{i}\Rightarrow N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

${\sf Merge}^{\sf c}$

$M,M^{\prime},P^{\prime}_{i}\Rightarrow N,Q^{\prime}_{i},N^{\prime}\,||\,{\cal H}\,|\,{\cal G}^{\prime}\,|\,P,P^{\prime}\Rightarrow Q,Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$M,M^{\prime},P,P^{\prime}\Rightarrow N,N^{\prime},Q,Q^{\prime}\,||\,{\cal H}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

2. Cut formula $\Box A$ is principal in both premises. Let $\Gamma=M,P$ and $\Delta=Q,N$, and let cut rule be as follows

$\mathcal{D}$

$M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q$

R$\Box$

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$A,\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H}^{\prime}$

L$\Box$

$\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H}^{\prime}$

Cut,

$M,P,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

This cut is transformed into

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$

5.15

$M,P\Rightarrow Q,N,A\,||\,{\cal H}$

     

$M,P\Rightarrow Q,N,\Box A\,||\,{\cal H}$       

$\mathcal{D}^{\prime}$

$A,\Box A,\Gamma^{\prime}\Rightarrow\Delta^{\prime}\,||\,{\cal H}^{\prime}$

Cut

$M,P,A,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

Cut

$M,P,M,P,\Gamma^{\prime}\Rightarrow Q,N,Q,N,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H}\,|\,{\cal H}^{\prime}$

LC, RC, EC

$M,P,\Gamma^{\prime}\Rightarrow Q,N,\Delta^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

**3. Cut formula $\Box A$ is not principal in the left premise.

**According to the last rule in the derivation of the left premise, we have subcases. For propositional rules the cut rule can be transformed into a derivation with cut(s) of lower cut-height.

Case 3.1. The left premise is derived by L$\Diamond$. Suppose that $\Gamma=\Diamond C,M,P$ and $\Delta=N,Q$ where $\Diamond C$ is the principal formula. Again, we have the following subcases according to the last rule in derivation of the right premise, and we only consider the modal rules.

Subcase 3.1.1 The right premise is derived by L$\Diamond$. Let $\Gamma^{\prime}=\Diamond B,M^{\prime},P^{\prime}\,$ and $\Delta^{\prime}=Q^{\prime},N^{\prime}$.

$\mathcal{D}$

$C,M\Rightarrow N,\Diamond A\,||\,{\cal H}\,|\,P\Rightarrow Q$

L$\Diamond$

$\Diamond C,M,P\Rightarrow Q,N,\Diamond A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Diamond A,B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

L$\Diamond$

$\Diamond A,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}$

Cut,

$\Diamond C,M,P,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

This cut is transformed into

$\mathcal{D}$

$C,M\Rightarrow N,\Diamond A\,||\,{\cal H}\,|\,P\Rightarrow Q$

     

$\mathcal{D}^{\prime}$

$\Diamond A,B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

L$\Diamond$

$\Diamond A,\Diamond B,M^{\prime}\Rightarrow N^{\prime}\,||\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

Cut

$C,M,\Diamond B,M^{\prime}\Rightarrow N,N^{\prime}\,||\,{\cal H}\,|\,P\Rightarrow Q\,|\,{\cal H}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}$

${\sf Merge}^{\sf c}$

$C,M,\Diamond B,M^{\prime}\Rightarrow N,N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}\,|\,P,P^{\prime}\Rightarrow Q,Q^{\prime}$

L$\Diamond$.

$\Diamond C,M,P,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

Subcase 3.1.2 The right premise is derived by R$\Box$ or Exch. Similar to Subcase 3.1.1.

Subcase 3.1.3 The right premise is derived by L$\Box$. Let $\Gamma^{\prime}=\Box B,M^{\prime},P^{\prime}\,$ and $\Delta^{\prime}=Q^{\prime},N^{\prime}$.

$\mathcal{D}$

$C,M\Rightarrow N,\Diamond A\,||\,{\cal H}\,|\,P\Rightarrow Q$

L$\Diamond$

$\Diamond C,M,P\Rightarrow Q,N,\Diamond A\,||\,{\cal H}$

     

$\mathcal{D}^{\prime}$

$\Diamond A,B,\Box B,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}$

L$\Box$

$\Diamond A,\Box B,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}$

Cut,

$\Diamond C,M,P,\Diamond B,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

This cut is transformed into

$\Diamond C,M,P\Rightarrow Q,N,\Diamond A\,||\,{\cal H}$       

$\mathcal{D}^{\prime}$

$\Diamond A,B,\Box B,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal H}^{\prime}$

Cut,

$\Diamond C,M,P,B,\Box B,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

L$\Box$

$\Diamond C,M,P,\Box B,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal H}\,|\,{\cal H}^{\prime}$

Subcase 3.1.3 The right premise is derived by R$\Diamond$. Similar to Subcase 3.1.2.

Case 3.2. The left premise is derived by R$\Box$. Similar to Case 3.1.

Case 3.2. The left premise is derived by Exch. Suppose the hypersequent ${\cal H}$ be as ${\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}$. We have the following subcases according to the last rule in derivation of the right premise, and we only consider modal rules here.

Subcase 3.2.1 The right premise is derived by Exch. Let the hypersequent ${\cal H}^{\prime}$ be as ${\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$, and let the last rules be as

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},N,\Diamond A\,||\,{\cal G}\,|\,P\Rightarrow Q\,|\,{\cal I}$

Exch

$M,P\Rightarrow Q,N,\Diamond A\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$\Diamond A,M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$\Diamond A,M^{\prime},P^{\prime}\Rightarrow Q^{\prime},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Cut,

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

where $\Gamma=M,P$, $\,\Delta=Q,N$, $\,\Gamma^{\prime}=M^{\prime},P^{\prime}$, and $\Delta^{\prime}=Q^{\prime},N^{\prime}$. This cut is transformed into

$\mathcal{D}$

$M,P_{i}\Rightarrow Q_{i},N,\Diamond A\,||\,{\cal G}\,|\,P\Rightarrow Q\,|\,{\cal I}$

     

$\mathcal{D}^{\prime}$

$\Diamond A,M^{\prime},P^{\prime}_{i}\Rightarrow Q^{\prime}_{i},N^{\prime}\,||\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Exch

$\Diamond A,M^{\prime}\Rightarrow N^{\prime}\,||\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

Cut,

$M,P_{i},M^{\prime}\Rightarrow Q_{i},N,N^{\prime},||\,{\cal G}\,|\,P\Rightarrow Q\,|\,{\cal I}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}\Rightarrow Q^{\prime}\,|\,{\cal I}^{\prime}$

${\sf Merge}^{\sf c}$

$M,P_{i},M^{\prime}\Rightarrow Q_{i},N,N^{\prime},||\,{\cal G}\,|\,P,P^{\prime}\Rightarrow Q,Q^{\prime}\,|\,{\cal I}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal G}^{\prime}\,|\,{\cal I}^{\prime}$

Exch.

$M,P,M^{\prime},P^{\prime}\Rightarrow Q,N,Q^{\prime},N^{\prime}\,||\,{\cal G}\,|\,P_{i}\Rightarrow Q_{i}\,|\,{\cal I}\,|\,{\cal G}^{\prime}\,|\,P^{\prime}_{i}\Rightarrow Q^{\prime}_{i}\,|\,{\cal I}^{\prime}$

Subcase 3.2.2 The right premise is derived by R$\Box$. Similar to Subcase 3.2.1.

Subcase 3.2.3 The right premise is derived by R$\Diamond$ or L$\Box$. Similar to Subcase 3.1.3.

∎

Theorem 6.3**.**

The following are equivalent.

• (1)

The sequent $\Gamma\Rightarrow A$ is S5-valid.

• (2)

$\Gamma\vdash_{\text{S5}}A$.

• (3)

The sequent $\Gamma\Rightarrow A$ is provable in $\cal{R}_{\sf S5}$.

Proof*.*

(1) implies (2) by completeness of S5. (3) implies (1) by soundness of $\cal{R}_{\sf S5}$. We show that (2) implies (3). Suppose $A_{1},\ldots,A_{n}$ is an S5-proof of $A$ from $\Gamma$. This means that $A_{n}$ is $A$ and that each $A_{i}$ is in $\Gamma$, is an axiom, or is inferred by modus ponens or necessitation. It is straightforward to prove, by induction on $i$, that $\vdash\Gamma\Rightarrow A_{i}$ for each $A_{i}$.

Case 1. $A_{i}\in\Gamma$: Let $\Gamma=A_{i},\Gamma^{\prime}$. It can be easily proved that $A_{i},\Gamma^{\prime}\Rightarrow A_{i}$ is derivable in $\cal{R}_{\sf S5}$ by induction on the complexity of $A_{i}$ and using weakening rules.

Case 2. $A_{i}$ is an axiom of S5: All axioms of S5 are easily proved in $\cal{R}_{\sf S5}$. As a typical example, in the following we prove the axiom 5:

$A\Rightarrow A,\Diamond A$

$\sf R\Diamond$

$A\Rightarrow\Diamond A$

$\sf L\Diamond$

$\Diamond A\Rightarrow\Diamond A$

R$\Box$

$\Diamond A\Rightarrow\Box\Diamond A$

R$\rightarrow$.

$\Rightarrow\Diamond A\rightarrow\Box\Diamond A$

Case 3. $A_{i}$ is inferred by modus ponens: Suppose $A_{i}$ is inferred from $A_{j}$ and $A_{j}\rightarrow A_{i}$, $j<i$, by use of the cut rule we prove $\Gamma\Rightarrow A_{i}$:

IH

$\Gamma\Rightarrow A_{j}$

     

IH

$\Gamma\Rightarrow A_{j}\rightarrow A_{i}$

     

$A_{j}\Rightarrow A_{i},A_{j}$         $A_{i}\Rightarrow A_{i}$

L$\rightarrow$

$A_{j}\rightarrow A_{i},A_{j}\Rightarrow A_{i}$

Cut

$A_{j},\Gamma\Rightarrow A_{i}$

Cut

$\Gamma,\Gamma\Rightarrow A_{i}$

LC.

$\Gamma\Rightarrow A_{i}$

Case 4. $A_{i}$ is inferred by necessitation: Suppose $A_{i}=\Box A_{j}$ is inferred from $A_{j}$ by necessitation. In this case, $\vdash_{S5}A_{j}$ (since the rule necessitation can be applied only to premises which are derivable in the axiomatic system) and so we have:

IH

$\Rightarrow A_{j}$

R$\Box$

$\Rightarrow\Box A_{j}$

LW.

$\Gamma\Rightarrow A_{i}$

∎

Corollary 6.4**.**

$\cal{R}_{\sf S5}$* is sound and complete with respect to the S5-Kripke frames.*

7 Concluding Remarks

We have presented the system $\cal{R}_{\sf S5}$, using rooted hypersequent, a sequent-style calculus for S5 which enjoys the subformula property. We have proved the soundness and completeness theorems, and the admissibility of the weakening, contraction and cut rules in the system. In the first draft of this paper, we wrote the rules L$\Diamond$ and R$\Box$ as follows:

$M,A\Rightarrow\Box\bigvee(\neg P,Q),N$

L$\Diamond$

$\Diamond A,M,P\Rightarrow Q,N$

      

$M\Rightarrow\Box\bigvee(\neg P,Q),N,A$

R$\Box$.

$M,P\Rightarrow Q,N,\Box A$

In these rules we can use $\Diamond\bigwedge(P,\neg Q)$ in the antecedent instead of $\Box\bigvee(\neg P,Q)$ in the succedent of the premises, since these formulae are equivalent and have the same role in derivations as storages; they equivalently can be exchanged, or be taken both of them. Taking each of them, one can prove the admissibility of the others. By applying these rules in a backward proof search, although the formulae in the premises are constructed from atomic formulae in the conclusions, the subformula property does not hold. Then, we decided to use semicolon in the middle part of the sequents. Using extra connective semicolon (;) we introduced a new sequent-style calculus for S5, and called it $\text{G3{\scriptsize S5}}^{;}$. Sequents in $\text{G3{\scriptsize S5}}^{;}$ are of the form $\Gamma;P_{1};\ldots;P_{n}\Rightarrow Q_{n};\ldots;Q_{1};\Delta$, where $\Gamma$ and $\Delta$ are multisets of arbitrary formulae, and $P_{i}$ and $Q_{i}$ are multisets of atomic formulae which serve as storages. For convenience, we used $H$ and $G$ to denote the sequence of multisets $P_{1};\ldots;P_{n}$ and $Q_{n};\ldots;Q_{1}$, respectively. Thus, we used $\Gamma;H\Rightarrow G;\Delta$ to denote the sequents. The main idea for constructing this sequent is to take an ordinary sequent $\Gamma\Rightarrow\Delta$ as a root and add two sequences of multisets of atomic formulae to it. The system $\text{G3{\scriptsize S5}}^{;}$ is obtained by extending G3c for propositional logic with the following rules:

[TABLE]

$M,P_{i};H_{1};P;H_{2}\Rightarrow G_{2};Q;G_{1};Q_{i},N$

Exch

$M,P;H_{1};P_{i};H_{2}\Rightarrow G_{2};Q_{i};G_{1};Q,N$

where $M$ and $N$ are multisets of modal formulae. In backward proof search, by applying the rules L$\Diamond$, R$\Box$, and Exch atomic formulae in $P$ and $Q$ in the conclusions move to the middle (storage) parts (between two semicolons) in the premises. These formulae, which are stored in the middle parts of sequents until applications of the rule Exch in a derivation, are called related formulae, and $(P,Q)$ is called a related pair of multisets. By applying the rule Exch, related formulae in $P_{i}$ and $Q_{i}$ come out from the middle part. In other words, multisets $P_{i}$ and $Q_{i}$ exist together from storages by applications of the rule Exch, if they have entered them together previously by applications of the rules L$\Diamond$, R$\Box$, or Exch.

In the following, we prove a simple sequent to show details of this system. Below, $H=G=\emptyset$ and so for convenience we omit their semicolons, also $P=\emptyset$ and $Q=p$ in the rules R$\Box$ and Exch:

$p,\Box p\Rightarrow p$

L$\Box$

$\Box p\Rightarrow p$

Exch

$\Box p;\Rightarrow p;$

R$\neg$

$;\Rightarrow p;\neg\Box p$

R$\Box$

$\Rightarrow p,\Box\neg\Box p$

This system has the subformula property, and we showed that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well.

The intended interpretation of the sequent $\Gamma;P_{1};\ldots;P_{n}\Rightarrow Q_{n};\ldots;Q_{1};\Delta$ is defined as follows

[TABLE]

This interpretation is similar to the standard formula interpretation of both nested sequents ([7]) and grafted hypersequents ([12]). A nested sequent is a structure $\Gamma\Rightarrow\Delta,[\mathcal{N}_{1}],\ldots,[\mathcal{N}_{n}]$, where $\Gamma\Rightarrow\Delta$ is an ordinary sequent and each $\mathcal{N}_{i}$ is again a nested sequent. A grafted hypersequent is a structure $\Gamma\Rightarrow\Delta\parallel\Gamma_{1}\Rightarrow\Delta_{1}\,|\cdots|\Gamma_{n}\Rightarrow\Delta_{n}$, where $\Gamma\Rightarrow\Delta$ is called root or trunk, and each $\Gamma_{i}\Rightarrow\Delta_{i}$ is called a component. The standard formula interpretation of nested sequent is given recursively as

[TABLE]

where $(\mathcal{N}_{i})^{tr}$ is the standard formula interpretation of the nested sequent $\mathcal{N}_{i}$. A grafted hypersequents is essentially the same as the nested sequent $\Gamma\Rightarrow\Delta,[\Gamma_{1}\Rightarrow\Delta_{1}],\ldots,[\Gamma_{n}\Rightarrow\Delta_{n}]$, and the interpretation of grafted hypersequents is adapted from the nested sequent setting as well. Therefore, the sequent $\Gamma;P_{1};\ldots;P_{n}\Rightarrow Q_{n};\ldots;Q_{1};\Delta$, can also be transformed into rooted hypersequent

[TABLE]

where all formulae in each component $P_{i}\Rightarrow Q_{i}$ are atomic formulae. Therefore, we decided to rewrite the sequent in the framework of grafted hypersequent and we call it rooted hypersequent.

Acknowledgement. The authors would like to thank Meghdad Ghari for useful suggestions.