1. Introduction
Our work aims to provide a topological presentation of coinductive predicates in dependent type theory by using tools of point-free topology developed in the field of formal topology.
Formal topology is the study of topology in a constructive and predicative setting (see e.g., Sambin, Reference Sambin2003). Its main object of investigation is a point-free notion of topological space, called basic topology, whose definition avoids impredicative uses of the power constructor. In particular, a basic topology consists of two relations, called basic cover and positivity relation, primitively representing its open and closed subsets, respectively. Powerful techniques for inductively generating basic covers and coinductively generating positivity relations have been developed by Coquand et al. (Reference Coquand, Sambin, Smith and Valentini2003) and Sambin (Reference Sambin2019), and have since been a cornerstone of the field.
In previous work with M.E. Maietti (Maietti and Sabelli, Reference Maietti and Sabelli2023), we compared the inductive generation of basic covers with other established inductive schemes in dependent type theory, obtaining in particular two equivalences. One equivalence was between inductive basic covers and inductive predicates – conceived within the context of axiomatic set theory by Aczel (Reference Aczel and Barwise1977) and Rathjen (Reference Rathjen2005) – over the Minimalist Foundation. The other equivalence was between inductive basic covers and the type of well-founded trees, known as
$\textsf{W}$
-types, over any Martin–Löf’s type theory validating either basic
$\eta$
-equalities or function extensionality.
In this work, we extend the comparison to their coinductive counterparts; namely, we relate the coinductive generation of positivity relations with coinductive predicates and non-wellfounded trees, known as
$\textsf{M}$
-types, defined in various extensions of Martin–Löf’s type theory, most notably in homotopy type theory (Ahrens et al., Reference Ahrens, Capriotti and Spadotti2015). In particular, we prove the equivalence of coinductive predicates and coinductive positivity relations in the Minimalist Foundation and, in turn, the definability of their proof-relevant versions as particular
$\textsf{M}$
-types in Martin–Löf’s type theory.
Since we were required to compare concepts taken from different foundational systems for mathematics, we extensively relied on the Minimalist Foundation as a ground theory to express those various notions. The Minimalist Foundation, conceived by Maietti and Sambin (Reference Maietti and Sambin2005) and finalised by Maietti (Reference Maietti2009), is a foundational system precisely designed to constitute a common core between all the most relevant foundations for constructive mathematics. This means that definitions, theorems and proofs written in its language can be exported soundly in a foundation of one’s choice.
With the exception of Corollary 13, all the results in this paper have been verified in Agda. Agda is a proof assistant implementing Martin–Löf’s type theory. The proofs performed in the Minimalist Foundation and in the Calculus of Constructions have been approximated in Agda in the former case by manually verifying that the types interpreted as propositions do not eliminate towards those interpreted as sets, and, in the latter case, by disregarding size issues for universe types. The source code is available at https://github.com/PietroSabelli/topological-co-induction; an HTML rendering is available at https://pietrosabelli.github.io/topological-co-induction-agda-html.
Structure of the paper
In Section 2, we make precise the dependent type theories in which we work and lay down some notation that we will exploit throughout the paper. In Section 3, working in the Minimalist Foundation, we introduce (co)inductive predicates and prove their equivalence with the (co)inductive methods of formal topology. In Section 4, working in the proof-relevant environment of Martin–Löf’s type theory, we relate the discussion on coinduction of the previous section to
$\textsf{M}$
-types. Finally, in Section 5, we draw some conclusions about the compatibility of the Minimalist Foundation extended with (co)inductive constructors.
2. Preliminaries
In this paper, we will mainly work within the Minimalist Foundation and Martin–Löf’s type theory. We will briefly recall them.
Minimalist foundation
The Minimalist Foundation is a two-level foundation, which consists of an extensional level
$\mathbf{emTT}$
, understood as the actual foundation in which constructive mathematics is formalised and developed, and an intensional level
$\mathbf{mTT}$
, which acts as a programming language enjoying a realisability interpretation à la Kleene in a fragment of second-order arithmetic (see Maietti and Maschio, Reference Maietti and Maschio2015; Ishihara et al. Reference Ishihara, Maietti, Maschio and Streicher2018). Both levels are formulated as dependent type theories à la Martin–Löf, and they are linked by a quotient interpretation, which implements types and terms of the extensional level into a quotient model built over the intensional one (Maietti, Reference Maietti2009).
In both
$\mathbf{emTT}$
and
$\mathbf{mTT}$
, there are four kinds of types: small propositions, propositions, sets and collections (denoted, respectively
$prop_s$
,
$prop$
,
$set$
and
$col$
). Sets are particular collections, just as small propositions are particular propositions. Moreover, we identify a proposition (respectively, a small proposition) with the collection (respectively, the set) of its proofs. Eventually, we have the following square of inclusions between kinds.
This fourfold cataloguing allows one to differentiate, on the one hand, between logical and mathematical entities. On the other, between different degrees of complexity (corresponding to the usual distinction between sets and classes in set theory – or the one between small and large types in Martin–Löf’s type theory with a universe), thus guaranteeing the predicativity of the theory.
Propositions are those of predicate logic with equality; a proposition is small if all its quantifiers and equality predicates are restricted to sets. The base sets are just the empty set
$\textsf{N}_0$
and the singleton set
$\textsf{N}_1$
; the set constructors are the dependent sum
$\Sigma$
, the dependent product
$\Pi$
, the disjoint sum
$+$
, the list constructor
$\textsf{List}$
and, only in the extensional level, a constructor
$-/-$
for quotienting a set by a small equivalence relation. The only base (proper) collection is, for the intensional level, a universe à la Russell of small propositions
${\textsf{Prop}_{s}}$
and, for the extensional level, a classifier
$\mathscr{P}({\textsf{N}_{1}})$
of small propositions up to equiprovability, which, as its notation suggests, can be thought of as the collection of the singleton’s subsets; collections are closed only under dependent sums and under function spaces of sets towards
${\textsf{Prop}_{s}}$
or
$\mathscr{P}({\textsf{N}_{1}})$
, depending on the level.
At the intensional level, propositions are proof-relevant, the equality predicate is intensional and the only computation rules are
$\beta$
-equalities. While at the extensional level, propositions are proof-irrelevant, the equality predicate is extensional, and also all
$\eta$
-equalities are valid.
A crucial remark is the fact that, in the Minimalist Foundation, elimination rules of propositional constructors act only towards propositions; in this way, neither the axiom of choice nor of unique choice is validated. In general, the system cannot internally extract a witness from a proof of an existential statement, as shown by Maietti (Reference Maietti2017).
Martin–Löf’s type theory
In this work, we consider a version
$\mathbf{MLTT_0}$
of intensional Martin–Löf’s type theory (see Nordström et al., Reference Nordström, Petersson and Smith1990) with the following type constructors: the empty type
${\textsf{N}_{0}}$
, the unit type
${\textsf{N}_{1}}$
, dependent sums
$\Sigma$
, dependent products
$\Pi$
, the list constructor
$\textsf{List}$
, identity types
$\textsf{Id}$
, disjoint sums
$+$
and a universe of small types
${\textsf{U}_{0}}$
à la Russell closed under all the above type constructors. Inductive type constructors are defined to allow elimination towards all (small and large) types.
The intensionality of the theory means that judgemental equality is not reflected by propositional equality. We instead assume
$\eta$
-equalities for
${\textsf{N}_{1}}$
,
$\Sigma$
-types, and
$\Pi$
-types. These weaker extensionality assumptions are justified in an intensional context since they do not break any computational property (see Abel et al. Reference Abel, Coquand and Dybjer2007), and they are largely supported by the Agda implementation of Martin–Löf’s type theory.
Finally, we will also consider extending the theory with the axiom of function extensionality
$\textsf{funext}$
, and the additional elimination scheme for the identity type known as axiom
$\textsf{K}$
introduced by Streicher (Reference Streicher1993).
Compatibility
As already mentioned, the Minimalist Foundation is compatible with various foundational theories. Formally, this means that there are interpretations of the former into the latter, which preserve the meaning of logical and mathematical entities. Moreover, when comparing the Minimalist Foundation with another foundational theory, one can choose, depending on the degree of abstraction of the latter, which of the two levels (intensional or extensional) to compare it with.
With the above provisions, we have the following compatibility results (Maietti, Reference Maietti2009):
-
•
$\mathbf{mTT}$
is compatible with
$\mathbf{MLTT_0}$
by identifying propositions and collections with types and small propositions and sets with small types; -
•
$\mathbf{mTT}$
is compatible with the Calculus of Inductive Constructions
$\mathbf{CIC}$
(Coquand, Reference Coquand1989); -
•
$\mathbf{emTT}$
is compatible with constructive set theory
$\mathbf{CZF}$
(see Aczel and Rathjen, Reference Aczel and Rathjen2010) by interpreting sets as (
$\mathbf{CZF}$
-)sets, collections as classes, propositions as first-order logic formulas and small propositions as bounded formulas. -
•
$\mathbf{emTT}$
is compatible with the calculus of topoi
$\mathscr{T}_{\mathbf{Topos}}$
as defined by Maietti (Reference Maietti2005) by identifying propositions with mono-types; -
• both
$\mathbf{mTT}$
and
$\mathbf{emTT}$
have been shown by Contente and Maietti (Reference Contente and Maietti2024) to be compatible with homotopy type theory by interpreting collections as h-sets, sets as h-sets in the first universe, propositions as h-propositions, small propositions as h-propositions in the first universe and judgemental equalities of
$\mathbf{emTT}$
as canonical propositional equalities of homotopy type theory.
Notation
We will use the following conventions and shorthands to improve readability.
Especially for writing elimination constructors, we will adopt Martin–Löf’s theory of higher-order expressions (see Nordström et al., Reference Nordström, Petersson and Smith1990, Chapter 3); for example, we can write
$\textsf{El}_{\mathbb{N}}(n,b,(x,y).c)$
in place of
$\textsf{El}_{\mathbb{N}}(n,b,c)$
to make explicit the variables bounded by the eliminator.
When writing inference rules, the piece of context common to all the judgements is omitted.
We reserve the arrow symbol
$\to$
(resp.
$\times$
) as a shorthand for a non-dependent function space (resp. for non-dependent product sets), and we denote the implication connective with the arrow symbol
$\Rightarrow$
.
If
$a \in A$
and
$f \in (\Pi x \in A)B(x)$
, we will often write
$f(a)$
as a shorthand for
$\textsf{Ap}(f,a)$
; analogously for
$\textsf{Ap}_\forall (f,a)$
and
$\textsf{Ap}_\Rightarrow (f,a)$
; when lambda abstracting, we omit the domain and the kind of lambda abstraction
$\lambda$
,
$\lambda _\forall$
,
$\lambda _\Rightarrow$
; analogously, we omit the kind of pair constructors
$\langle -,- \rangle$
,
$\langle -,_\wedge - \rangle$
and
$\langle -,_\exists - \rangle$
.
We define the two projections from a dependent sum in the usual way
$\pi _{\textsf{1}}(z) :\equiv \textsf{El}_{\Sigma }(z,(x,y).x)$
and
$\pi _{\textsf{2}}(z) :\equiv \textsf{El}_{\Sigma }(z,(x,y).y)$
.
Negation, the true constant and logical equivalence are defined in the following way
$\neg \varphi :\equiv \varphi \Rightarrow \bot$
,
$\top :\equiv \neg \bot$
and
$\varphi \Leftrightarrow \psi :\equiv \varphi \Rightarrow \psi \wedge \psi \Rightarrow \varphi$
.
We will sometimes render the judgement
$\textsf{true} \in \varphi \; [\Gamma ]$
as
$\varphi \; true \; [\Gamma ]$
.
As usual in
$\mathbf{emTT}$
, we let
$\mathscr{P}(A) :\equiv A \to \mathscr{P}({\textsf{N}_{1}})$
to denote the power-collection of a set
$A$
, and
$a \,\varepsilon \, V :\equiv \textsf{Ap}(V,a)$
to denote the (propositional) relation of membership between terms
$a \in A$
and subsets
$V \in \mathscr{P}(A)$
; moreover, given
$\varphi (x) \; prop \; [x \in A]$
we will use the following set-theoretic shorthands
\begin{align*} \{ x \in A \,|\, \varphi (x) \}& :\equiv \lambda x.[\varphi ] \in \mathscr{P}(A)\\[3pt] \emptyset & :\equiv \{ x \in A \,|\, \bot \} \in \mathscr{P}(A) \\[3pt] A & :\equiv \{ x \in A \,|\, \top \} \in \mathscr{P}(A) \\[3pt] (\forall x \,\varepsilon \, V)\varphi (x) & :\equiv (\forall x \in A)(x \,\varepsilon \, V \Rightarrow \varphi (x)) \; prop \\[3pt] (\exists x \,\varepsilon \, V)\varphi (x) & :\equiv (\exists x \in A)(x \,\varepsilon \, V \,\wedge \; \varphi (x)) \; prop \end{align*}
Encodings
The following notion formally says when a theory is expressive enough to define a given constructor.
If
$\mathscr{T}$
is a dependent type theory and
$\textsf{C}$
is a type constructor – intended as the sets of its rules defined in the language of
$\mathscr{T}$
– we say that
$\mathscr{T}$
encodes
$\textsf{C}$
if each new symbol appearing in
$\textsf{C}$
can be interpreted in
$\mathscr{T}$
in such a way that all the rules of
$\textsf{C}$
are valid under this interpretation. Given two type constructors
$\textsf{C}$
and
$\textsf{D}$
, we say that they are mutually encodable over
$\mathscr{T} \, $
if
$\mathscr{T}+\textsf{C}$
encodes
$\textsf{D}$
and
$\mathscr{T}+\textsf{D}$
encodes
$\textsf{C}$
.
As an example, consider the set of natural numbers
$\mathbb{N}$
defined as an inductive type in the usual way; it is encodable in
$\mathbf{emTT}$
using the following interpretation.
\begin{align*} \mathbb{N} & :\equiv \textsf{List}({\textsf{N}_{1}}) \\ 0 & :\equiv \epsilon \\ \textsf{succ}(n) & :\equiv \textsf{cons}(n,\star ) \\ \textsf{El}_{\mathbb{N}}(n,b,(x,z).c) & :\equiv \textsf{El}_{\textsf{List}}(n,b,(x,y,z).c) \end{align*}
3. (Co)Induction in the Minimalist Foundation
The basic idea behind any (co)inductive construction is to generate an object by proceeding from a given set of rules – intuitively, with induction following them in the forward direction and coinduction backwards. How such rules are specified and used heavily depends on (1) the kind of mathematical object to be constructed and (2) the setting in which this construction is formalised.
Concerning the first point, a key distinction is between the (co)inductive generation of sets and the (co)inductive generation of predicates (or, equivalently, subsets). The present section focuses on the latter. Paradigmatic examples of inductively and coinductively defined sets are lists (of which natural numbers are the most basic instance) and streams, respectively. On the other hand, the fundamental example of (co)inductive predicates we would consider is given by deductive systems, which, through their rules, for a given set of formluae inductively declare which are the theorems, and coinductively declare which are the refutables.
Concerning the second point, each foundational theory has its peculiar ways of expressing (co)induction, e.g. in extensional Martin–Löf’s type theory (non-)wellfounded trees are usually expressed via category theory (Moerdijk and Palmgren, Reference Moerdijk and Palmgren2000; van den Berg and De Marchi, Reference van den Berg and De Marchi2007); homotopy type theory assumes higher inductive types, a very general form of induction (The Univalent Foundations Program, 2013), while Ahrens et al. (Reference Ahrens, Capriotti and Spadotti2015) define non-wellfounded trees in it through an internal universal property; finally, in (constructive) set theory, generalised inductive definitions are usually considered (see Rathjen, Reference Rathjen2005). We are interested in adapting to dependent type theory the latter scheme, first introduced by Aczel (Reference Aczel and Barwise1977) and then adapted to a constructive setting by Aczel and Rathjen (Reference Aczel and Rathjen2010) (see also Rathjen, Reference Rathjen2005). To this end, in this section, we define it in the Minimalist Foundation; while, in the next section, we will consider it in Martin–Löf’s type theory.
3.1 (Co)Inductive predicates
We first consider (co)inductive predicates in the extensional level of the Minimalist Foundation. The starting point is to specify how to declare rules for the (co)inductive generation of predicates. We do so using a notion that goes under two names, corresponding to two different – although related – interpretations. The first name is rule set,Footnote 1 and, in this sense, it is a straightforward adaptation in the setting of the Minimalist Foundation of the notion of the same name by Aczel (Reference Aczel and Barwise1977). The second name is axiom set, defined in the context of (co)inductive generation of formal topologies in (Coquand et al. Reference Coquand, Sambin, Smith and Valentini2003). We will focus on the first interpretation (and hence use the first name) while presenting (co)inductive predicates. We will turn to the second name when recalling (co)inductive methods in formal topology.
Definition 1.
A rule or axiom set over a given set
$A$
consists of the following data:
-
(1) a dependent family of sets
$I(x) \; set \; [x \in A]$
; -
(2) a dependent family of
$A$
-subsets
$C(x,y) \in \mathscr{P}(A) \; [x \in A,y \in I(x)]$
.
Given two elements
$a \in A$
and
$i \in I(a)$
, we say that
$i$
is a rule with premises
$C(a,i)$
and conclusion
$a$
; sometimes it is represented pictorially as
\begin{equation*} \frac {C(a,i)}{a}\;i \end{equation*}
Hopefully, the chosen terminology makes the logical interpretation of a rule set as an internally defined deduction system transparent. Furthermore, notice that an axiom set can be equivalently read as a proof-irrelevant version of an indexed container (Altenkirch et al. Reference Altenkirch, Ghani, Hancock, McBride and Morris2015). In turn, an indexed container contains exactly the parameters for forming dependent (non-)well founded trees, as will be recalled in Section 4.
The constructors for inductive and coinductive predicates are quite straightforward to define, the former being close to a proof-irrelevant version of
$\textsf{W}$
-types in Martin–Löf’s type theory, and the latter being a simple dualisation of the former, known as
$\textsf{M}$
-types. In the following, we report their formal rules, where we left the judgements formalising the rule set implicit in their premises.
Rules for inductive predicates in emTT
\begin{equation*} \textsf{F}-\textsf{Ind}\;\frac {a \in A}{\textsf{Ind}_{I,C}(a) \; prop_s} \end{equation*}
\begin{equation*} \textsf{I}\textsf{-}\textsf{Ind}\;\frac {a \in A \quad i \in I(a) \quad (\forall x \,\varepsilon \, C(a,i))\textsf{Ind}_{I,C}(x) \; true}{\textsf{Ind}_{I,C}(a) \; true} \end{equation*}
\begin{equation*} \textsf{E}\textsf{-}\textsf{Ind}\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & P(x) \; true \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(x,y))P(z)] \\ & a \in A \quad \textsf{Ind}_{I,C}(a) \; true \end{aligned}} {P(a) \; \; true} \end{equation*}
Rules for coinductive predicates in emTT
\begin{equation*} \textsf{F}-\textsf{CoInd}\;\frac {a \in A}{\textsf{CoInd}_{I,C}(a) \; prop_s} \end{equation*}
\begin{equation*} \textsf{E}\textsf{-}\textsf{CoInd}\;\frac {a \in A \quad i \in I(a) \quad \textsf{CoInd}_{I,C}(a) \; true}{(\exists x \,\varepsilon \, C(a,i))\textsf{CoInd}_{I,C}(x) \; true} \end{equation*}
\begin{equation*} \textsf{I}\textsf{-}\textsf{CoInd}\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & (\exists z \,\varepsilon \, C(x,y))P(z) \; true \; [x \in A, y \in I(x), w \in P(x)] \\ & a \in A \quad P(a) \; true \end{aligned}}{ \textsf{CoInd}_{I,C}(a) \; true} \end{equation*}
We now proceed to link the above type-theoretical formulation of the constructors
$\textsf{Ind}$
and
$\textsf{CoInd}$
with the set-theoretical treatment of inductive definitions (Aczel, Reference Aczel and Barwise1977; Rathjen, Reference Rathjen2005). In the following, assume we have fixed a set
$A$
and a rule set
$(I,C)$
over it.
Recall that a predicate
$P$
on
$A$
is just a proposition depending on
$A$
\begin{equation*} P(x) \; prop \; [x \in A] \end{equation*}
A rule set induces two ways of transforming predicates.
Definition 2.
Given a predicate
$P$
on
$A$
, we define two other predicates
$\textsf{Der}_{I,C}(P)$
and
$\textsf{Conf}_{I,C}(P)$
on
$A$
as follows:
\begin{align*} \textsf{Der}_{I,C}(P)(x) & :\equiv (\exists y \in I(x))(\forall z \,\varepsilon \, C(x,y))P(z) \; prop \; [x \in A]\\[3pt] \textsf{Conf}_{I,C}(P)(x) & :\equiv (\forall y \in I(x))(\exists z \,\varepsilon \, C(x,y))P(z) \; prop \; [x \in A] \end{align*}
We call them derivability and confutability from
$P$
, respectively.
As the names suggest, derivability from
$P$
tells which elements of
$A$
can be derived with exactly one rule application, assuming as axioms the elements for which
$P$
holds – in fact, the definition of
$\textsf{Der}_{I,C}(P)(x)$
explicitly reads as there exists a rule with conclusion
$x$
such that all its premises are satisfied by
$P$
. Dually, confutability from
$P$
tells which elements of
$A$
can be confuted after exactly one step of backward search, assuming that the elements for which
$P$
holds are already refuted – the definition of
$\textsf{Conf}_{I,C}(P)(x)$
reads all rules with conclusion
$x$
have at least one premise for which
$P$
holds.
In accordance with the above interpretation, the constructions
$\textsf{Der}_{I,C}(\! - \!)$
and
$\textsf{Conf}_{I,C}(\! - \!)$
can be read meta-theoretically as two endomorphisms of the preorder of predicates on
$A$
: the preorder relation is formally given by
\begin{equation*} P \leq _A Q :\equiv (\forall x \in A)(P(x) \Rightarrow Q(x)) \; prop \end{equation*}
and it is straightforward to check that
$P \leq _A Q$
implies both
\begin{equation*} \textsf{Der}_{I,C}(P) \leq _A \textsf{Der}_{I,C}(Q) \quad \text{and}\quad \textsf{Conf}_{I,C}(P) \leq _A \textsf{Conf}_{I,C}(Q)\text{.}\end{equation*}
Finally, notice that if
$P$
is a small proposition, then so
$\textsf{Der}_{I,C}(P)$
and
$\textsf{Conf}_{I,C}(P)$
are.
Thanks to these observations, we can exploit the theory and terminology of monotone operators; in particular, notice that the operator
$\textsf{Der}_{I,C}$
corresponds to the class operator
$\Gamma _\Phi$
defined by Rathjen (Reference Rathjen2005, Definition 2.2).
Definition 3.
We say that a predicate
$P$
on
$A$
is closed with respect to the rule set
$(I,C)$
, if it is closed with respect to
$\textsf{Der}_{I,C}(\! - \!)$
, namely if
\begin{equation*} \textsf{Der}_{I,C}(P) \leq _A P \; true \end{equation*}
Dually, we say that
$P$
is correct with respect to the rule set
$(I,C)$
, if it is correct with respect to
$\textsf{Conf}_{I,C}(\! - \!)$
, namely if
\begin{equation*} P \leq _A \textsf{Conf}_{I,C}(P) \; true \end{equation*}
Once again, notice how a closed (resp. correct) predicate with respect to an axiom set
$(I,C)$
corresponds to the notion of a
$\Phi$
-closed (resp.
$\Phi$
-correct) class as defined by Rathjen (Reference Rathjen2005).
Thanks to monotonicity, if
$P$
is a closed predicate, we have the following chain of inequalities.
\begin{equation*} \cdots \leq _A \textsf{Der}_{I,C}(\textsf{Der}_{I,C}(P)) \leq _A \textsf{Der}_{I,C}(P) \leq _A P \end{equation*}
Dually, if
$P$
is correct it, follows that
\begin{equation*} P \leq _A \textsf{Conf}_{I,C}(P) \leq _A \textsf{Conf}_{I,C}(\textsf{Conf}_{I,C}(P)) \leq _A \cdots \end{equation*}
Therefore, to be a closed predicate means that the deductive system cannot derive anything new from it. Dually, to be a correct predicate means that if the elements satisfying it are assumed not to be derivable by the deduction system, then they will always be underivable even after any number of backward search steps. We are then naturally led to interpret the smallest closed predicate as expressing derivability and the greatest correct predicate as confutability. The following lemma formalises this intuition and links the type-theoretical formulation of the constructors
$\textsf{Ind}$
and
$\textsf{CoInd}$
with the above interpretation.
Proposition 4.
The introduction and elimination rules for inductive predicates in
$\mathbf{emTT}$
can equivalently be formulated in the following way (as before, the rule set is left implicit in the premises of each rule).
\begin{equation*} \textsf{I}-\mathrm{\textsf{Ind}}'\;\frac {}{ \textsf{Der}_{I,C}(\textsf{Ind}_{I,C}) \leq _A \textsf{Ind}_{I,C} \; true} \end{equation*}
\begin{equation*} \textsf{E}-\mathrm{\textsf{Ind}}'\;\frac {P(x) \; prop \; [x \in A]\quad \textsf{Der}_{I,C}(P) \leq _A P \; true}{\textsf{Ind}_{I,C} \leq _A P \; true} \end{equation*}
Analogously, for coinductive predicates.
\begin{equation*} \textsf{E}-\mathrm{\textsf{CoInd}}'\;\frac {}{ \textsf{CoInd}_{I,C} \leq _A \textsf{Conf}_{I,C}(\textsf{CoInd}_{I,C}) \; true} \end{equation*}
\begin{equation*} \textsf{I}-\mathrm{\textsf{CoInd}}'\;\frac {P(x) \; prop \; [x \in A]\quad P \leq _A \textsf{Conf}_{I,C}(P) \; true}{P \leq _A \textsf{CoInd}_{I,C} \; true} \end{equation*}
Proof.
Let
$\varphi (x) \; prop \; [x \in A]$
be a predicate and
$\psi \; prop$
and
$\chi \; prop$
two propositions; it is easy to show that in
$\mathbf{emTT}$
the following pairs of inference rules are mutually derivable.
\begin{align*} \textit {(Existential quantifier)} & \qquad \frac {(\exists x \in A)\varphi (x) \; true }{\psi \; true} \qquad \frac {a \in A \quad \varphi (a) \; true}{\psi \; true} \\ \textit {(Universal quantifier)} & \qquad \frac { }{(\forall x \in A)\varphi (x) \; true} \qquad \frac {a \in A}{\varphi (a) \; true} \\ \textit {(Implication)} & \qquad \frac { }{\psi \Rightarrow \chi \; true} \qquad \frac {\psi \; true}{\chi \; true} \end{align*}
The equivalences in the statement can be derived by repeatedly apply the above equivalences. As an example, the equivalence of
$\textsf{I-Ind}$
and
$\textsf{I-Ind}'$
is obtained through the following chain of equivalent inference rules.
\begin{align*} & \frac {}{(\forall x \in A)((\exists y \in I(x))(\forall z \,\varepsilon \, C(x,y))\textsf{Ind}_{I,C}(z) \Rightarrow \textsf{Ind}_{I,C}(x)) \; true} \\ & \frac {a \in A}{(\exists y \in I(a))(\forall z \,\varepsilon \, C(a,y))\textsf{Ind}_{I,C}(z) \Rightarrow \textsf{Ind}_{I,C}(a) \; true} \\ & \frac {a \in A \quad (\exists y \in I(a))(\forall z \,\varepsilon \, C(a,y))\textsf{Ind}_{I,C}(z) \; true}{ \textsf{Ind}_{I,C}(a) \; true} \\ & \frac {a \in A \quad i \in I(a) \quad (\forall z \,\varepsilon \, C(a,i))\textsf{Ind}_{I,C}(z) \; true}{\textsf{Ind}_{I,C}(a) \; true} \end{align*}
Remark 5. Classically, i.e. assuming the principle of excluded middle, inductive and coinductive predicates are complementary subsets and thus mutually encodable as
\begin{align*} \textsf{CoInd}_{I,C}(x) & :\equiv \neg \textsf{Ind}_{I,C}(x) \; prop \; [x \in A] \\[2pt] \textsf{Ind}_{I,C}(x) & :\equiv \neg \textsf{CoInd}_{I,C}(x) \; prop \; [x \in A] \end{align*}
To see this, observe that the following equivalences between predicates hold.
\begin{align*} \neg \textsf{Der}_{I,C}(P)(x) & \Leftrightarrow \textsf{Conf}_{I,C}(\neg P)(x) \; true \; [x \in A] \\[2pt] \neg \textsf{Conf}_{I,C}(P)(x) & \Leftrightarrow \textsf{Der}_{I,C}(\neg P)(x) \;\,true \; [x \in A] \end{align*}
Therefore, a predicate
$P$
is closed (resp. correct) if and only if its complement
$\neg P$
is correct (resp. closed) – and the smallest closed predicate is the complement of the greatest correct one, and vice versa.
On another note, both inductive and coinductive predicates are impredicatively encodable in the internal language of toposes
$\mathscr{T}_{\mathbf{Topos}}$
defined by Maietti (Reference Maietti2005). This is done in the usual way, interpreting them as the intersection of all closed predicates and the union of all correct predicates, respectively.
\begin{align*} \textsf{Ind}_{I,C}(x) &:\equiv (\forall P \in \mathscr{P}(A))(\textsf{Der}_{I,C}(P) \leq _A P \Rightarrow x\,\varepsilon \,P) \; prop \; [x \in A] \\[2pt ] \textsf{CoInd}_{I,C}(x) &:\equiv (\exists P \in \mathscr{P}(A))(P \leq _A \textsf{Conf}_{I,C}(P) \wedge x\,\varepsilon \,P) \; prop \; [x \in A] \end{align*}
In the intensional level
$\mathbf{mTT}$
, the intensional counterparts of the rules for inductive and coinductive predicates are easily obtained from the extensional ones by adding the now-relevant proof terms. On the one hand, we set the computational behaviour of inductive proofs simply drawing on the usual pattern of inductive types in Martin–Löf’s type theory. On the other hand, we stayed close to the choice in (Maietti et al. Reference Maietti, Maschio and Rathjen2022) of not postulating any computation rule for coinduction. We intend, in future works, to consider the addition of such rules, following their description by Giménez (Reference Giménez1996). The complete rules are listed in Appendix A.
The Calculus of Constructions can be regarded as an impredicative version of
$\mathbf{mTT}$
. Thus, we can adapt the impredicative encoding described in Remark 5 to the intensional case; crucially, we need to check that the computation rule in the inductive case is satisfied.
Proposition 6. Inductive and coinductive predicates are encodable in the Calculus of Constructions.
Proof. In light of Remark 5, we check that the computation rule of the inductive case is satisfied.
\begin{align*} \textsf{Ind}_{I,C}(a) :\equiv \; & (\forall P \in A \to \textsf{Prop}) \\ & \quad ((\forall x \in A)(\forall y \in I(x))((\forall z \,\varepsilon \,C(x,y))P(z)\Rightarrow P(x)) \\ & \qquad \Rightarrow P(a)) \\ \textsf{ind}(a,i,p) :\equiv \;& \lambda P.\lambda c.c(a,i,\lambda xq.p(x,q,P,c)) \\ \textsf{El}^P_{\textsf{Ind}}(a,p,(x,y,w).c) :\equiv \; & p(\lambda x . P(x),\lambda xyw.c(x,y,w)) & \end{align*}
In the next section, we will prove that (co)inductive predicates are equivalent to the (co)inductive topological methods described in (Maietti et al. Reference Maietti, Maschio and Rathjen2021, Reference Maietti, Maschio and Rathjen2022); moreover, there, an extension of the quotient interpretation and an extension of the realisability interpretation supporting those constructors have been defined. Therefore, we will obtain as an immediate corollary that such interpretations also support (co)inductive predicates.
3.2 Topological (co)induction
The development of the Minimalist Foundation has always been closely tied to that of formal topology, and it is no surprise that (co)induction has been first considered in that context. Maietti et al. (Reference Maietti, Maschio and Rathjen2021, Reference Maietti, Maschio and Rathjen2022) proposed extensions of the Minimalist Foundation supporting the formalisation of (co)inductive generation methods developed in formal topology. Moreover, they showed that both the setoid interpretation of the extensional level into the intensional one and the realisability interpretation of the latter can be adapted to support those extensions.
We now recall those extensions, starting with the basic definitions of formal topology formalised in the Minimalist Foundation.
Definition 7. A basic topology consists of the following data:
-
(1) A set
$A$
, whose elements are called basic opens;
-
(2) A small binary relation
$\vartriangleleft$
, called basic cover, between elements of
$A$
and subsets of
$A$
, satisfying the following properties:
-
− (reflexivity) if
$a \,\varepsilon \, V$
, then
$a \vartriangleleft V$
; -
− (transitivity) if
$a \vartriangleleft U$
and
$(\forall x \,\varepsilon \, U)x \vartriangleleft V$
, then
$a \vartriangleleft V$
.
-
-
(3) A small binary relation
$\ltimes$
, called positivity relation, between elements of
$A$
and subsets of
$A$
, satisfying the following properties:
-
− (coreflexivity) if
$a \ltimes V$
, then
$a \,\varepsilon \, V$
; -
− (cotransitivity) if
$a \ltimes U$
and
$(\forall x \in A)(x \ltimes V \Rightarrow x \,\varepsilon \, U)$
, then
$a \ltimes V$
; -
− (compatibility) if
$a \ltimes V$
and
$a \vartriangleleft U$
, then
$(\exists x \,\varepsilon \,V )(x \ltimes U)$
.
-
The intuitive meaning of the above definition is the following: the set
$A$
, as its name suggests, is a base for the topology; the relation
$a \vartriangleleft V$
means that the basic open
$a$
is covered by the family of basic opens
$V$
; and finally,
$a \ltimes V$
means that there is a point of
$a$
all of whose basic neighbourhoods belong to
$V$
.
Coquand et al. (Reference Coquand, Sambin, Smith and Valentini2003) devised a way to inductively generate a basic cover on a given set
$A$
, starting from an axiom set
$(I,C)$
over it. In this sense, an axiom set is the set-indexed family of axioms
$ a \vartriangleleft C(a,i)$
for each
$a \in A$
and
$i \in I(a)$
. The inductively generated basic cover is then the smallest one which satisfies them. Later, Sambin (Reference Sambin2003) dualised this idea to coinductively generate the greatest positivity relation satisfying
$ a \ltimes C(a,i)$
for each
$a \in A$
and
$i \in I(a)$
, which, moreover, turns out to be compatible with the inductively generated basic cover on the same axiom set, and thus gives rise to a basic topology.
Example 8.
Consider the set
$\textsf{List}(\mathbb{N})$
, and the following axiom set over it.
\begin{align*} I(s) & :\equiv {\textsf{N}_{1}} + (\Sigma l \in \textsf{List}(\mathbb{N}))(\exists t \in \textsf{List}(\mathbb{N}))[l,t] =_{\textsf{List}(\mathbb{N})} s \\ C(s,\textsf{inl}(\star )) & :\equiv \{ \textsf{cons}(s,n) \,|\, n \in \mathbb{N} \} \\ C(s,\textsf{inr}(z)) & :\equiv \{ \pi _1(z) \} \end{align*}
where
$[-,-]$
is the concatenation operator. The basic topology obtained by (co)induction from the above axiom set is the Baire space topology. In particular, in this situation, the positivity relation
$a \ltimes _{I,C} V$
precisely states that there exists a spread containing
$a$
and contained in
$V$
; see (Ciraulo and Sambin, Reference Ciraulo and Sambin2019).
When the positivity relation
$a \ltimes V$
is specialised to the case where
$V = A$
, one talks of the positivity predicate
$\textsf{Pos}(a) :\equiv a \ltimes A$
. Sometimes, coinductively generated positivity predicates have been considered on their own, as in Maietti and Valentini (Reference Maietti and Valentini2004), where the authors used them to constructively and predicatively construct the coreflection of locales in open locales.
These methods were formalised in the Minimalist Foundation in the form of an inductive constructor
$\vartriangleleft$
in Maietti et al. (Reference Maietti, Maschio and Rathjen2021) and a coinductive constructor
$\ltimes$
in Maietti et al. (Reference Maietti, Maschio and Rathjen2022), both having as parameters an axiom set
$(I,C)$
over a set
$A$
(formalised as in the case of (co)inductive predicates), and a subset
$V \in \mathscr{P}(A)$
in the extensional level, or a propositional function
$V \in A \to {\textsf{Prop}_{s}}$
in the intensional one. Their precise rules in the extensional and the intensional level are both recalled in the Appendix B. In both cases, their rules are very close to those of (co)inductive predicates presented as in Proposition 4, the only difference being the presence of the parameter
$V$
, which implies, for each of the two constructors, an additional (co)reflection rule and the additional condition
$x \,\varepsilon \, V$
on the predicates involved in the universal properties. To account for the clause induced by the parameter
$V$
, we modify the derivability and confutability endomorphisms in the following way.
\begin{align*} \textsf{Der}_{I,C,V}(P)(x) & :\equiv x \,\varepsilon \,V \vee \textsf{Der}_{I,C}(P)(x) \; prop \; [x \in A] \\[2pt] \textsf{Conf}_{I,C,V}(P)(x) & :\equiv x \,\varepsilon \,V \wedge \textsf{Conf}_{I,C}(P)(x) \; prop \; [x \in A] \end{align*}
The next lemma shows indeed that, at the extensional level, the predicates
$- \vartriangleleft _{I,C} V$
and
$- \ltimes _{I,C}\,V$
can be seen as the smallest closed predicate of
$\textsf{Der}_{I,C,V}$
and the greatest correct predicate of
$\textsf{Conf}_{I,C,V}$
, respectively.
Proposition 9.
The introduction and elimination rules for extensional inductive basic covers in
$\mathbf{emTT}$
can equivalently be formulated in the following way.
\begin{equation*} \textsf{I}\textsf - {\vartriangleleft }'\;\frac {}{ \textsf{Der}_{I,C,V}(- \vartriangleleft _{I,C} V) \leq _A - \vartriangleleft _{I,C} V \; true} \end{equation*}
\begin{equation*} \textsf{E}\textsf-{\vartriangleleft }'\;\frac {P(x) \; prop \; [x \in A]\quad \textsf{Der}_{I,C,V}(P) \leq _A P \; true}{- \vartriangleleft _{I,C} V \leq _A P \; true} \end{equation*}
For extensional coinductive positivity relations, we have the analogous result:
\begin{equation*} \textsf{E}\textsf-{\ltimes }'\;\frac {}{ - \ltimes _{I,C} V \leq _A \textsf{Conf}_{I,C,V}(\! - \ltimes _{I,C} V) \; true} \end{equation*}
\begin{equation*} \textsf{I}\textsf-{\ltimes }'\;\frac {P(x) \; prop \; [x \in A]\quad P \leq _A \textsf{Conf}_{I,C,V}(P) \; true}{ P \leq _A - \ltimes _{I,C} V \; true} \end{equation*}
Proof. The proof is entirely analogous to that of Proposition 4.
The following result shows that the two flavours of (co)induction considered so far are equivalent. Regarding induction at the intensional level, it is a refinement of the second item of (Maietti and Sabelli, Reference Maietti and Sabelli2023, Theorem 6.3) in that it discharges the additional hypothesis of
$\eta$
-equalities. Indeed, we observed that since in the Minimalist Foundation inductive basic covers do not eliminate towards types that depend on them, we do not need to refine our encodings with a canonical predicate, as done in the proof of (Maietti and Sabelli, Reference Maietti and Sabelli2023, Theorem 5.1) for Martin–Löf’s type theory. It is precisely this trick that required principles of extensionality to work.
Theorem 10. In both levels of the Minimalist Foundation, inductive basic covers and inductive predicates are mutually encodable, and so are coinductive positivity relations and coinductive predicates. In particular, coinductive predicates coincide with positivity predicates.
Proof. (i.) Topological (co)induction encodes (co)inductive predicates in the extensional level.
To see that topological (co)induction encodes (co)inductive predicates, it is enough to choose
$V$
to be irrelevant. Indeed, notice that, for each axiom set
$(I,C)$
over
$A$
, the operator
$\textsf{Der}_{I,C}$
(resp.
$\textsf{Conf}_{I,C}$
) is equivalent to the operator
$\textsf{Der}_{I,C,\emptyset }$
(resp.
$\textsf{Conf}_{I,C,A}$
). The following interpretations satisfy the rules of (co)inductive predicates.
\begin{align*} \textsf{Ind}_{I,C}(x) & :\equiv x \vartriangleleft _{I,C} \emptyset \\ \textsf{CoInd}_{I,C}(x) & :\equiv \textsf{Pos}(x) \equiv x \ltimes _{I,C} A \end{align*}
(ii.) Inductive predicates encode basic inductive covers in the extensional level.
Assume to have an axiom set
$(I,C)$
over
$A$
, and a subset
$V \in \mathscr{P}(A)$
; to prove that inductive predicates encode basic inductive covers we define an enlarged rule set by encoding the additional reflexivity clause given by
$V$
.
\begin{align*} I_V(x) & :\equiv x \,\varepsilon \, V + I(x) \\ C_V(x,\textsf{inl}(p)) & :\equiv \emptyset \\ C_V(x,\textsf{inr}(y)) & :\equiv C(x,y) \end{align*}
where, formally, we set
$C_V(x,y,z) : \equiv \textsf{T}(\textsf{El}_{+}(y,(w).[\bot ],(w).[C(x,w,z)]))$
. We claim that the following interpretation satisfy the rules of inductive basic covers.
\begin{equation*} x \vartriangleleft _{I,C} V :\equiv \textsf{Ind}_{I_V,C_V}(x) \end{equation*}
(iii.) Coinductive predicates encode coinductive positivity relations in the extensional level.
To show that coinductive predicates encode coinductive positivity relations, we define a restriction of both the set
$A$
and the axiom set
$(I,C)$
by comprehension on those elements for which
$V$
already holds.
\begin{align*} A^V & :\equiv (\Sigma w \in A)w\,\varepsilon \, V \\ I^V(x) & :\equiv I(\pi _1(x)) \\ C^V(x,y) & :\equiv \{ z \in A^V \,|\, \pi _1(z) \,\varepsilon \, C(\pi _1(x),y) \} \end{align*}
We claim that the following interpretation satisfies the rules of coinductive positivity relations as presented in Proposition 9.
\begin{equation*} x \ltimes _{I,C} V :\equiv (\exists p \in x \,\varepsilon \, V)\textsf{CoInd}_{I^V,C^V}(\langle x , p \rangle ) \end{equation*}
(iv.) The encodings in the intensional level.
We prove that the above encodings works also in the intensional level, where, as an additional difficulty, we have to also make the interpretation of the proof terms explicit and, in the inductive case, check that the computation rules are satisfied. Refer to Appendices A and B for the constructors used in the following.
Case (i.) is trivial to check.
For case (ii.) we set
\begin{align*} \textsf{rf}(a,r) & :\equiv \textsf{ind}(a,\textsf{inl}(r),\lambda x.\lambda y.\textsf{El}_\bot (y)) \\[2pt] \textsf{tr}(a,i,p) & :\equiv \textsf{ind}(a,\textsf{inr}(i),p) \\[2pt] \textsf{El}_\vartriangleleft (a,p,q_1,q_2) & :\equiv \textsf{El}_{ \textsf{Ind}}(a,p,(x,y,w).\textsf{Ap}(f(x,y),w)) \end{align*}
where
$f(x,y) :\equiv \textsf{El}_+(y,(u).\lambda w.q_1(x,u),(u).\lambda w.q_2(x,u,w))$
. Notice in particular that the encoding of the elimination term
$\textsf{El}_\vartriangleleft$
has been performed by bearing in mind the corresponding computation rule it will need to satisfy, which is then trivially verified.
For case (iii.) we have the following
\begin{align*} \textsf{corf}(a,p) & :\equiv \textsf{El}_\exists (p,(x,y).x) \\[2pt] \textsf{cotr}(a,i,p) & :\equiv \textsf{El}_\exists (d,(z,w).\langle \pi _1(z), \pi _1(w) , \pi _2(z) , \pi _2(w) \rangle ) \end{align*}
where
$d$
is a shorthand for the following proof-term.
\begin{align*} & \textsf{El}_{\textsf{CoInd}}(\langle a, \textsf{El}_\exists (p,(x,y).x)\rangle , i, \textsf{El}_\exists (p,(x,y).y)) \\[2pt] & \qquad \in (\exists z \in (\Sigma x \in A)x \,\varepsilon \, V)(C(a,i,\pi _1(z)) \wedge \textsf{CoInd}_{I^V,C^V}(z)) \end{align*}
For the coinduction term, assume to have a predicate
$P(x) \; prop \; [x \in A]$
and terms
$q_1$
and
$q_2$
as in the premise of the rule
$\textsf{I-}\ltimes$
; we define the auxiliary predicate
\begin{equation*} P'(x) :\equiv P(\pi _1(x)) \; prop \; [x \in A^V] \end{equation*}
together with the auxiliary term
\begin{align*} & c'(x,y,w) \in (\exists z \in C^V(x,y))P'(z) \; [x \in A^V, y \in I^V(x), w \in P'(x)] \\[2pt] & c'(z,y,w) :\equiv \textsf{El}_\exists (q_2(\pi _1(z),y,w),(u,v).\langle \langle u , q_1(u,\pi _2(v)) \rangle , \pi _1(v) , \pi _2(v) \rangle ) \end{align*}
finally, we can write the following proof-term
\begin{equation*} \textsf{coind}(a,p,q_1,q_2) :\equiv \langle q_1(a, p) , \textsf{coind}(\langle a , q_1(a, p) \rangle , p , (x,y,w).c')\rangle \end{equation*}
In Maietti et al. (Reference Maietti, Maschio and Rathjen2021), the two-level foundation obtained by extending the Minimalist Foundation (i.e. extending both its levels, lifting the quotient interpretation between them and lifting the realisability interpretation of the intensional one) with inductive basic covers was called
$\mathbf{MF}_{\textsf{ind}}$
; then, in Maietti et al. (Reference Maietti, Maschio and Rathjen2022), the extension of the Minimalist Foundation with both inductive basic covers and coinductive positivity relations was called
$\mathbf{MF}_{\textsf{cind}}$
. Here, in light of Theorem 10, we overload the notation by using the same names
$\mathbf{MF}_{\textsf{ind}}$
and
$\mathbf{MF}_{\textsf{cind}}$
to refer to the extensions of the Minimalist Foundation with inductive predicates and with both inductive and coinductive predicates, respectively.
4. Coinduction in Martin–Löf’s Type Theory
Induction and coinduction in Martin–Löf’s type theory and its extensions can assume many forms; two of the most common schemes are given by a pair of dual constructions called
$\textsf{W}$
-types (or wellfounded trees) for induction and
$\textsf{M}$
-types (or non-wellfounded trees) for coinduction. They both come with respective dependent versions. In this section, we examine the relationship of (dependent)
$\textsf{M}$
-types with the forms of coinduction introduced in the previous section. The dual discussion comparing
$\textsf{W}$
-types with the other forms of induction has been carried out in Maietti and Sabelli (Reference Maietti and Sabelli2023).
Throughout this section, we work in the theory
$\mathbf{MLTT_0}$
extended with
$\textsf{funext}$
. The parameters of an
$\textsf{M}$
-type consist of a type
$A \in \textsf{U}_{0}$
and a dependent type
$B \in A \to \textsf{U}_{0}$
, which together are often referred to as a container, see (Altenkirch et al. Reference Altenkirch, Ghani, Hancock, McBride and Morris2015). The
$\textsf{M}$
-type constructed using such parameters is intuitively understood as the set of non(-necessarily)-wellfounded trees with nodes labelled by elements of
$A$
and with branching function given by
$B$
. Dependent
$\textsf{M}$
-types are interpreted again as sets of non-necessarily-well founded trees with labelled nodes; however, each label now has a set of possible options for the branching function; moreover, each branching function not only indicates the number of direct subtrees but also dictates how each of their roots is to be labelled. The formal rendering of this intuition in Martin–Löf’s type theory goes as follows. The parameters of a dependent
$\textsf{M}$
-type, which, analogously to the non-dependent case, are referred to as indexed container, consist of a small type
$A \in \textsf{U}_{0}$
of nodes’ labels and a family of sets
$I \in A \to \textsf{U}_{0}$
indexing the possible branching functions associated with each label. The branching functions are usually formalised with two arity functions
\begin{align*} Br(x,y) \in \textsf{U}_{0} & \; [x \in A,y\in I(x)] \\[2pt] ar(x,y) \in Br(x,y) \to A & \; [x \in A,y\in I(x)] \end{align*}
that respectively describe the number of direct subtrees and for each subtree, the label of its root. Notice how the type names
$A$
and
$I$
are swapped with respect to presentation of indexed containers as that of (Ahrens et al., Reference Ahrens, Capriotti and Spadotti2015). We chose this naming to stay close to the presentation of axiom-sets.
Usually (van den Berg and De Marchi, Reference van den Berg and De Marchi2007; Ahrens et al., Reference Ahrens, Capriotti and Spadotti2015), (dependent)
$\textsf{M}$
-types are not presented through explicit inference rules; instead, they are characterised semantically by the universal property of being terminal coalgebras for certain endofunctors, which we recall in the following. Consider the category
$\mathbf{Set}$
as formalised in
$\mathbf{MLTT_0}+\textsf{funext}$
; its objects are closed types
$A \; type$
(considered up to judgemental equality); an arrow between objects
$A$
and
$B$
is a function
$f \in A \to B$
; and two arrows
$f,g \in A \to B$
are considered equal if there exists a term
$p \in \textsf{Id}(A \to B,f,g)$
. Then,
$\textsf{M}$
-types are defined as the terminal coalgebras for the following, so-called polynomial endofunctors
$\textsf{P}_{A,B}$
on the category
$\mathbf{Set}$
\begin{equation*} \textsf{P}_{A,B}(X) :\equiv (\Sigma x \in A)(B(x) \to X) \end{equation*}
with
$A \; type$
intuitively giving the set of nodes’ labels and
$B(x) \; type \; [x \in A]$
representing the tree’s branching function.
For dependent
$\textsf{M}$
-types, fix a closed type
$A$
and consider the category
$\mathbf{Fam}(A)$
of
$A$
-dependent families; its objects are dependent types
$ B(x) \; type \; [x \in A]$
; an arrow between objects
$B(x)$
and
$C(x)$
is a function
$f \in (\Pi x \in A)(B(x) \to C(x))$
; as before, the arrows are considered up to propositional equality. Given an indexed container
$(A,I,Br,ar)$
, the associated dependent
$\textsf{M}$
-type is defined as the terminal coalgebra of the following dependent polynomial endofunctor on the category
$\mathbf{Fam}(A)$
.
\begin{equation*} \textsf{Der}_{Br,ar}(P)(x) :\equiv (\Sigma y \in I(x))(\Pi z \in Br(x,y))P(ar(x,y,z)) \end{equation*}
In these forms, plain and dependent
$\textsf{M}$
-types have been shown to exist in Martin–Löf’s type theory with
$\textsf{W}$
-types, function extensionality
$\textsf{funext}$
and axiom
$\textsf{K}$
(Altenkirch et al. Reference Altenkirch, Ghani, Hancock, McBride and Morris2015). By internally approximating the above categories and endofunctors in
$\mathbf{HoTT}$
, Ahrens et al. (Reference Ahrens, Capriotti and Spadotti2015) have defined an analogous notion of internal M-type and proved its existence.
Here, to define coinductive predicates and positivity relations in Martin–Löf’s type theory, we follow instead the axiomatic approach taken by Maietti et al. (Reference Maietti, Maschio and Rathjen2022), where rules for proof-relevant coinductive positivity relations have been explicitly introduced in Martin–Löf’s type theory by just taking the corresponding rules for the intensional level of the Minimalist Foundation after identifying propositions with sets. The explicit rules are listed in Appendix C. Notice that, in this way, coinductive predicates and positivity relations in Martin–Löf’s type theory can be interpreted as the greatest fixed points of the following endomorphisms on the preorder reflection of the category
$\mathbf{Fam}(A)$
, respectively.
\begin{align*} \textsf{Conf}_{I,C}(P)(x) &:\equiv (\Pi y \in I(x))(\Sigma z \in A)(C(x,y,z) \times P(z)) \\[2pt] \textsf{Conf}_{I,C,V}(P)(x) & :\equiv V(x) \times (\Pi y \in I(x))(\Sigma z \in A)(C(x,y,z) \times P(z)) \end{align*}
The way they are defined, coinductive predicates are not related to inductive predicates in the same way that
$\textsf{M}$
-types are related to
$\textsf{W}$
-types: while coinductive predicates are defined as the greatest fixed points of the operator
$\textsf{Conf}$
, which is itself dual to the defining operator
$\textsf{Der}$
for inductive predicates,
$\textsf{M}$
-types and
$\textsf{W}$
-types (and their dependent versions) are terminal coalgebras and initial algebras, respectively, of the same endofunctors. For this reason, it seems we cannot get a fully symmetrical result to the one obtained in the inductive case. Nonetheless, thanks to the distributivity of
$\Pi$
over
$\Sigma$
(which, in a propositions-as-types reading, amounts to the axiom of choice), we can prove that the class of constructors
$\textsf{Der}_{Br,ar}$
can encode the class
$\textsf{Conf}_{I,C}$
.
Proposition 11.
In
$\mathbf{MLTT_0}+\textsf{funext}$
, each functor of the form
$\textsf{Conf}_{I,C}$
is naturally isomorphic to one of the form
$\textsf{Der}_{Br,ar}$
.
Proof.
Assume to have a rule set
$(I,C)$
over
$A$
and consider the following parameters.
\begin{align*} I'(x) & :\equiv (\Pi y \in I(x))(\Sigma z \in A)C(x,y,z) \\ Br(x,f) & :\equiv I(x) \\ ar(x,f,y) & :\equiv \pi _1(f(y)) \end{align*}
For each
$A$
-dependent type
$P(x) \; type \; [x \in A]$
, we obtain the following isomorphism between
$A$
-dependent types.
\begin{align*} \textsf{Der}_{Br,ar}(P)(x) & = (\Sigma f \in (\Pi y \in I(x))(\Sigma z \in A)C(x,y,z))(\Pi y \in I(x))P(\pi _1(f(y))) \\[2pt] & \cong (\Pi y \in I(x))(\Sigma w \in (\Sigma z \in A)C(x,y,z))P(\pi _1(w)) \\[2pt] & \cong (\Pi y \in I(x))(\Sigma z \in A)(C(x,y,z) \times P(z)) \\[2pt] & = \textsf{Conf}_{I,C}(P)(x) \end{align*}
Notice that the first isomorphism above is precisely an application of the distributivity of
$\Pi$
over
$\Sigma$
. It is easy to check that such a family of isomorphisms is natural in
$P$
.
Corollary 12.
Coinductive predicates and coinductive positivity relations are mutually encodable in
$\mathbf{MLTT_0}+\textsf{funext}$
. Moreover, they are both encodable in any theory extending
$\mathbf{MLTT_0}+\textsf{funext}$
in which dependent
$\textsf{M}$
-types exist.
Proof.
Since the rules of the constructors are identical, the proof of the first statement is entirely analogous to that of the intensional level of the Minimalist Foundation. Then, it is enough to prove the second statement for coinductive predicates. If the theory admits a terminal coalgebra for every endofunctor of the form
$\textsf{Der}_{Br,ar}$
, then, by Proposition 11, it equivalently admits one also for every endofunctor of the form
$\textsf{Conf}_{I,C}$
, which can be used to interpret the coinductive predicates
$\textsf{CoInd}_{I,C}(x)$
.
We observe that in the previous results, the assumption of function extensionality is required only to make sure that the operators involved are actually functors.
5. Compatibility of (co)induction
The results obtained in the previous sections allow us to extend some of the compatibility results described in Section 2 to (co)inductive definitions.
Corollary 13.
-
(1)
$\mathbf{mTT}_{\textsf{ind}}$
is compatible with
$\mathbf{MLTT_0}+\textsf{W}$
; -
(2)
$\mathbf{mTT}_{\textsf{ind}}$
is compatible with
$\mathbf{HoTT}$
; -
(3)
$\mathbf{mTT}_{\textsf{cind}}$
is compatible with
$\mathbf{MLTT_0}+\textsf{W}+\textsf{funext}+\textsf{K}$
; -
(4)
$\mathbf{mTT}_{\textsf{cind}}$
is compatible with
$\mathbf{CIC}$
; -
(5)
$\mathbf{emTT}_{\textsf{cind}}$
is compatible with
$\mathscr{T}_{\mathbf{Topos}}$
. -
(6)
$\mathbf{emTT}_{\textsf{ind}}$
is compatible with
$\mathbf{CZF}+\textsf{REA}$
; -
(7)
$\mathbf{emTT}_{\textsf{cind}}$
is compatible with
$\mathbf{CZF}+\textsf{RRS}\text{-}\bigcup \textsf{REA}$
.
In the last two points above,
$\textsf{REA}$
and
$\textsf{RRS}\text{-}\bigcup \textsf{REA}$
are the Extension Axiom schemes defined for Regular sets and Strongly Regular sets satisfying the Relation Reflection Scheme, respectively. They were introduced to accommodate inductive and coinductive definitions in constructive set theory; for their precise statements see (Aczel and Rathjen, Reference Aczel and Rathjen2010).
Proof.
-
(1) Inductive predicates can be interpreted straightforwardly into their analogues defined in Martin–Löf’s type theory. In turn, thanks to (Maietti and Sabelli, Reference Maietti and Sabelli2023, Theorem 6.1), we know how to construct the latter using
$\textsf{W}$
-types. -
(2) Despite
$\mathbf{HoTT}$
being an extension of
$\mathbf{MLTT_0}+\textsf{W}$
, the interpretation defined in the previous point ceases to be compatible once the target theory is changed to
$\mathbf{HoTT}$
. This is because propositions of the Minimalist Foundation should be interpreted as h-propositions. Moreover, as already observed in the case of inductive basic covers in (Coquand and Tosun, Reference Coquand and Tosun2020), the propositional truncation of a dependent
$\textsf{W}$
-type does not produce an inductive predicate. The solution adopted there, which also works for interpreting inductive predicates, is to make more use of the expressive power of Higher Inductive Types which allow postulating an introduction constructor
$\textsf{ind}$
at the same time with another constructor whose function is to trivialise the identity type. -
(3) The target theory satisfies the hypotheses of Corollary 12, since dependent
$\textsf{M}$
-types can be constructed from
$\textsf{W}$
-types in Martin–Löf’s type theory with function extensionality and Axiom K (Altenkirch et al. Reference Altenkirch, Ghani, Hancock, McBride and Morris2015). Therefore, it is enough to extend the interpretation described in the first point by interpreting
$\mathbf{mTT}$
-coinductive predicates as
$\mathbf{MLTT_0}$
-coinductive predicates, once we know that the target theory supports them. -
(4) By Proposition 6.
-
(5) By Remark 5.
-
(6) By Theorem 4.6 of Maietti et al. (Reference Maietti, Maschio and Rathjen2022).
-
(7) By Theorem 13.2.3 of Aczel and Rathjen (Reference Aczel and Rathjen2010).
Remark 14. Maietti et al. (Reference Maietti, Maschio and Rathjen2022) proved a compatibility result for coinductive predicates in Martin–Löf’s type theory alternative to the one in point 3 of the above Corollary. There, instead of constructing coinductive predicates as
$\textsf{M}$
-types, they assumed in the target theory the existence of a Palmgren’s superuniverse and encoded them directly.
Remark 15.
The compatibility of
$\mathbf{mTT}_{\textsf{cind}}$
with
$\mathbf{HoTT}$
remains an open problem. This is because, to adapt the interpretation described by Contente and Maietti (Reference Contente and Maietti2024) to account for coinductive predicates, one should be able to coinductively generate h-propositions in
$\mathbf{HoTT}$
, but it is not evident how and if this can be done.
6. Conclusion and Future Work
We have shown that (co)inductive methods of formal topology are equivalent to (co)inductive predicates and that, in turn, they can all be constructed in Martin–Löf’s type theory using (non)-wellfounded trees.
In the future, we aim to motivate the addition of
$\textsf{M}$
-types in the Minimalist Foundation and implement them in the quotient model (Maietti, Reference Maietti2009) and the realisability interpretation (Maietti et al. Reference Maietti, Maschio and Rathjen2022). Further goals would be to deepen the categorical semantics of coinductive predicates in a locally cartesian closed category.
Acknowledgements
Thanks to Francesco Ciraulo, Milly Maietti and Giovanni Sambin for helpful discussions on the topic of this article. This work was supported by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA-INDAM).
Competing interests
The author declares that he has no competing interests.
Appendix A. Rules for (co)inductive predicates in mTT
Rules for inductive predicates in mTT
\begin{equation*} \textit {Parameters:}\quad A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in A \to {\textsf{Prop}_{s}} \; [x \in A, y \in I(x)] \end{equation*}
\begin{equation*} \textsf{F}-\textsf{Ind}\;\frac {a \in A}{\textsf{Ind}_{I,C}(a) \; prop_s} \end{equation*}
\begin{equation*} \textsf{I}-\textsf{Ind}\; \frac { a \in A \quad i \in I(a) \quad p \in (\forall x \,\varepsilon \, C(a,i))\textsf{Ind}_{I,C}(x) } { \textsf{ind}(a,i,p) \in \textsf{Ind}_{I,C}(a) } \end{equation*}
\begin{equation*} \textsf{E}-\textsf{Ind}\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & c(x,y,w) \in P(x) \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(x,y))P(z)] \\ & a \in A \quad p \in \textsf{Ind}_{I,C}(a) \end{aligned}} {\textsf{El}_{\textsf{Ind}}(a,p,(x,y,w).c) \in P(a)} \end{equation*}
\begin{equation*} \textsf{C}-\textsf{Ind}\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & c(x,y,w) \in P(x) \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(a,i))P(z)] \\ & a \in A \quad i \in I(a) \quad p \in (\forall x \,\varepsilon \, C(a,i))\textsf{Ind}_{I,C}(x) \end{aligned}} {\textsf{El}_{\textsf{Ind}}(a,\textsf{ind}(a,i,p),(x,y,w).c) = c(a,i,\lambda z.\lambda q.\textsf{El}_{\textsf{Ind}}(z,p(z,q),(x,y,w).c)) \in P(a)} \end{equation*}
Rules for coinductive predicates in mTT
\begin{equation*} \textit {Parameters:}\quad A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in A \to {\textsf{Prop}_{s}} \; [x \in A, y \in I(x)] \end{equation*}
\begin{equation*} \textsf{F}-\textsf{CoInd}\;\frac {a \in A}{\textsf{CoInd}_{I,C}(a) \; prop_s} \end{equation*}
\begin{equation*} \textsf{E}-\textsf{CoInd}\;\frac {a \in A \quad i \in I(a) \quad p \in \textsf{CoInd}_{I,C}(a)}{ \textsf{El}_{\textsf{CoInd}}(a,i,p) \in (\exists x \,\varepsilon \, C(a,i))\textsf{CoInd}_{I,C}(x)} \end{equation*}
\begin{equation*} \textsf{I}-\textsf{CoInd}\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & c(x,y,w) \in (\exists z \,\varepsilon \, C(x,y))P(z) \; [x \in A, y \in I(x), w \in P(x)] \\ & a \in A \quad p \in P(a) \end{aligned}}{\textsf{coind}(a,p,(x,y,w).c) \in \textsf{CoInd}_{I,C}(a)} \end{equation*}
Appendix B. Rules for topological constructors in
$\mathbf{MF}$
Rules for inductive basic covers in emTT
\begin{equation*} \textit {Parameters:}\quad A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in \mathscr{P}(A) \; [x \in A, y \in I(x)] \quad V \in \mathscr{P}(A) \end{equation*}
\begin{equation*} \textsf{F}-\vartriangleleft \;\frac {a \in A}{a \vartriangleleft _{I,C} V\; prop_s} \end{equation*}
\begin{equation*} \textsf{I}_{\textsf{rf}}-\vartriangleleft \;\frac {a \,\varepsilon \, V\; true}{a \vartriangleleft _{I,C} V\; true} \end{equation*}
\begin{equation*} \textsf{I}_{\textsf{tr}}-\vartriangleleft \;\frac {a \in A \quad i \in I(a) \quad (\forall x \,\varepsilon \, C(a,i))x \vartriangleleft _{I,C} V \; true}{a \vartriangleleft _{I,C} V\; true} \end{equation*}
\begin{equation*} \textsf{E}-\vartriangleleft \;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & (\forall x \in A)(x \,\varepsilon \, V \vee (\exists y \in I(x))(\forall z \,\varepsilon \, C(x,y))P(z) \Rightarrow P(x)) \; true \\ & a \in A \quad a \vartriangleleft _{I,C} V\; true \end{aligned} } {P(a) \; true} \end{equation*}
Rules for coinductive positivity relations in emTT
\begin{equation*} \textit {Parameters:}\quad A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in \mathscr{P}(A) \; [x \in A, y \in I(x)] \quad V \in \mathscr{P}(A) \end{equation*}
\begin{equation*} \textsf{F}-{\ltimes }\;\frac {a \in A}{a \ltimes _{I,C} V\; prop_s} \end{equation*}
\begin{equation*} \textsf{E}_\textsf{corf}-\ltimes \;\frac {a \ltimes _{I,C} V\; true}{a \,\varepsilon \, V\; true} \end{equation*}
\begin{equation*} \textsf{E}_\textsf{cotr}-\ltimes \;\frac {a \in A \quad i \in I(a) \quad a \ltimes _{I,C} V \; true}{(\exists x \,\varepsilon \, C(a,i))x \ltimes _{I,C} V\; true} \end{equation*}
\begin{equation*} \textsf{I-}\ltimes \;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & (\forall x \in A)(P(x) \Rightarrow x \,\varepsilon \, V \wedge (\forall y \in I(x))(\exists z \,\varepsilon \, C(x,y))P(z)) \; true \\ & a \in A \quad P(a) \; true \end{aligned} } {a \ltimes _{I,C} V \; true} \end{equation*}
Rules for inductive basic covers in mTT
\begin{align*} & \textit {Parameters:} \\ & A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in A \to \textsf{Prop}_\textsf{s} \; [x \in A, y \in I(x)] \quad V \in A \to \textsf{Prop}_\textsf{s} \end{align*}
\begin{equation*} \textsf{F}-\vartriangleleft \;\frac {a \in A}{a \vartriangleleft _{I,C}V \; prop_s} \end{equation*}
\begin{equation*} \textsf{I}_\textsf{rf}-\vartriangleleft \; \frac { a \in A \quad r \in a\,\varepsilon \,V } { \textsf{rf}(a,r) \in a \vartriangleleft _{I,C}V } \end{equation*}
\begin{equation*} \textsf{I}_\textsf{tr}-\vartriangleleft \; \frac { a \in A \quad i \in I(a) \quad p \in (\forall x \,\varepsilon \, C(a,i))x \vartriangleleft _{I,C}V } { \textsf{tr}(a,i,p) \in a \vartriangleleft _{I,C}V } \end{equation*}
\begin{equation*} \textsf{E}-{\vartriangleleft }\;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & q_1(x,y) \in P(x) \; [x \in A, y \in x \,\varepsilon \,V] \\ & q_2(x,y,w) \in P(x) \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(x,y))P(z)] \\ & a \in A \quad p \in a \vartriangleleft _{I,C}V \end{aligned}} {\textsf{El}_\vartriangleleft (a,p,(x,y).q_1,(x,y,w).q_2) \in P(a)} \end{equation*}
\begin{equation*} \textsf{C}_\textsf{rf} -\vartriangleleft \;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & q_1(x,y) \in P(x) \; [x \in A, y \in x \,\varepsilon \,V] \\ & q_2(x,y,w) \in P(x) \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(x,y))P(z)] \\ & a \in A \quad r \in a \,\varepsilon \,V \end{aligned}} {\textsf{El}_\vartriangleleft (a,\textsf{rf}(a,r),(x,y).q_1,(x,y,w).q_2) = q_1(a,r) \in P(a)} \end{equation*}
\begin{equation*} \textsf{C}_\textsf{tr}-\vartriangleleft \;\frac { \begin{aligned} & P(x) \; prop \; [x \in A] \\ & q_1(x,y) \in P(x) \; [x \in A, y \in x \,\varepsilon \,V] \\ & q_2(x,y,w) \in P(x) \; [x \in A, y \in I(x), w \in (\forall z \,\varepsilon \, C(x,y))P(z)] \\ & a \in A \quad i \in I(a) \quad p \in (\forall x \,\varepsilon \, C(a,i))x \vartriangleleft _{I,C}V \end{aligned}}{\textsf{El}_\vartriangleleft (a,\textsf{tr}(a,i,p),q_1,q_2) = q_2(a,i,\lambda z.\lambda w. \textsf{El}_\vartriangleleft (z,p(z,w),q_1,q_2) \in P(a)} \end{equation*}
Rules for coinductive positivity relation in mTT
\begin{align*} & \textit {Parameters:} \\ & A\; set \quad I(x) \; set \; [x \in A] \quad C(x,y) \in A \to \textsf{Prop}_\textsf{s} \; [x \in A, y \in I(x)] \quad V \in A \to \textsf{Prop}_\textsf{s} \end{align*}
\begin{equation*} \textsf{F}-\ltimes \;\frac {a \in A}{a \ltimes _{I,C} V \; prop_s} \end{equation*}
\begin{equation*} \textsf{E}_\textsf{corf}-\ltimes \;\frac { a \in A \quad p \in a \ltimes _{I,C} V }{\textsf{corf}(a,p) \in a \,\varepsilon \, V} \end{equation*}
\begin{equation*} \textsf{E}_\textsf{cotr}-\ltimes \;\frac { a \in A \quad i \in I(a) \quad p \in a \ltimes _{I,C} V}{\textsf{cotr}(a,i,p) \in (\exists x \,\varepsilon \, C(a,i))x \ltimes _{I,C} V}\end{equation*}
\begin{equation*} \textsf{I}-\ltimes \;\frac {\begin{aligned} & P(x) \; prop \; [x\in A]\\ & q_1(x,y) \in x \,\varepsilon \, V \; [x \in A,y \in P(x)]\\ & q_2(x,y,w) \in (\exists z \,\varepsilon \, C(x,y))P(z) \; [x \in A,y \in I(x),w\in P(x)]\\ & a \in A \quad p \in P(a) \end{aligned}} {\textsf{coind}(a,p,q_1,q_2) \in a \ltimes _{I,C} V} \end{equation*}
Appendix C. Rules for coinductive constructors in
$\mathbf{MLTT_0}$
Coinductive predicates
\begin{align*} & \textit {Parameters:} \\ & A \in \textsf{U}_{0} \quad I \in A \to \textsf{U}_{0} \quad C(x,y) \in A \to \textsf{U}_{0} \; [x \in A, y \in I(x)] \quad V \in A \to \textsf{U}_{0} \end{align*}
\begin{equation*} \textsf{F}-\textsf{CoInd}\;\frac {}{\textsf{CoInd}_{I,C} \in A \to \textsf{U}_{0}} \end{equation*}
\begin{equation*} \textsf{E}-\textsf{CoInd}\;\frac {a \in A \quad i \in I(a) \quad p \in \textsf{CoInd}_{I,C}(a)}{ \textsf{El}_{\textsf{CoInd}}(a,i,p) \in (\Sigma x \in A)(C(a,i,x) \times \textsf{CoInd}_{I,C}(x))} \end{equation*}
\begin{equation*} \textsf{I}-\textsf{CoInd}\;\frac { \begin{aligned} & M(x) \; type \; [x \in A] \\ & d(x,y,w) \in (\Sigma z \in A)(C(x,y,z) \times M(z)) \; [x \in A, y \in I(x), w \in P(x)] \\ & a \in A \quad m \in M(a) \end{aligned}}{\textsf{coind}(a,m,(x,y,w).d) \in \textsf{CoInd}_{I,C}(a)} \end{equation*}
Coinductive positivity relations
\begin{align*} & \textit {Parameters:} \\ & A \in \textsf{U}_{0} \quad I \in A \to \textsf{U}_{0} \quad C(x,y) \in A \to \textsf{U}_{0} \; [x \in A, y \in I(x)] \quad V \in A \to \textsf{U}_{0} \end{align*}
\begin{equation*} \textsf{F}-{\ltimes }\;\frac {}{- \ltimes _{I,C} V \in A \to \textsf{U}_{0}} \end{equation*}
\begin{equation*} \textsf{E-corf}-{\ltimes }\;\frac { a \in A \quad p \in a \ltimes _{I,C} V }{\textsf{corf}(a,p) \in V(a)} \end{equation*}
\begin{equation*} \textsf{E-cotr}-{\ltimes }\;\frac { a \in A \quad i \in I(a) \quad p \in a \ltimes _{I,C} V}{\textsf{cotr}(a,i,p) \in (\Sigma x \in A)(C(a,i,x) \times x \ltimes _{I,C} V)}\end{equation*}
\begin{equation*} \textsf{I}-\ltimes \;\frac {\begin{aligned} & M(x) \; type \; [x \in A]\\ & q_1(x,y) \in V(x) \; [x \in A,y \in M(x)]\\ & q_2(x,y,w) \in (\Sigma z \in A)(C(x,y,z) \times x \ltimes _{I,C} V) \; [x \in A,y \in I(x),w\in M(x)]\\ & a \in A \quad p \in M(a) \end{aligned}} {\textsf{coind}(a,p,q_1,q_2) \in a \ltimes _{I,C} V} \end{equation*}
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