Maehara-style modal nested calculi

We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of non-termination in the proof search present in the case of transitive and Euclidean logics.


Introduction
Ever since Gentzen's LK and LI1 , it is almost considered common knowledge that sequent systems for intuitionistic logic are single conclusion, in other words, one must restrict the succedent to (no more than) one formula.Gentzen [16] himself obtained this as a natural consequence of the natural deduction presentation, which has only one conclusion.In effect, the ability to have several formulas in the succedent was an additional feature introduced by Gentzen to incorporate the principle of the excluded middle.Despite this near-consensus, a multi-conclusion sequent calculus for intuitionistic propositional (and predicate) logic is almost as old as Gentzen's LI.It was proposed by Maehara [17] as the auxiliary calculus L 1 J used to translate intuitionistic reasoning into classical one.It is hard to divine Maehara's thinking: the terse language of results, the whole results, and nothing but results was much in vogue at the time.But on the face of it, his system amounts to an observation that most classical sequent rules remain valid intuitionistically, with only a couple of propositional rules requiring the singleton-succedent restriction in the intuitionistic case.Thus, the blanket restriction of succedent to (at most) singleton sets can be seen as an overreaction.Even the interpretation of the succedent as the disjunction of its formulas is retained in Maehara's calculus.
One possible criticism of this calculus could be that it was introduced as an auxiliary, artificial construct bridging the gap between the natural(-deduction) inspired LI and the fully symmetric LK.This criticism is, however, unfounded.It has been noted (see, for instance, the excellent in-depth survey of various intuitionistic calculi by Dyckhoff [8]) that Maehara's calculus is essentially a notational variant of tableaux from Fitting's PhD thesis [14] (which Fitting himself attributes to Beth [3]).In fact, the same system can be found in [7] and, according to von Plato [25], a similar system was considered by Gentzen himself.
In other words, this calculus is quite natural, has been discovered by several researchers independently, and has a distinction of correlating with the semantic presentation of intuitionistic reasoning much better than LI.Indeed, tableau rules are typically read from the semantics, and Beth-Fitting's destructive tableaux match intuitionistic Kripke models perfectly.It should also be noted that Egly and Schmitt [9] demonstrated that LI cannot polynomially simulate Maehara's calculus, meaning that the latter is more efficient with respect to proof search.
The idea of extending intuitionistic reasoning with modalities is equally natural but less straightforward.There have been multiple approaches over the years with each classical modal logic receiving several alternative intuitionizations.We refer the reader to Simpson's PhD [23] for the discussion of these approaches and concentrate on what eventually became officially known as intuitionistic modal logics.Just like for their classical counterparts, one can talk about the intuitionistic modal cube consisting of 15 logics.And just like their classical brethren, ordinary sequent systems seem inadequate to describe these logics, but nested sequent systems [5] exist for all of them [24].A nested sequent is a tree of ordinary sequents, referred to as sequent nodes.The tableau analog of nested sequents is prefixed tableaux [13], the underlying idea being that the tree structure of a sequent is homomorphically embedded into the accessibility structure of a Kripke model.
The nested sequent calculi in [24,18,6] for intuitionistic modal logics are all globally singleconclusion: exactly one of the sequent nodes is allowed to have a non-empty succedent, and it contains exactly one formula.One can say that they are the modalizations of LI in that the propositional rules are local and identical to those in LI if the rest of the nested structure is ignored.The goal of this paper is to construct a modalization of the Maehara-style calculus with propositional rules conforming to the birelational semantics of intuitionistic modal logics.We formulate the calculi for the 15 intuitionistic modal logics and prove their completeness in two alternative ways: 1. by a syntactic translation from the single-conclusion calculi of [24] and 2. by a direct semantic proof of cut admissibility.
The syntactic-translation method originally employed by Maehara in [17] for translating from multi-conclusion systems to single-conclusion systems is not applicable in the setting of nested d : A Ą 3A @w.Dv. wRv pserialq t : pA Ą 3Aq ^p A Ą Aq @w.wRw preflexiveq b : pA Ą 3Aq ^p3 A Ą Aq @w.@v.wRv Ą vRw psymmetricq 4 : p33A Ą 3Aq ^p A Ą Aq @w.@v.@u.wRv ^vRu Ą wRu ptransitiveq 5 : p3A Ą 3Aq ^p3 A Ą Aq @w.@v.@u.wRv ^wRu Ą vRu pEuclideanq sequents because formulas in the succedent of the sequent can occur in various places in the nested sequent tree, and there is no immediate equivalence to their disjunction as it is the case with ordinary sequents.Also the method by [9] that is based on rule permutations does not work in our case, again, because the formulas in the succedent do not necessarily occur in the same sequent node.For this reason, we prove soundness for our multi-conclusion systems by a direct semantic argument.This paper is organized as follows.First, in Section 2 we recall the syntax and semantics of intuitionistic modal logics, and in Section 3 we present our nested sequent systems, and we show completeness of the multi-conclusion systems by using completeness of the single-conclusion systems shown in [24].Then we show semantically in Sections 4 the soundness of the multiconclusion systems, and finally, in Section 5, we give a semantic argument for the completeness of the multi-conclusion system with respect to the birelational Kripke models.

Syntax and Semantics of Intuitionistic Modal Logics
Definition 2.1 (Language of intuitionistic modal logic).We start from a countable set A of propositional variables (or atoms).Then the set M of formulas of intuitionistic modal logic (IML) is generated by the grammar We denote atoms by lowercase Latin letters, like a, b, c, and formulas by capital Latin letters, like A, B, C. Negation can be defined as A :" A Ą K, and the constant J is defined as J :" K.

Definition 2.2 (Logic IK).
A formula is a theorem of IK, an intuitionistic variant of the modal logic K, if it can be derived from the axioms of intuitionistic propositional logic (IPL) extended with the k-axioms called necessitation and modus ponens respectively.
Note that in the classical case the axioms k 2 -k 5 from (1) would follow from k 1 , but due to the lack of De Morgan duality, this is not the case in intuitionistic logic.
This variant of IK has first been studied in [11,21] and investigated in detail in [23].There exist other intuitionistic variants of K, e.g., [12,22,4,20], the most prominent being the one which has only the axioms k 1 and k 2 from (1).There is now consensus in the literature to call this variant constructive modal logic, e.g., [1,19,2].
Besides the axioms (1) we also consider the axioms d, t, b, 4, and 5 shown in the left column of Figure 1.By adding a subset of these five axioms, we can a priori define 32 different logics.But some of them coincide, so that we get (as in the classical case) only 15 different logics that can be organized in the intuitionistic version of the "modal cube" [15], which is shown in Figure 2. Definition 2.3 (Logics IK `X).For any X Ď td, t, b, 4, 5u, the logic IK `X is obtained from IK by adding all axioms in X.We typically simplify the name of the logic by dropping the plus and capitalizing the names of axioms that are letters.For example, the logic ID45 in Figure 2 is IK `td, 4, 5u.Additionally, IS4 :" IK `tt, 4u and IS5 :" IK `tt, 4, 5u.We write IK `X $ A to state that A is a theorem of IK `X.
The 45-closure of X, denoted by X, is the smallest 45-closed set that contains X.
Let us now recall the birelational models [21,10] for intuitionistic modal logics, which are a combination of the Kripke semantics for propositional intuitionistic logic and for classical modal logic.Definition 2.5 (Birelational semantics).A frame xW, ď, Ry is a non-empty set W of worlds together with two binary relations ď, R Ď W ˆW , where ď is a preorder (i.e., reflexive and transitive), such that the following two conditions hold: (F1) For all w, v, v 1 , if wRv and v ď v 1 , then there is a w 1 such that w ď w 1 and w 1 Rv 1 .
(F2) For all w 1 , w, v, if w ď w 1 and wRv, then there is a v 1 such that w 1 Rv 1 and v ď v 1 .
A (birelational) model M is a quadruple xW, ď, R, V y, where xW, ď, Ry is a frame, and V , called the valuation, is a monotone function from the ď-ordered set xW, ďy of worlds into the set x2 A , Ďy of subsets of propositional variables ordered by inclusion.Valuation V maps each world w to the set of propositional variables that are true in w.We write M, w , a if a P V pwq.The relation , is extended to all formulas as follows: M, w , A Ą B iff for all w 1 ě w we have that M, w 1 , A implies M, w 1 , B; M, w , A iff for all w 1 , v 1 P W with w 1 ě w and w 1 Rv 1 we have M, v 1 , A; M, w , 3A iff there is a v P W such that wRv and M, v , A.
When M, w , A we say that w forces A in M. We write M, w .A (w does not force A in M) if M, w , A does not hold.Whenever omit the name of the model M when it does not create confusion.
In particular, note that w , J for all worlds w in all models.It is easy to show that: Proposition 2.6 (Monotonicity).If w ď w 1 and M, w , A, then M, w 1 , A. Definition 2.7 (Validity for formulas).We say that a formula A is valid in a model M and write M , A if every world in M forces A. Definition 2.8 (X-model and X-validity for formulas).Let X Ď td, t, b, 4, 5u and M " xW, ď, R, V y be a birelational model.If the relation R obeys all frame conditions in the second column of Figure 1 that correspond to the axioms in X, then we call M an X-model.We say that a formula A is X-valid and write X , A if A is valid in every X-model.
The raison d'être of the birelational models is the following theorem, for which a proof can be found in [23].
Theorem 2.9 (Soundness and completeness).For any X Ď td, t, b, 4, 5u, a formula A is a theorem of IK `X if and only if it is valid in all X-models, i.e., If we collapse the relation ď by letting w ď v iff w " v we obtain the standard Kripke models for classical modal logics.

Nested Sequents for Modal Logics
Ordinary one-sided sequents are usually multisets of formulas separated by commas: The intended meaning of such a sequent is given by its corresponding formula Ordinary two-sided sequents are pairs of such multisets of formulas usually separated by the sequent arrow ñ.The corresponding formula of a two-sided sequent is the formula In their original formulation for classical modal logics, nested sequents are a generalization of ordinary one-sided sequents: a nested sequent is a tree whose nodes are multisets of formulas.More precisely, it is of the form A 1 , . . ., A n , rΓ 1 s, . . ., rΓ m s where A 1 , . . ., A n are formulas and Γ 1 , . . ., Γ m are nested sequents.The corresponding formula for the sequent in (5) in the classical case is where fmpΓ i q is the corresponding formula of Γ i for i P t1, . . ., mu.In the following, we just write sequent for nested sequent.
Definition 3.1 (Sequent tree).For a sequent Γ we write tr pΓq to denote its sequent tree whose nodes (called sequent nodes from now on and denoted by lowercase Greek letters, like γ, δ, σ) are multisets of formulas.We slightly abuse the notation and write γ P Γ instead of γ P tr pΓq.
tr pΓq :" A 1 , . . ., A k tr pΓ 1 q tr pΓ 2 q ¨¨¨tr pΓ n q (6) The depth of a sequent is defined to be the depth of its tree.

Inria
For capturing intuitionistic logic, we need "two-sided" nested sequents.For this, we follow [24] and assign each formula in the nested sequent a unique polarity that can be either ‚ for input/left polarity (representing "being in the antecedent of the sequent" or "on the left of the sequent arrow, if there were a sequent arrow"), and ˝for output/right polarity (representing "being in the succedent of the sequent" or "on the right of the sequent arrow, if there were a sequent arrow").Definition 3.2 (Two-sided nested sequent).A two-sided nested sequent is of the shape where B 1 , . . ., B h , C 1 , . . ., C l are formulas and Γ 1 , . . ., Γ m are two-sided nested sequents.
In a classical setting, the corresponding formula of ( 7) is simply However, in the intuitionistic setting, the situation is not as simple.The systems presented in [24,18,6] follow Gentzen's idea of having exactly one formula of output polarity in the sequent.Such a sequent is generated by the grammar Definition 3.3 (Single-conclusion two-sided nested sequent).
In ( 9), Λ stands for a sequent that contains only formulas with input polarity, and Γ for a sequent that contains exactly one formula with output polarity.The corresponding formula of a sequent in ( 9) is defined as follows: fmp∅q :" K , fmpΛ, B ‚ q :" fmpΛq ^B , fmpΛ, rΛ 1 sq :" fmpΛq ^3fmpΛ 1 q , fmpΛ, C ˝q :" fmpΛq Ą C , fmpΛ, rΓsq :" fmpΛq Ą fmpΓq .(10) Unfortunately, it seems impossible to give such a corresponding formula for a multi-conclusion sequent in the intuitionistic setting.For this reason, in the next section we provide an alternative definition of validity for multi-conclusion sequents.The next step is to show the inference rules.But before we can do so, we need to introduce an additional notation.Definition 3.4 (Sequent context).A (sequent) context is a nested sequent with a hole t u, taking the place of a formula.Contexts are denoted by Γt u, and Γt∆u is the sequent obtained from Γt u by replacing the occurrence of t u with ∆.We write ΓtHu for the sequent obtained from Γt u by removing the t u (i.e., the hole is filled with nothing).Definition 3.5 (Multi-conclusion nested sequent calculi NKK `X for classical modal logics).Figure 3 shows the system for the classical modal logic K, which is just the two-sided version of Brünnler's system [5] (see also [18]), extended with the rules for K and Ą. 2 Then, Figure 4 shows the rules for the axioms d, t, b, 4, and 5 from Figure 1.For X Ď td, t, b, 4, 5u we write X ‚ and X ˝to be the corresponding subsets of td ‚ , t ‚ , b ‚ , 4 ‚ , 5 ‚ u and td ˝, t ˝, b ˝, 4 ˝, 5 ˝u respectively.Then we write As usual, we denote derivability in these and other nested sequent calculi by using $.
Theorem 3.6 (Brünnler [5]).For a 45-closed set X Ď td, t, b, 4, 5u, NKK `X is sound and complete w.r.t. the classical modal logic K extended with the axioms X.We can now straightforwardly obtain an intuitionistic variant of the system NKK by demanding that each sequent occurring in a proof contains exactly one output formula.Note that almost all rules in Figures 3 and 1 preserve this property when going from conclusion to premise, and can therefore remain unchanged.There are only two rules that violate this condition: c ˝and Ą ‚ .We therefore forbid the use of c ˝and change Ą ‚ in that we delete the old output formula in the left premise.
with Γ Ó t u standing for the context Γt u with all output formulas removed.For each X Ď td, t, b, 4, 5u, we define Another rule that we use in this paper is Γtr su d rs ´´´Γ t∅u .
We write NIKs `X1 for the system obtained from NIKs `X by replacing the two rules d ‚ and d with the rule d rs if they are present: (In a similar way we can define NKK `X1 .) Theorem 3.8 (Straßburger, [24]).For a 45-closed set X Ď td, t, b, 4, 5u, NIKs `X1 is sound and complete w.r.t. the intuitionistic modal logic IK `X. 3he proof in [24] is done via cut elimination where the cut rule is shown on the left below: The variant of the cut rule on the right above is the version for the systems without the restriction of having only one output formula in a sequent.This brings us to the actual purpose of this paper: multiple-conclusion systems for the logics IK `X, in the style of Maehara [17].Definition 3.9 (Multi-conclusion nested sequent calculi NIKm `X and NIKm `X1 for intuitionistic modal logics).As before, we start from the classical system and define where the rules Ą m and m are given below and Γ Ó t u is defined in Definition 3.7: Then, the systems NIKm `X and NIKm `X1 are defined analogously to NIKs `X and NIKs `X1 .
In all these systems, the weakening rule ΓtHu w ´´´Γ t∆u is depth-preserving admissible: Lemma 3.10 (dp-admissibility of weakening).Let X Ď td, t, b, 4, 5u.Then the weakening rule w is depth-preserving admissible in NKK `X, in NIKs `X, and in NIKm `X, i.e., if ΓtHu has a proof, then Γt∆u has a proof of at most the same depth.
Proof.The proof is a straightforward induction on the depth of the derivation (see [5] for details).
The following lemma clarifies the relationship between the rule d rs and the rules d ‚ and d ˝.
Lemma 3.11.Let X Ď td, t, b, 4, 5u.If d P X then d rs is admissible in NKK `X and in NIKm `X.Furthermore, d ‚ and d ˝are derivable in td rs , ‚ , 3 ˝u.
Proof.The proof of the first statement is by induction on the derivation depth with case distinction based on the last rule used in this derivation.It is obvious that the empty bracket can be removed from any initial sequent.For most rules, the statement for the conclusion easily follows from the IH for the premises.If the bracket to be removed became empty because the last rule was ‚ with ∆ " ∅, then these ‚ followed by d rs can be replaced with d ‚ .This also proves the derivability of d ‚ from ‚ and d rs .Similarly, 3 ˝with ∆ " ∅ followed by d rs can be replaced with d ˝, making the latter derivable from 3 ˝and d rs .The cases for the rules 4 ˝, 4 ‚ , 5 ˝, and 5 ‚ are similar.We only show the transformation for 5 ˝: Note that the first transition is by weakening, which is admissible in all our systems by Lemma 3.10, and that the proviso for both applications of 5 ˝in the transformed derivation is satisfied whenever it is satisfied in the original derivation.
Remark 3.12.Lemma 3.11 fails to hold for NIKs `X because of the absence of c ˝. Below is an example of a derivation from which d rs cannot be eliminated: In all systems presented so far, the identity rule id is restricted to atomic formulas, but the general form is derivable.Proposition 3.13 (Non-atomic initial sequents).For every formula A and every appropriate context Γt u, the sequent ΓtA ‚ , A ˝u is derivable in NKK, in NIKs, and in NIKm.
Proof.By a straightforward induction on A.
Remark 3.14.The appropriateness of the context only plays a role for NIKs, where Γt u is not allowed to contain output formulas.
Maehara shows [17] the equivalence of his multiple conclusion system to Gentzen's single conclusion system from [16] by translating a multiple conclusion sequent into a single conclusion sequent whereby the multiple formulas on the right are replaced by one, their disjunction.This is not possible in the nested sequent setting because "the formulas on the right" are generally scattered all over the sequent tree.
However, one direction is straightforward: Theorem 3.15 (Translation from single-to multi-conclusion).Let X Ď td, t, b, 4, 5u and Γ be a single-conclusion sequent.
Proof.The only rule in NIKs `X (resp.NIKs `X1 ) that is not an instance of a rule in NIKm `X (resp.NIKm `X1 ) is Ą ‚ s .But it can be derived using Ą ‚ and weakening.Thus, the theorem follows from Lemma 3.10.Corollary 3.16.Let X Ď td, t, b, 4, 5u.If a sequent Γ is provable in NIKs `X or in NIKs `X1 , then Γ is also provable both in NIKm `X and in NIKm `X1 .
Proof.This follows immediately from Theorem 3.15 using Lemma 3.11.
Note that in Corollary 3.16, it is implicitly assumed that the sequent Γ has exactly one output formula because otherwise it could not be the endsequent of a correct derivation in NIKs `X or NIKs `X1 .Corollary 3.17 (Formula-level completeness of NIKm `X and NIKm `X1 ).For a 45-closed set X Ď td, t, b, 4, 5u, Proof.If B is X-valid, then B ˝is derivable in NIKs `X1 by Theorem 3.8.Thus, B ˝is derivable both in NIKm `X and in NIKm `X1 by Corollary 3.16.

Semantic Proof of Soundness
In this section we show that every rule in NIKm `X is sound with respect to X-models.For this, we first have to extend the notion of validity from formulas to sequents.
Definition 4.1 (M-map).For a sequent Γ and a birelational model M " xW, ď, R, V y, an M-map for Γ is a map f : tr pΓq Ñ W from nodes of the sequent tree to worlds in the model such that, whenever δ is a child of γ in tr pΓq, then f pγqRf pδq.
Definition 4.2 (Forcing for sequents).A sequent Γ is satisfied by an M-map f for Γ, written f , Γ, iff M, f pγq , A for all A ‚ P γ P Γ ùñ M, f pδq , B for some B ˝P δ P Γ .
If Γ is not satisfied by f , it is refuted by it.
Remark 4.3.This definition works for both single-and multi-conclusion sequents.
Definition 4.4 (X-validity for sequents).For every X Ď td, t, b, 4, 5u, a sequent Γ is X-valid, written X , Γ, iff it is satisfied by all M-maps for Γ for all X-models M. A sequent is X-refutable iff there is an M-map for an X-model M that refutes it.
Lemma 4.5 (Sequent validity extends formula validity).A formula B is X-valid in all X-models if and only if the sequent B ˝is: Proof.This follows immediately from the definition of validity.
Proof.We prove the contrapositive: if Σ is X-refutable, then NIKm `X does not prove Σ.To demonstrate this, it is sufficient to show that, whenever the conclusion of a rule from NIKm `X is X-refutable, then so is at least one of the premises of this rule.Let M " xW, ď, R, V y be an arbitrary X-model and f be an arbitrary M-map for the conclusion Γ of a given rule.Let γ P Γ be the node with the hole of this rule.Since the model M is never modified, we omit its mentions in this proof.Note that an M-map refutes a sequent iff it maps its nodes into worlds of M in a way that makes all input formulas forced and all output formulas not forced.Initial sequents.The statement is vacuously true for K ‚ and id because neither ΓtK ‚ u nor Γta ‚ , a ˝u can be refuted in any birelational model.Local propositional rules _ ‚ , ^‚, Ą ‚ , _ ˝, ^˝.Since propositional rules (including contraction rules) are local in that they act within one node of the sequent tree, the node we called γ, the proof for them is analogous to the case of propositional intuitionistic logic.Namely, for all propositional rules except Ą m, for any birelational model M, any M-map refuting the conclusion must refute one of the premises.Consider, for instance, an instance of the rule _ ‚ and an M-map f that refutes its conclusion ΓtA _ B ‚ u.In particular, it forces all input formulas from Γt u, forces none of output formulas from Γt u (each formula at the world assigned by f ), and satisfies f pγq , A _ B. For the latter to happen, either f pγq , A, making f refute the left premise ΓtA ‚ u, or f pγq , B, in which case it is the right premise ΓtB ‚ u that is refuted by f .Rule Ą m.Assume that all input formulas in the conclusion of the rule ˝u are forced and all output formulas are not forced by an M-map f in their respective worlds, in particular, f pγq .B Ą C. Then there exists a world w ě f pγq where w , B and w .C. It is easy to show using (F1) and (F2) that there exists another M-map g for the conclusion such that gpγq " w and gpδq ě f pδq for each node δ P ΓtB Ą C ˝u.By monotonicity (Proposition 2.6), all input formulas in the conclusion are also forced by g in their respective worlds.Since additionally gpγq , B and gpγq .C, it follows that in the premise all input formulas are forced and the only output formula, C, is not forced by g in their respective worlds.Thus, the constructed g refutes the premise in the same model.Rules ‚ , 3 ˝, t ‚ , t ˝, b ‚ , b ˝, 4 ˝, and 5 ˝.Although these rules are not local in that they affect two nodes of the sequent tree, their treatment is much the same as that of propositional rules: any M-map refuting the conclusion must also refute the premise.Consider, for instance, Assume that all input formulas in the conclusion are forced and all output formulas are not forced by an M-map f in their respective worlds of a transitive model M, in particular, f pγq .3A.Let δ be the node corresponding to the displayed bracket.Note that f pγqRf pδq.Consider any world w such that f pδqRw.Then, by transitivity, f pγqRw and w .A. We have shown that w .A whenever f pδqRw.Thus, f pδq .3A, which is sufficient to demonstrate that f refutes the premise of the rule.We also show the argument for ΓtHut3A ˝u 5 ˝´´´´´´´Γ t3A ˝utHu .
Assume that all input formulas in the conclusion are forced and all output formulas are not forced by an M-map f in their respective worlds of a Euclidean model M, in particular, f pγq .3A for the displayed 3A in the conclusion (here γ is node with the hole containing the principle formula).Let δ be the nodes containing the other hole and ρ be the root of the sequent tree.Then f pρqR k f pγq and f pρqR l f pδq for some k, l ě 0.Moreover, the proviso for the rule demands that k ą 0. Consider any world w such that f pδqRw.Then both f pγq and w are accessible from f pρq in one or more R steps.It is an easy corollary of Euclideanity that f pγqRw, meaning that w .A. We have shown that w .A whenever f pδqRw.Thus, f pδq .3A, which is sufficient to demonstrate that f refutes the premise of the rule.Rules 4 ‚ and 5 ‚ are similar in nature but require an additional consideration in the proof.We explain it on the example of Γtr A ‚ , ∆su 4 ‚ ´´´´´´´Γ t A ‚ , r∆su .
As in the case of 4 ˝, we deal with two nodes: parent γ and its child δ, the latter corresponding to the displayed bracket.We assume that f pγq , A and need to show that f pδq , A. The difference lies in the fact that apart from worlds accessible from f pδq itself, as in the case of 4 we have to consider also worlds accessible from futures of f pδq.However, the condition (F1) and transitivity ensure that any world accessible from a future of f pδq is also accessible from some future of f pγq making it possible to apply the assumption.Rules 3 ‚ , d ‚ , and d ˝.All these rules are similar to the majority of modal rules, except for the fact that one needs to choose a new world for the premise.For rules d ‚ and d ˝, this world is chosen as any world accessible from f pγq by seriality.For the rule 3 ‚ , the assumption is that 3A is forced at f pγq, which implies that there exists an accessible world forcing A, and it is this world that is chosen for the extra node in the sequent tree of the premise.Consider, e.g., an instance of 3 ‚ and assume that f refutes its conclusion Γt3A ‚ u.In particular, f pγq , 3A.Thus, there exists a world w P W such that f pγqRw and w , A. We define an M-map g for the premise ΓtrA ‚ su to act like f on all nodes that are present in the conclusion and to map the node δ corresponding to the displayed bracket to w.Then, just like f , the map g forces all input formula in Γt u and none of output formulas in Γt u and, in addition, gpδq , A, meaning that g refutes the premise.Rule m.Assume that all input formulas in the conclusion of the rule Inria are forced and all output formulas are not forced by an M-map f in their respective worlds, in particular, f pγq .A. Then there exist worlds u and w such that u ě f pγq, and uRw, and w .A.
It is easy to show using (F1) and (F2) that there exists an M-map g for the premise such that gpγq " u, gpδq " w for the node δ present in the premise but not in the conclusion, and gpϑq ě f pϑq for each node ϑ P Γt A ˝u.By monotonicity (Proposition 2.6), all input formulas in the conclusion are also forced by g in their respective worlds.Since additionally gpδq .A, it follows that in the premise all input formulas are forced and the only output formula, A, is not forced by g in their respective worlds.Thus, the constructed g refutes the premise in the same model.
This completes the proof of soundness.

Semantic Proof of Completeness
In this section we show the completeness of our multiple conclusion systems semantically.To simplify the argument, we work with a modified system cNIKm `X1 , that is defined as follows.
For every inference rule in NIKm `X (and NIKm `X1 ), except for Ą m and m, we can define its contraction variant, denoted by the subscript c, that keeps the principal formula of the conclusion in all premises.Below are three examples: Note that K ‚ and K ‚ c are identical (as are d rs and d rs c ).We denote by cNIKm `X1 the system obtained from NIKm `X1 by removing c ‚ and c ˝, and by replacing every rule, except for Ą m and m, with its contraction variant.Definition 5.1 (Equivalent derivations).Two derivations are equivalent if they have the same endsequent.Two systems S 1 and S 2 are equivalent if for every derivation in S 1 , there is a derivation in S 2 of the same endsequent, and vice versa.Proof.Every rule r c is derivable via r and c ‚ or c ˝, and conversely, every rule r is derivable from r c and w.Hence, the statement follows from Lemmas 3.10 and 3.11.
We can now state the completeness theorem: Theorem 5.3 (Completeness).Let X Ď td, t, b, 4, 5u be a 45-closed set, and let Υ be a sequent.
Remark 5.4.Note that this is stronger than the completeness result proved syntactically in Corollary 3.17 which was formulated for single formulas rather than arbitrary sequents.While the argument used to prove Corollary 3.17 extends as is to single-conclusion sequents, the result in this section shows completeness for all multi-conclusion sequents.
The rest of this section is dedicated to the proof of Theorem 5.3, and we let X and Υ be fixed.We prove the contrapositive: if NIKm `X & Υ, then Υ is X-refutable.By Lemma 5.2 we can work with the system cNIKm `X1 , which is equivalent to NIKm `X.We work with the (almost) complete proof search tree T in cNIKm `X1 that is constructed as follows: the nodes of T are sequents, and the root of T is the endsequent Υ.For each possible unary rule application r to a sequent Γ in T the premise of r is a child of Γ in T, and for each possible binary rule application to Γ, both premises of r are children of Γ in T. (Recall that we mean here upward rule applications.)There are only two exceptions: along each branch of the proof search tree 1. each formula 3A ‚ is used no more than once as the principal formula of the rule 3 ‚ c (different occurrences of the same formula in the same sequent node are considered the "same" here) and 2. each sequent node is used no more than once as the principal node of the rule d rs (but, of course, the rule d rs is applied to every node in a sequent Γ).
The countermodel that we are going to construct will be based on the tree T that is obtained from T by removing all subtrees that have derivable sequents as roots.In the following, we use Γ, ∆, etc. to denote sequent occurrences in T rather than sequents.We distinguish three types of unary rules: 1. the leveling rules Ą m and m, which are non-invertible, 2. the node creating rules 3 ‚ c and d rs , and 3. all other unary rules (which are invertible), that we call simple.
Definition 5.5 (Level).The level of Γ (and of every γ P Γ) is the total number of leveling-rule instances on the path from Υ to Γ in T. Sequents Γ and ∆ with the same level are equilevel.
In contrast to the soundness proof, we now distinguish between nodes in the premise and conclusion of the rule, which necessitates the following Definition 5.6 (Corresponding nodes).Let Ω be the set of all sequent nodes of all sequent occurrences in T. We define the correspondence relation « on Ω recursively: if γ and δ can be traced to the same sequent node in the endsequent Υ, then γ « δ; if γ and δ are created by instances of 3 ‚ c with the same principal formula 3A ‚ in nodes γ 1 « δ 1 respectively, then γ « δ; if γ and δ are created by instances of d rs from nodes γ 1 « δ 1 respectively, then γ « δ; if γ and δ are created by applications of m with the same principal formula A ˝in equilevel nodes γ 1 « δ 1 respectively, then γ « δ.
If γ « δ, we also say that γ and δ are corresponding.
Clearly, « is an equivalence relation.It is easy to see that distinct nodes of the same sequent occurrence cannot be corresponding.For a sequent node γ P Ω and a sequent occurrence ∆ we denote by γ ∆ the unique sequent node of ∆ corresponding to γ (if it exists).If γ is the parent of δ in tr pΓq and both γ Σ and δ Σ exist for some sequent occurrence Σ, then γ Σ is the parent of δ Σ in tr pΣq.Definition 5.7 (Superior sequent).We call ∆ a superior of Γ, written Γ Ď ∆, if γ ∆ exists for all γ P Γ and satisfies γ Ď γ ∆ as multisets of formulas.
It is clear that Ď is reflexive and transitive.Definition 5.8 (Corresponding rules).Two instances of the same rule r are called corresponding if they are applied to nodes γ « δ, to the same principal formula in γ and δ (this requirement is dropped for d rs , which has no principal formulas), to corresponding children of γ and δ (for ‚ c , 3 c , 4 ‚ c , and 4 c ), and to corresponding second nodes (for 5 ‚ c and 5 c ).
Clearly, rule correspondence is also an equivalence relation.
Definition 5.9 (Rule transfer).Let r be a rule instance and ∆ be the conclusion of a corresponding rule instance.We denote the first premise of this corresponding rule instance by rp∆q and, in case of binary rules, the second premise by r 1 p∆q.
The following lemma is a direct consequence of the definition: Lemma 5.10 (Corresponding rules for superior sequents).Let Γ be the conclusion of a rule instance r and ∆ be a superior of Γ.

Inria
If r is not node-creating, then ∆ is the conclusion of a corresponding rule instance and rpΓq Ď rp∆q (also r 1 pΓq Ď r 1 p∆q for binary rules).
If r is node-creating, then either 1. a corresponding rule instance has already been used on the path from Υ to ∆ and rpΓq Ď ∆, or 2. ∆ is the conclusion of a corresponding rule instance and rpΓq Ď rp∆q.
In the former case, we define rp∆q :" ∆ to unify the notation.
In the following, we use G to denote an arbitrary subset of the set of sequent occurrences in T. We write G Ď T if all occurrences are taken from T. Definition 5.11 (Confluent sets).A set G Ď T is called confluent iff the following condition is satisfied: for any Γ, ∆ P G, the sequent occurrences Γ and ∆ are equilevel and there is a sequent occurrence Σ P G that is a superior of both Γ and ∆.The set G is maximal confluent if it is confluent and has no proper confluent supersets in T.
It is an immediate corollary of Zorn's Lemma that Lemma 5.12 ("Lindenbaum").Each confluent set can be extended to a maximal confluent set.Definition 5.13.Let G Ď T be a set of sequent occurrences, and let r be a rule instance with conclusion in G. Then we define rpGq :" trp∆q | ∆ P G and rp∆q is definedu .
For binary rules, we use r 1 pGq for the second premises.
Lemma 5.14 (Properties of confluent sets).Let the set G Ď T be confluent, and let r be a rule instance with conclusion in G.
1.If r is unary, then rpGq is confluent, and rpGq Ď T. If r is not a leveling rule, then G Y rpGq is also confluent, and G Y rpGq Ď T.
2. If r is simple and G is maximal confluent, then rpGq Ď G, in other words, maximal confluent sets are closed with respect to applications of simple rules.
3. If r is a binary rule, then at least one of rpGq and r 1 pGq is a confluent set and contained in T, and additionally must be a subset of G if the latter is maximal confluent. Proof.
1. Let rpΠq, rp∆q P rpGq, where Π, ∆ P G.By the confluence of G, there is Σ P G such that Π, ∆ Ď Σ.By Lemma 5.10, rpΠq, rp∆q Ď rpΣq P rpGq . ( This demonstrates that rpGq is confluent. For a non-leveling r, take two sequents from G Y rpGq.If both belong to G or both belong to rpGq, the two sequents have a superior in the same set by its confluence.If Π P G and rp∆q P rpGq, then there is a superior Σ Ě Π, ∆ in G by its confluence.By Lemma 5.10, (14) holds again.Given that rpΠq Ě Π because r is not a leveling rule, rpΣq P rpGq is a superior of both Π and rp∆q.
2. Follows from Clause 1 and the maximality of G.
3. We prove that either rpGq Ď T or r 1 pGq Ď T by contradiction.Otherwise, there would have been Π, ∆ P G such that r T pΠq and r 1 T p∆q are both derivable.For a superior Σ Ě Π, ∆, which would have existed by the confluence of G, both rpΣq and r 1 pΣq would have been derivable by admissibility of weakening (Lemma 3.10), making Σ P G Ď T derivable by r, in contradiction to our assumptions.Whichever of rpGq or r 1 pGq is within T must be confluent (and contained in G for maximal confluent sets) as in Clause 1. Definition 5.15 (Limit).For a confluent set G we define its limit Ĝ as a (possibly infinite) nested sequent tree obtained by taking the quotient of all sequent nodes in G with respect to the equivalence relation «, i.e., tr p Ĝq consists of equivalence classes rγs G over all γ P Γ P G.We define rγs G to be the parent of rδs G in tr p Ĝq iff there are γ 1 P rγs G and δ 1 P rδs G such that γ 1 is the parent of δ 1 in tr pΓq for some Γ P G.For formulas inside, we define Lemma 5.16.Ĝ is well-defined, i.e., tr p Ĝq is a tree.Furthermore, if ρ is the root of Γ P G, then rρs G is the root of tr p Ĝq.
Proof.It immediately follows from Definition 5.6 that ρ 1 « ρ 2 for the roots ρ 1 and ρ 2 of any two sequents Σ 1 , Σ 2 P G because ρ 1 and ρ 2 can be traced down to the root ρ Υ of the endsequent Υ.Thus, rρs G includes all roots of all sequents in G.To show that there is a path from rρs G to any element rγs G P Ĝ for a node γ from Σ P G, it remains to note that there is a path from ρ Σ to γ in tr pΣq and that ρ Σ « ρ.
Further, it is easy to show that for γ « δ with γ from a sequent Σ P G and δ from a sequent Π P G, the path from ρ Σ to γ has the same length as the path from ρ Π to δ.In other words, each edge in tr p Ĝq increases the distance from the root (in each member sequent), which prevents directed cycles.
Finally, we show that each node in tr p Ĝq has at most one parent.Indeed, assume rγs G and rγ 1 s G are both parents of rδs G in tr p Ĝq.This means that γ Σ is the parent of δ Σ and γ 1 Π is the parent of δ Π for some sequent occurrences Σ and Π from the confluent set G. They must have a superior Λ P G. Since all nodes in δ Σ « δ Π « δ Λ , and γ Σ « γ Λ , and γ 1 Π « γ 1 Λ exist by superiority of Λ and since both γ Λ and γ 1 Λ must coincide with the unique parent of δ Λ in tr pΛq, it follows that γ Λ " γ 1 Λ and, consequently, γ Σ « γ 1 Π .In other words, rγs G " rγ 1 s G .
The binary relation R 0 on W is the (disjoint) union of all parent-child relations on Ĝ over all maximal confluent sets G Ď T. The binary relation R on W is defined as the closure of R 0 with respect to the frame properties corresponding to the axioms from the 45-closed X, except for seriality.
For rγs G , rδs H P W , where G and H are maximal confluent sets of sequents, we define rγs G ď 0 rδs H iff for some conclusion Γ P G and premise ∆ P H of a leveling-rule instance r with rpGq Ď H we have γ Γ « δ ∆ .The binary relation ď on W is the reflexive and transitive closure of ď 0 .
For rγs G P W , we define V prγs G q :" ta | a ‚ P rγs G u.
Note that this model construction is a distant relative of the canonical models.Indeed, the structure of the proof-search tree is almost completely ignored: only levels are used to prevent maximal confluent sets from reaching over leveling rules.We use the completeness of the (infinite) proof search to demonstrate the properties of maximal confluent trees, there is no direct translation of rule applications in the proof search to accessibility relation in the model.Proof.The statement for ď follows from that for ď 0 .Assume A ‚ P rγs G and rγs G ď 0 rδs H . Then 1.A ‚ P γ Π for some Π P G and 2. for some conclusion Γ P G and premise ∆ P H of a leveling-rule instance r with rpGq Ď H we have γ Γ « δ ∆ .

Inria
By confluence of G, there is a superior Σ to both Π and Γ.We have A ‚ P γ Π Ď γ Σ and rpΣq P rpGq Ď H. Thus, A ‚ P γ rpΣq .Since Monotonicity.To show monotonicity of V along ď, assume a P V prγs G q and rγs G ď rδs H . Then a ‚ P rγs G and a ‚ P rδs H by Lemma 5.18.Hence, a P V prδs H q.
(F1)-(F2).Since the proofs of these two properties are similar, we only show (F2).We first show (F2) for ď 0 and R 0 .Assume that rγs G ď 0 rδs H and rγs G R 0 rσs G for some rγs G , rσs G , rδs H P W .This means that: 1. for some conclusion Γ P G and premise ∆ P H of a leveling-rule instance r with rpGq Ď H we have γ Γ « δ ∆ and 2. for some Π P G, the node γ Π is the parent of σ Π in tr pΠq.
By confluence of G, there is a superior Σ to both Π and Γ.In tr pΣq, the node γ Σ is the parent of σ Σ .Further rpΣq P rpGq Ď H and γ rpΣq is the parent of σ rpΣq in tr prpΣqq.Since σ rpΣq « σ Σ « σ, Extending (F2) to ď and R 0 is straightforward.It remains to note that, if (F1)-(F2) hold for ď and R 0 , then they hold for ď and R, which is the closure of R 0 with respect to the frame properties of X.This is proved by induction on the length of derivation of an R-link from the R 0 -links.( We can now complete the proof of Theorem 5.3.
Proof of Theorem 5.3.By Lemma 5.12, the endsequent Υ belongs to some maximal confluent set G.
The map f : γ Þ Ñ rγs G embeds tr pΥq into M.By the Truth Lemma 5.21, this map refutes the endsequent.
Proof.If NIKm `X `cut m $ Γ, then Γ is X-valid by Soundness Theorem 4.6 and the obvious fact that cut m preserves validity.Thus, NIKm `X $ Γ by Completeness Theorem 5.3.

Conclusion
In this paper we have presented a multiple-conclusion calculus for all intuitionistic modal logics in the intuitionistic S5-cube, using nested sequents.The observation made by Egly and Schmitt [9], that multiple conclusion calculi for intuitionistic logic can provide exponentially shorter proofs than single-conclusion calculi, does also apply to our case, which makes our calculi interesting for possible applications in proof search.This raises the question whether we can obtain a focused variant for the multiple-conclusion calculus, in the same way as for the single-conclusion calculus in [6].The answer is not as easy as one might expect: due to the non-invertibility of the rules Ą m and m, we have to make the connectives Ą and positive.But in a focused system also 3 has to be positive.On the one hand, due to the absence of De Morgan duality, we certainly can make both modalities positive.But on the other hand, this is against the "spirit of focusing" which demands to make as much as possible negative-the more connectives are negative, the less choices we have to make and the less backtracking is needed.Having a "focused system" in which every connective is positive is trivial and not interesting.
However, there is something more to say about 3. It can be seen as "morally negative" because when the 3 ˝and X ˝-rules are applicable, they can be applied such that no backtracking is needed (using contraction and the multiple conclusion setting).But this is not "negative" in the sense of focusing: we cannot dispose of 3 after the rule application because we might have to wait for an instance of to unfold first.This is a topic of ongoing research.
It remains an open problem whether the multi-conclusion calculi have a formula interpretation.It would also be useful to remove the condition of 45-closure from the completeness proves, in the style of [18].
The semantic proof of completeness required a novel method of model construction.However, both the original proof-search tree and the constructed model are generally infinite.It is an interesting task to attempt to finitize the construction.
it follows that γ rpΣq P rδs H and A ‚ P rδs H .Proof.Clearly, W ‰ ∅.By construction, R satisfies all requisite frame conditions other than seriality, and ď is a preorder.Seriality.If seriality is explicitly required, then d rs is a rule of cNIKm `X1 .By Lemma 5.14, for every rγs G P W , where γ P Γ P G, we have that rpΓq P G for the instance r of d rs applied to γ.Since γ rpΓq must have a child δ in tr prpΓqq, we have rγs G Rrδs G .
Recall that no seriality closure is performed.)Weshowonly(F1)fortheEuclideanclosure:assumingbyIH that the R-links Let rδs G ď rδ 1 s H .By (F1) for rγs G Rrδs G , there is rγ1s H such that rγ 1 s H Rrδ 1 s H and rγs G ď rγ 1 s H .By (F2) for rγs G Rrσs G , there is rσ 1 s H such that rγ 1 s H Rrσ 1 s H and rσs G ď rσ 1 s H . .If A ‚ P rγs G , then A ‚ P rδs G .In addition, if R was obtained by applying (among others) transitive closure to R 0 , then A ‚ P rδs G . 2. If 3A ˝P rγs G , then A ˝P rδs G .In addition, if R was obtained by applying (among others) transitive closure to R 0 , then 3A ˝P rδs G .(transitive case) Both additional statements also hold when the closure included Euclideanity and rγs G is not the root of Ĝ. (Euclidean case) Proof.The last claim is a direct consequence of the closure of G w.r.t. 5 ‚ c and 5 c .The other claims are proved by induction on the length of a minimal derivation of an R-link from the R 0 -links.For the remaining statements, consider A ‚ because 3A ˝is completely analogous.R 0 -links.For R 0 -links, A ‚ P rδs G follows from the closure of maximal confluent sets w.r.t.‚ c .For transitive logics, additionally A ‚ P rδs G because of 4 ‚ c .Reflexive closure.If the link rγs G Rrγs G is obtained by reflexivity, then A ‚ P rγs G trivially and A ‚ P rγs G because of t ‚ c .Symmetric closure.If the link rγs G Rrδs G is obtained by symmetry from rδs G Rrγs G , then by minimality it is neither an R 0 -link nor a reflexive loop.In the absence of transitivity closure, rδs G R 0 rγs G by minimality: if rδs G Rrγs G were added by Euclideanity from rσs G Rrδs G and rσs G Rrγs G , then adding rγs G Rrδs G instead would have been shorter.Thus, A ‚ P rδs G because of b ‚ c .In the transitive case, 5 P X by 45-closure.If neither rγs G nor rδs G is the root of Ĝ, then A ‚ is in rδs G and its parent by 5 ‚ c and A ‚ P rδs G by ‚ c .If rγs G is the root, both A ‚ and A ‚ belong to all nodes of Ĝ by ‚ c , 4 ‚ c , and b ‚ c (for A ‚ P rγs G ).If rδs G but not rγs G is the root, then A ‚ is in rδs G and all its children, which exist, by 5 ‚ c and A ‚ is in rδs G by b ‚ c .Transitive closure.If the link rγs G Rrδs G is obtained by transitivity from rγs G Rrσs G and rσs G Rrδs G , then both of them have shorter derivations and, by IH, A ‚ , A ‚ P rσs G .Hence, by IH, both A ‚ , A ‚ P rδs G .Euclidean closure.Assume the link rγs G Rrδs G is obtained by Euclideanity from rσs G Rrγs G and rσs G Rrδs G .It is sufficient to show that A ‚ is present in all nodes of Ĝ, including rσs G and rδs G , from which the main statement follows by IH from rσs G Rrδs G :If rγs G is not the root of Ĝ, then A ‚ is in all nodes by 5 ‚ c .If rγs G is the root, then we claim that transitivity must also hold.Otherwise, by 45-closure of X, none of reflexive, symmetric or transitive closure would apply to R and Euclidean closure alone would not have added any incoming links into the root rγs G .This means that transitive closure has also been applied.Therefore, from A ‚ P rγs G it immediately follows by 4 ‚ c that A ‚ is present in all nodes.Lemma 5.21 (Truth Lemma).If C ‚ P rγs G , then rγs G ( C; if C ˝P rγs G , then rγs G * C.Proof.The proof is reasonably standard and relies on Lemma 5.20 for input 's and output 3's, as well as on Lemma 5.18 for input Ą's and 's.The cases for C ‚ " a ‚ , C ‚ " K ‚ , andC ˝" K Case C ˝" a ˝.If a ˝P rγs G , then a ‚ R rγs G .Indeed, if a ‚ P γ Π and a ˝P γ ∆ for some Π, ∆ P G, then any superior Σ of Π and ∆, would be derivable due to both a ‚ , a ˝P γ Σ , whereas the confluent G must contain a non-derivable superior of Π and ∆.Case C ‚ " A ^B‚ .If A ^B‚ P rγs G , then both A ‚ ,B ‚ P rγs G by Lemma 5.14.Thus, rγs G ( A and rγs G ( B by IH, making rγs G ( A ^B. Case C ˝" A _ B ˝is similar.Case C ˝" A ^B˝.If A ^B˝P rγs G , then either A ˝P rγs G or B ˝P rγs G by Lemma 5.14.Thus, either rγs G * A or rγs G * B by IH, making rγs G * A ^B. ‚ " A _ B ‚ is similar.Case C ˝" A Ą B ˝.If A Ą B ˝P rγs G , then there is a maximal confluent set H Ě rpGq with rγ rpΓq s H ě 0 rγs G for the application r of Ą ˝to A Ą B ˝in some γ Γ for some Γ P G.In that case, A ‚ , B ˝P rγ rpΓq s H . Thus, by IH rγ rpΓq s H ( A and rγ rpΓq s H * B making rγs G * A Ą B. Case C ‚ " A Ą B ‚ .Let A Ą B ‚ P rγs G and rδs H ě rγs G .By monotonicity of input formulas (Lemma 5.18), A Ą B ‚ P rδs H .By Lemma 5.14, either A ˝P rδs H or B ‚ P rδs H . Thus, for any rδs H ě rγs G , we have by IH that either rδs H * A or rδs H ( B. Thus, rγs G ( A Ą B. Case C ‚ " 3A ‚ .If 3A ‚ P rγs G , then by Lemma 5.14, there is another sequent ∆ P G with A ‚ P δ for the child δ of γ ∆ in tr p∆q.Thus, rδs G ( A by IH.Since rγs G R 0 rδs G , we have rγs G ( 3A.Case C ˝" 3A ˝.Let 3A ˝P rγs G and rγs G Rrδs G .Then, by Lemma 5.20, we have A ˝P rδs G .Thus, by IH rδs G * A whenever rγs G Rrδs G .We have shown that rγs G * 3A.Case C ˝" A ˝.If A ˝P rγs G , then there is a maximal confluent set H Ě rpGq with rγ rpΓq s H ě rγs G for the application r of ˝to A ˝in some γ Γ for some Γ P G.In that case, A ˝P rδs H for the child δ of γ rpΓq in tr prpΓqq.Thus, by IH rδs H * A. Since rγs G ď 0 rγ rpΓq s H R 0 rδs H , we have rγs G * A. Case C ‚ " A ‚ combines the monotonicity argument for input implications with the use of Lemma 5.20 for output diamonds. Finally, we have rδ 1 s H Rrσ 1 s H by Euclideanity.The cases for transitivity, reflexivity, and symmetry are similar.Lemma 5.20 (Modal saturation).Assume that rγs G Rrδs G in the constructed model M.1are trivial.Inria Case C