Coalgebraic Simulations and Congruences
Abstract
In a recent article Gorín and Schröder [3] study \(\lambda \)simulations of coalgebras and relate them to preservation of positive formulae. Their main results assume that \(\lambda \) is a set of monotonic predicate liftings and their proofs are settheoretical. We give a different definition of simulation, called strong simulation, which has several advantages:

\(\lambda \) is monotonic

every simulation is strong

every bisimulation is a (strong) simulation

every Fcongruence is a (strong) simulation.

if \(\lambda \) is a separating set, then each difunctional strong simulation is an \(F\)congruence,

if \(\lambda \) is monotonic, then the converse is true: if each difunctional strong simulation is an \(F\)congruence, then \(\lambda \) is separating.
1 Introduction
Coalgebraic logic as introduced by D. Pattinson [6] and refined by L. Schröder [9], has been very successful in providing a common framework for quite a variety of modal logics, see for instance [2, 5], or [11]. In many cases, the type functor, used to model such coalgebras preserves weak pullbacks, so logical equivalence can be modeled by structural relations called bisimulations. Two states related by a bisimulation are equivalent. In a recent paper Gorín and Schröder have introduced a notion of \(\lambda \)simulation, where \(\lambda \) is a (set of) predicate lifting(s). Their definition is set theoretical and their proofs are calculational. In all of their results they assumed that all predicate liftings are monotonic.
Here we offer a different notion of simulation, which we call strong simulation. The definition is amenable to diagrammatical reasoning, whose utility we show in a number of proofs. Moreover, we show that under the assumption of monotonicity our definition coincides with that of [3]. Since they used monotony as a general hypothesis in their work, their results could be proved as well with our definition. We relate our strong simulations to the notion of AczelMendler bisimulation (called \(F\)bisimulation) and to (generalized) congruences.
2 Basic Notions and Preparations
Given a binary relation \(R\subseteq A\times B\), let \(R^{}\subseteq B\times A\) be the converse relation. If \(S\subseteq B\times C\) is another relation, then \(R\circ S:=\{(a,c)\mid \exists b\;{\in }\; B.aRb\wedge bSc\}\) is called the composition of \(R\) and \(S\). Obviously, \(\circ \) is associative and \((R\circ S)^{}=S^{}\circ R^{}\).
2.1 Difunctionality
Difunctional relations are generalizations of equivalence relations, for the case of relations \(R\subseteq A\times B\) between possibly different sets. Reflexivity, symmetry and transitivity make no sense for such relations, so a possible generalization is:
Definition 1
Immediately from the definition we see ([7]):
Lemma 1
\(R\) is difunctional \(\iff R\circ R^{}\circ R\subseteq R \iff R^{}\circ R\circ R^{}\subseteq R^{} \iff R^{}\) is difunctional. The difunctional closure of a relation \(R\) is obtained as \(R^{d}:=R\circ (R^{}\circ R)^{\star }=(R\circ R^{})^{\star }\circ R\).
Lemma 2
A relation \(R\subseteq A\times B\) is difunctional, if and only if there are maps \(f:A\rightarrow C\), \(g:B\rightarrow C\) with \(R=ker(f,g)\).
Proof
More generally, if \(f:A\rightarrow C\) and \(g:B\rightarrow D\), then any difunctional relation \(R\subseteq C\times D\) gives rise to a difunctional relation \(ker(f,g)_{R}:=\{(a,b)\mid f(a)\, R\, g(b)\}\subseteq A\times B\}\).
2.2 Directed Diagrams
It is sometimes convenient to write \(a\;{\models }\;\theta \) rather than \(\theta (a)=1\) or \(a\;{\in }\;\theta \). Similarly, \(A\;{\models }\;\theta \) means that \(a\;{\models }\;\theta \) for each \(a\;{\in }\; A.\)
Definition 2
Given a relation \(S\) between sets \(A\) and \(B\) and predicates \(\theta :A\rightarrow 2\) and \(\psi :B\rightarrow 2\), we introduce
Lemma 3
 1.
\(\theta '\subseteq \theta \mathop {\Longrightarrow }\limits ^{S}\psi \) implies \(\theta '\mathop {\Longrightarrow }\limits ^{S}\psi \)
 2.
\(\theta \mathop {\Longrightarrow }\limits ^{S}\psi \subseteq \psi '\) implies \(\theta \mathop {\Longrightarrow }\limits ^{S}\psi '\)
 3.
\(\theta \mathop {\implies }\limits ^{R}\varphi \) and \(\varphi \mathop {\implies }\limits ^{S}\psi \) implies \(\theta \mathop {\implies }\limits ^{R\circ S}\psi \)
Proof
3 Functors, Coalgebras and Bisimulations
Let \(F:Set\rightarrow Set\) be an endofunctor on the category of sets. We shall write \(F(X)\) for the action of \(F\) on an object \(X\) and \(Ff\) for the action of \(F\) on a map \(f\).

\(F\) defines a type of “constructions”.

Elements of \(F(X)\) are those “constructions” whose elements are drawn from a set \(X\); we will call them \(Xpatterns\).

Given a map \(f:X\rightarrow Y,\) the map \(Ff:F(X)\rightarrow F(Y)\) acts on an \(X\)pattern \(p\;{\in }\; F(X)\) by replacing in \(p\) each \(x\) by \(f(x)\).

A pattern \(p\;{\in }\; F(X)\) is finite, if there is a subset \(\{x_{1},\ldots ,x_{n}\}\subseteq X\) such that \(p\;{\in }\; F(\{x_{1},\ldots ,x_{n}\})\). In this case, we write \(p=p(x_{1},\ldots ,x_{n})\) and we let \(p(f(x_{1}),\ldots ,f(x_{n}))\) denote \((Ff)p(x_{1},\ldots ,x_{n})\).

In particular, if \(\theta :X\rightarrow 2\) is a predicate, then \(F\theta \) acts on an element \(p\;{\in }\; F(X)\) by replacing in \(p\) each \(x\) by \(1\) if \(x\;{\models }\;\theta \) and by \(0\) otherwise.

If \(p=p(x_{1},\ldots ,x_{n})\), then \((F\theta )p(x_{1},\ldots ,x_{n})=p(\theta (x_{1}),\ldots ,\theta (x_{n}))\) is called a \(01pattern\).
3.1 Coalgebras
Definition 3
An \(F\)coalgebra \(\mathcal {A}=(A,\alpha )\) consists of a set \(A\) and a map \(\alpha :A\rightarrow F(A)\). \(A\) is called the base set and \(\alpha \) the structure map. The functor \(F\) is called the type of coalgebra \(\mathcal {A}\).
We shall keep \(F\) fixed and consider only coalgebras of that given type \(F\).
Definition 4
A map \(\varphi :A\rightarrow B\) between two coalgebras \(\mathcal {A}=(A,\alpha )\) and \(\mathcal {B}=(B,\beta )\) is called a homomorphism, if \(\beta \circ \varphi =F\varphi \circ \alpha \).
The functor properties immediately guarantee that the class of all \(F\)coalgebras with homomorphisms as morphisms forms a category \(Set_{F}\). The forgetful functor \(U:Set_{F}\rightarrow Set\) which associates with every coalgebra \(\mathcal {A}\) its underlying set \(A\) and with every homomorphism its underlying map is known to create and preserve colimits [8], so in particular the category \(Set_{F}\) is cocomplete and colimits have the same underlying set and mappings as the corresponding colimits in \(Set.\)
Example 1
Kripke frames are coalgebras of type \(\mathbb {P}\) where \(\mathbb {P}\) is the covariant powerset functor, acting on a map \(f:X\rightarrow Y\) as \(\mathbb {P}f:\mathbb {P}(X)\rightarrow \mathbb {P}(Y)\) where \((\mathbb {P}f)(U):=f[U]:=\{f(u)\mid u\;{\in }\; U\}\) for any \(U\;{\in }\;\mathbb {P}(X)\).
Kripke structures come with a fixed set \(V\) of atomic properties, so they are modeled as coalgebras of type \(\mathbb {P}()\times \mathbb {P}(V)\), where the second component is simply a constant. A coalgebra of type \(\mathbb {P}()\times \mathbb {P}(V)\) is therefore a base set \(A\) with a structure map \(\alpha :A\rightarrow \mathbb {P}(A)\times \mathbb {P}(V)\). Its first component associates to a state \(a\;{\in }\; A\) the set of its successors \(succ_{A}(a):=(\pi _{1}\circ \alpha )(a)\) and its second component yields the set of all atomic values \(val_{A}(a):=(\pi _{2}\circ \alpha )(a)\) which are true for \(a\).
Homomorphisms \(\varphi :A\rightarrow B\) between Kripke frames, resp. Kripke structures are also known as bounded morphisms. They are maps preserving and reflecting successors and atomic values in the following sense: \(\varphi [succ_{A}(a)]=succ_{B}(\varphi (a))\) and \(val_{A}(a)=val_{B}(\varphi (a))\).
3.2 Bisimulations
In the structure theory of coalgebras, bisimulations play the role of compatible relations.
Definition 5
([1]) A bisimulation between coalgebras \(\mathcal {A}\) and \(\mathcal {B}\) is a relation \(R\subseteq A\times B\) for which there exists a coalgebra structure \(\rho :R\rightarrow F(R)\) such that the projections \(\pi _{A}^{R}:R\rightarrow A\) and \(\pi _{B}^{R}:R\rightarrow B\) are homomorphisms.
Typical bisimulations are graphs of homomorphisms \(G(\varphi ):=\{(a,\varphi (a))\mid a\;{\in }\; A\}\). In fact, a map \(f:A\rightarrow B\) is a homomorphism iff its graph is a bisimulation ([8]). If \(R\subseteq A\times B\) is a bisimulation between coalgebras \(\mathcal {A}\) and \(\mathcal {B}\), then there could be several possible structure maps \(\rho :R\rightarrow F(R)\) establishing that \(R\) is a bisimulation.
The empty relation \(\emptyset \subseteq A\times B\) is always a bisimulation and (more generally) the union of bisimulations is a bisimulation, so that bisimulations between \(\mathcal {A}\) and \(\mathcal {B}\) form a complete lattice with largest element called \(\sim _{\mathcal {A},\mathcal {B}}.\)
The following proposition will be needed later in the proof of Theorem 3. It shows that bisimulations can be enlarged as long as the structure maps are not affected in the following sense:
Proposition 1
Let \(\mathcal {A}_{1}\) and \(\mathcal {A}_{2}\) be coalgebras with corresponding structure maps \(\alpha _{1}\) and \(\alpha _{2}\). Let \(R\subseteq \mathcal {A}_{1}\times \mathcal {A}_{2}\) be a bisimulation and \(R'\) an enlargement i.e. \(R\subseteq R'\subseteq ker\,\alpha _{1}\circ R\circ ker\,\alpha _{2}\). Then \(R'\) is also a bisimulation.
Proof
Corollary 1
Let \(\mathcal {A}=(A,\alpha )\) be a coalgebra, then every reflexive relation \(R\subseteq ker\,\alpha \) is a bisimulation.
Proof
Since \(\varDelta \subseteq A\) is always a bisimulation, we have \(\varDelta \subseteq R\subseteq ker\,\alpha =ker\,\alpha \circ ker\,\alpha =ker\,\alpha \circ \varDelta \circ ker\,\alpha \), because \(ker\,\alpha \) is transitive.
3.3 Predicate Liftings and Boxes
We denote the contravariant powerset functor by \(2^{}.\) Thus \(2^{X}\) is the set of all subsets of \(X\) and a map \(f:X\rightarrow Y\) induces a map \(2^{f}:2^{Y}\rightarrow 2^{X}\) via \(2^{f}(V):=f^{1}[V]\). If we consider the elements of \(2^{Y}\) as predicates \(\tau :Y\rightarrow 2\), we can write \(2^{f}(\tau )=\tau \circ f\), or \(2^{f}=()\circ f\).
The classical Kripke style modal logic introduces formulae expressing properties holding for all successors of a point \(x\). If \(\varphi \) is a state formula then \(\square \varphi \) holds at \(x\) if \(\varphi \) holds for each successor \(x'\) of \(x.\) The set of all successors of a point \(x\) is \(\alpha (x)\;{\in }\;\mathbb {P}(X)\), in the case of Kripke frames. Thus \(\square \) can be understood as lifting a property \(\varphi \) from the base set \(A\) to a property \(\lambda _{A}(\varphi )\subseteq \mathbb {P}(A)\), so \(x\;{\models }\;\square \varphi \) iff \(\alpha (a)\) satisfies the lifted property \(\lambda _{A}(\varphi )\). Generalizing this observation, Pattinson [6] introduced predicate liftings \(\lambda _{A}:2^{A}\rightarrow 2^{F(A)}\) as natural transformations between the contravariant powerset functors \(2^{()}\) and \(2^{F()}.\)
Definition 6
A predicate lifting \(\lambda \) for \(F\) is a natural transformation \(\lambda :2^{}\rightarrow 2^{F()}\) where the latter is the composition of the functor \(F\) with \(2^{}\). For each \(X\) denote by \(\lambda _{X}\) its \(X\)component \(\lambda _{X}:2^{X}\rightarrow 2^{F(X)}\).
The idea is that every property for elements of a set \(X\) is transformed to a property for elements of \(F(X)\).
By the Yoneda lemma, such a natural transformation \(\lambda \) is uniquely determined by the action of \(\lambda _{2}\) on the input \(id_{2}\) where \([\![id_{2}]\!]=\{1\}\subseteq 2\), i.e. by \(\lambda _{2}(id_{2}):F(2)\rightarrow 2,\) which is a predicate on \(F(2)\). This was observed in [9]. We shall from now on write \([\lambda ]\) or simply \(\square \), if \(\lambda \) is understood, for this predicate.
Conversely, given a predicate \(\square :F(2)\rightarrow 2\) on \(F(2)\), then \(\theta \mapsto \square \circ F\theta \) defines a predicate transformer, and it is easy to see that \(id_{2}\) is sent to \(\square \) again.
Intuitively, we think of \(\square \subseteq F(2)\) as a selection of \(01patterns.\) The map \(\lambda _{A}\) of the corresponding predicate transformer \(\lambda \), when applied to \(\theta \;{\in }\;2^{A}\) takes an \(A\)pattern \(p(a_{1},\ldots ,a_{n})\;{\in }\; F(A)\) to \(1\) if \(p(\theta (a_{1}),\ldots ,\theta (a_{n}))\;{\in }\;\square \), and to \(0\) otherwise.
Definition 7
Given a predicate \(\theta \) on \(\mathcal {A}=(A,\alpha )\), denote by \(\square \theta \) the predicate \(\square \circ F\theta \circ \alpha \), that is for any \(a\;{\in }\; A\) we define
3.4 Coalgebraic Modal Logic
4 Simulations

if \(\lambda \) is monotonic then bisimulations are \(\lambda \)simulations

if \(\lambda \) is monotonic, then each \(\lambda \)simulation preserves positive formulae.

proofs are diagrammatical

monotonicity need not be assumed.
4.1 Strong Simulations
Lemma 4
Strong simulations are closed under unions and relational composition, i.e. if \(R\subseteq A\times B\) and \(S\subseteq B\times C\) are strong simulations, then so is \(R\circ S\subseteq A\times C\).
Proof
Closure under unions is easily checked. For closure under relational composition, let \(\theta \mathop {\Longrightarrow }\limits ^{R\circ S}\psi \) be given. Obviously, then \(\theta \mathop {\Longrightarrow }\limits ^{R}R[\theta ]\) and \(R[\theta ]\mathop {\Longrightarrow }\limits ^{S}\psi \). Assuming that \(R\) and \(S\) are simulations, we obtain \(\square \theta \mathop {\Longrightarrow }\limits ^{R}\square R[\theta ]\) as well as \(\square R[\theta ]\mathop {\Longrightarrow }\limits ^{S}\square \psi \), so Lemma 3 yields \(\square \theta \mathop {\Longrightarrow }\limits ^{R\circ S}\square \psi \).
Simulations have a preferred direction. This is emphasized by the following logical fact:
Theorem 1
Strong simulations preserve positive formulae.
Proof
Let \(S\) be a strong simulation between coalgebras \(\mathcal {A}\) and \(\mathcal {B}\), and \((x,y)\;{\in }\; S\). By structural induction, we show that for any positive formula \(\phi \) we have: \(x\;{\models }\;\phi \implies y\;{\models }\;\phi ,\) that is we need to show \(\phi _{\mathcal {A}}\mathop {\implies }\limits ^{S}\phi _{\mathcal {B}}\). The only interesting case is when \(\phi =\square \psi \) with \(\psi \) another positive formula. Let \(\psi _{\mathcal {A}}\), resp \(\psi _{\mathcal {B}}\) be the predicates defined by \(\psi \) in \(\mathcal {A}\), resp \(\mathcal {B}\). By assumption then, \(\psi _{A}\mathop {\implies }\limits ^{S}\psi _{B}\), whence the definition of simulation yields \(\square \psi _{A}\mathop {\implies }\limits ^{S}\square \psi _{B}\), hence (\(\square \psi )_{A}\mathop {\implies }\limits ^{S}(\square \psi )_{B}\).
By a (strong) bidirectional simulation we understand a (strong) simulation \(S\) for which \(S^{}\) is also a simulation. We must be careful not to confuse this with the notion of bisimulation.
From Lemmas 1 and 4 we obtain:
Lemma 5
Let \((S_{i})_{i\;{\in }\; I}\) be a family of bidirectional simulations, then their difunctional closure is again a bidirectional simulation.
4.2 Monotonicity
Definition 8
A predicate lifting \(\lambda \) is called monotonic, if for all sets \(U,V,A\) with \(U\subseteq V\subseteq A\) one has \(\lambda _{A}(U)\subseteq \lambda _{A}(V)\). We say that \(\square :F(2)\rightarrow 2\) is monotonic, if the predicate lifting given by \(\square \) is monotonic.
We get the following characterization:
Lemma 6
\(\square :F(2)\rightarrow 2\) is monotonic, iff for any \(A\) and any predicates \(\theta ,\psi \) on \(A\) with \(\theta \implies \psi \), we obtain \(\square \circ F\theta \implies \square \circ F\psi \).
Proof
Suppose \(\lambda _{A}=\square \circ F()\) is monotonic, \(\theta \implies \psi \) and \(\square \circ F\theta =1\), that is \(\lambda _{A}(\theta )=1\). By monotonicity, \(\lambda _{A}(\psi )=1\), i.e. \(\square \circ F\psi =1.\) Conversely, assume \(U\subseteq V\subseteq A\) and \(u\;{\in }\;\lambda _{A}(U)\), where \(\lambda _{A}(U)=[\![\square \circ F\chi _{U}]\!]\). Then \(\chi _{U}\implies \chi _{V}\) and \((\square \circ F\chi _{U})(u)=1\) whence by assumption \((\square \circ F\chi _{V})(u)=1\), meaning \(u\;{\in }\;\lambda _{A}(V).\) Thus \(\lambda _{A}\) is monotonic.
Lemma 7
Proof
This means that monotonicity needs only be checked for \(\theta =\chi _{\{x\}}\) and \(\psi =\chi _{\{x,y\}}\), which translates immediately into the statement \(p(1,0,0)\;{\in }\;\square \implies p(1,1,0)\;{\in }\;\square \) for each \(p\;{\in }\; F(\{x,y,z\})\).
Theorem 2
\(\square \) is monotonic iff each simulation is strong.
Proof
Theorem 3
 1.
\(\square \) is monotonic
 2.
each bisimulation is a simulation
 3.
each bisimulation is a strong simulation
Proof
5 Congruences and Separability
5.1 Congruences
In classical examples of coalgebras, such as Kripke structures, deterministic and nondeterministic automata, etc., observational equivalence is definable via bisimulations. The reason is that the corresponding type functors preserve weak pullbacks (see [4]). This in turn has many structural consequences. In particular the largest bisimulation is always the same as the largest congruence relation, where a congruence is defined as the kernel of a homomorphism. Thus a congruence is a relation on a single coalgebra. Since we want to study relations between different coalgebras, we have to widen the notion of congruence and therefore introduce the notion of \(F\)congruence. This notion has been studied by Sam Staton under the name kernel bisimulation [12]:
Definition 9
Theorem 4
The following are equivalent:
 1.
\(\square \) is monotonic
 2.
each congruence is a simulation
 3.
each \(F\)congruence is a strong simulation.
Proof
(1.\(\rightarrow \)3.): An \(F\)congruence \(\theta =ker(\varphi ,\psi )\) can be obtained as a composition of relations: \(\theta =G(\varphi )\circ G(\psi )^{}\) where \(G(\varphi )\) and \(G(\psi )\) are the graphs of \(\varphi \) and \(\psi .\) The graphs of homomorphisms are bisimulations ([8]) and the converse of a bisimulation is a bisimulation. Assuming monotonicity of \(\square \), Theorem 3 tells us that they are strong simulations. By Lemma 4, their composition is a strong simulation. In particular, each congruence is a simulation, too. (3.\(\rightarrow \)2) is of course trivial, since each congruence is an \(F\)congruence and each strong simulation is a simulation.
For (2.\(\rightarrow \)1.), assuming that each congruence is a simulation, we can reuse the proof of (3\(\rightarrow \)2) in Theorem 3. This time, we only need to observe that \(R\) happens to be a congruence relation, since it is the kernel of the obvious homomorphism from \(\mathcal {A}_{p}=\mathcal {A}_{p(x,y,z)}\) to the constant coalgebra \(\mathcal {A}_{p(x,x,z)}\) on \(\{x,z\}\).
5.2 Separability
In this section we need to work with a family of boxes \((\square _{i})_{i\;{\in }\; I}\). Such is usually required in order to render coalgebraic modal logic expressive. Separability is usually expressed for the functor and for the boxes separately. A functor is called \(2\)separable, if for any \(X\) and any \(p,q\;{\in }\; F(X)\) with \(p\ne q\) there is a predicate \(\phi :X\rightarrow 2\) such \((F\phi )(p)\ne (F\phi )(q)\). Next, we call a family \((\square _{i})_{i\;{\in }\; I}\) of predicate liftings separating, if the functor \(F\) is 2separating and the predicates \(\square _{i}:F(2)\rightarrow 2\) combined with the unary boolean operations \(\theta :2\rightarrow 2\) form a monosource. We can equivalently define this as follows:
Definition 10
Theorem 5
If \((\square _{i})_{i\;{\in }\; I}\) is separating then every difunctional bidirectional strong simulation is an \(F\)congruence.
Proof
Theorem 6
If each difunctional simulation is an \(F\)congruence, then \((\square _{i})_{i\;{\in }\; I}\) is separating.
Proof
Assume \(p,q\;{\in }\; FX\) such that \(p\;{\models }\;\square _{i}\theta \ \Longleftrightarrow \ q\;{\models }\;\square _{i}\theta \) for each \(i\;{\in }\; I\) and each \(\theta :X\rightarrow 2\). We must show \(p=q\).
Case 1
Case 2
\(X=\emptyset \): According to our general assumption, \(F\iota :F\emptyset \rightarrow F1\) is injective. Thus in order to separate \(p,q\;{\in }\; F\emptyset ,\) it is enough to separate \((F\iota )(p)\;{\in }\; F(1)\) from \((F\iota )(q)\;{\in }\; F(1)\) which is possible due to the previous case.
Corollary 2
If \(\square \) is monotonic and separating then every difunctional simulation is an \(F\)congruence.
As a further corollary, we obtain a converse to another result found in [3].
Corollary 3
Let \((\square _{i})_{i\;{\in }\; I}\) be monotonic. Then \((\square _{i})_{i\;{\in }\; I}\) are separating and \(F\) weakly preserves pullbacks if and only if each difunctional simulation is an \(F\)bisimulation.
Proof
The direction from left to right is from [3]. For the converse, suppose that each difunctional simulation is an \(F\)bisimulation. Then by monotony each \(F\)congruence is an \(F\)bisimulation. This is the same as saying that \(F\) weakly preserves pullbacks. Similarly, every difunctional simulation is an \(F\)congruence, hence by the above proposition, \((\square _{i})_{i\;{\in }\; I}\) is separating.
6 Conclusion and Further Work
We have given a new definition of coalgebraic simulation, which has the advantage to be amenable to diagrammatic reasoning. We have demonstrated its use with a number of results and related our definition to that of Gorín and Schröder in [3]. In the case where our boxes (respectively predicate liftings) are monotonic, a general assumption in the paper [3], our definition agrees with that of the authors. We have related our simulations to 2dimensional congruences (so called \(F\)congruences). We suspect that the set of all \(F\)congruences between fixed coalgebras \(\mathcal {A}\) and \(\mathcal {B}\) forms a complete lattice with the natural ordering. However we were only able to show it under the additional assumption that there exists a set of separating monotonic boxes \((\square _{i})_{i\;{\in }\; I}\). In that case, \(F\)congruences are bidirectional simulations and their supremum is given by difunctional closure. We leave it open whether the existence of a separating set \((\square _{i})_{i\;{\in }\; I}\) is needed.
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