From type theory to setoids and back

Abstract

A model of Martin-Löf extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-Löf intensional type theory. This may be understood, we claim, as a solution to the old problem of modeling the full extensional theory in the intensional theory. Types are interpreted as setoids, and the model is therefore a setoid model.We solve the problem of interpreting type universes by utilizing Aczel’s type of iterative sets and show how it can be made into a setoid of small setoids containing the necessary setoid constructions. In addition, we interpret the bracket types of Awodey and Bauer. Further quotient types should be interpretable.

1. Foreword

The present paper was submitted by Erik Palmgren in September 2019 and was under review when he sadly passed away in November the same year. After discussions between the main editor (Pierre-Louis Curien), the managing editor (Thierry Coquand), and the reviewer (Peter Dybjer), it was decided to publish the paper as submitted. We think that it is an important paper, since it provides a formalized interpretation of the extensional version of Martin-Löf type theory in an intensional version and since such an interpretation has foundational significance. The draft paper is also well-written and will be published as submitted with only a few minor changes, indicated by footnotes in the text. Nevertheless, the reader should be aware that it still is a draft version.

The publication of this paper is a tribute to our dear friend and valued colleague Erik Palmgren. It is sad to realize that it will be the last in the long list of Erik’s important contribution to type theory and logic.

2. Introduction

In this paper, we present an interpretation of full extensional Martin-Löf type theory (Martin-Lof, 1982) into intensional Martin-Löf type theory via setoid constructions. There are several qualifications to this statement. We actually formalize this interpretation in the Agda proof assistant, where the fragment of the system used is considered as the intensional type theory. Our system of extensional type theory is not a syntactically defined system, but rather a system of closure rules for judgments about setoids in Agda. The fragment of Agda used is limited to certain kinds of inductive-recursive definitions and record types. The K rule of Agda is not used. We believe that the proofs carried out in this fragment may also be carried out in a Logical Framework presentation of Martin-Löf type theory with a super universe (Palmgren, 1998). A super universe is closed under construction of universes, in addition to the standard type constructions introduced in Martin-Lof, (1982).

A first approach that may come to mind when interpreting extensional type theory using setoid constructions is to interpret a family of types over a context as a family of setoids over a setoid that interprets the context. (For background on setoids, see Section 3.) The basic judgment forms of Martin-Löf type theory (Martin-Lof, 1984) are displayed to the left in the table below.

$$\begin{array}{llll}&\Gamma \Longrightarrow A \; \textrm{type} &\qquad & A :\textrm{Fam}(\Gamma) \\&\Gamma \Longrightarrow A = B && ? \\&\Gamma \Longrightarrow a : A && a : \Pi(\Gamma, A) \\& \Gamma \Longrightarrow a = b : A && a =_{\Pi(\Gamma, A)} b\end{array}$$

We may now try to interpret the forms of judgments as the statements about setoids to the right. But we do not yet have any obvious interpretation of the type equality. We need to compare A and B as setoid families over $\Gamma$ . A crucial problem is how to interpret the type equality rule

(1)

A solution is to embed all dependent families of setoids in to a big universal setoid (or as we will call it classoid). To obtain a setoid model without coherence problems, we may seek inspiration from type-free interpretations of (extensional) type theory; see Aczel (1977), Smith (1984), Beeson (1985, Ch. XI). But instead of using combinators or recursive realizers as type-free objects, we use constructive iterative sets in the sense of Aczel (1978).

Aczel’s type of iterative sets V (Aczel, 1978) consists of well-founded trees where the branching f can be indexed by any type A in a universe U of small types. The introduction rule tells how to build a set $\alpha=\textrm{sup}(A,f)$ from a family f(x) ( $x: A$ ) of previously constructed sets

Equality $=_V$ is defined by the smallest bisimulation, and then, membership is given by

$$x \in_V \textrm{sup}(A,f) := (\exists a: A) (x =_V f(a)).$$

The classoid $ {\mathbb V} = (V,=_V)$ forms, together with the membership relation $\in_V$ , a model of Constructive Zermelo-Fraenkel set theory (CZF) with Dependent Choice (DC), and possibly further axioms, depending on the type theory. It is thus expected to be a rich universe of sets. In fact, each set $\alpha= \textrm{ sup}(A,f)$ may also be understood as a setoid on the type A

(2) \begin{equation} \kappa(\alpha) = (A, =_f)\end{equation}

where $a =_f b$ is defined as $ f(a) =_V f(b)$ . The assignment $\kappa$ may be extended to a full and faithful functor from the category of sets in V to the category of small setoids. Using $\kappa,$ we can also construct a bijection of classoids

$$ {\mathbb V} \cong \textrm{Sub}({\mathbb V}).$$

Following Aczel (1982), one can see that there are internal versions in V of the setoid construction for $\Pi$ , $\Sigma$ and extensional identity, which commute with $\kappa$

$$ \kappa(\sigma(\alpha,f)) \cong \Sigma(\kappa(\alpha), \kappa \circ f) \qquad\kappa(\pi(\alpha,f)) \cong \Pi(\kappa(\alpha), \kappa \circ f)$$

Thus, ${\mathbb V}$ is suitable for interpreting both terms and types of dependent type theory. A type A in a context $\Gamma$ will be interpreted as an extensional function $A : \kappa(\Gamma) \rightarrow {\mathbb V}$ . Now any two types in the same context can be compared. A raw term a in a context $\Gamma$ will likewise be interpreted as extensional function $a : \kappa(\Gamma) \rightarrow {\mathbb V}$ . The judgment $a : A$ will be interpreted as membership judgment. The new setoid interpretation is on the right in the table below.

(3) \begin{equation} \begin{array}{llll}&\Gamma \Longrightarrow A \; \textrm{type} &\qquad & A : [\kappa(\Gamma) \rightarrow {\mathbb V}]\\&\Gamma \Longrightarrow A = B && A =_\textrm{ext} B: [\kappa(\Gamma) \rightarrow {\mathbb V}]\\&\Gamma \Longrightarrow a : A && \forall x : \kappa(\Gamma), a(x) \in_V A(x) \\& \Gamma \Longrightarrow a = b : A && \forall x : \kappa(\Gamma), a(x)=_V b(x) \in_V A(x)\end{array}\end{equation}

The interpretation of the problematic type equality rule (1) is now direct. Further, the basic rules in type theory for $\Sigma$ , $\Pi$ , $+$ , extensional identity types, and the basic types $N_0$ and N can now be interpreted; see Section 4. Some further considerations are necessary to interpret the hierarchy of type universes. Here we use a superuniverse (Palmgren, 1998; Rathjen, 2000, 2001), which is a type universe closed under the operation of building a universe over a family of base types. This makes it possible to build the hierarchy of setoid universes internally to the superuniverse and interpret the universe rules à la Russell. This is covered in Section 5. Bracket type constructions are defined in Section 6. The interpretation of the judgment forms is fixed in Section 7. Section 8 lists all the rules interpreted together with references to the formalization. Section 9 contains links to the actual formalization which is available online. A comparison between Agda and the Logical Framework is made in Section 10.

Related work

Though it was widely recognized from the beginning of Martin-Löf type theory that it had a natural classical set-theoretic interpretation, a proof seems only to have been written down and published in detail by Salvesen in her 1986 MSc thesis (Salvesen, 1986). Closely related to the work of the present paper is that of Aczel (1999) who interprets extensional Martin-Löf type theory with universes in an extension of CZF with a hierarchy of inaccessible sets. Earlier Werner (1997) had modeled a Coq system in ZFC and vice versa ZFC in Coq using Aczel’s encoding of sets. A refinement by Barras models a Coq system in intuitionistic ZF and formalizes the model in Coq (Barras, 2010). Rathjen and Tupailo (2006) make a close analysis of the interpretation of CZF into Martin-Löf type theory, and the question about what general classes of set-theoretic statements are validated in type theory.

Hofmann (1997) modeled an extensional Martin-Löf type theory $\textrm{TT}_\textrm{E}$ (without universes) in an “intensional” version of the theory $\textrm{TT}_\textrm{I}$ . A conservativity theorem (for type inhabitation) of $\textrm{TT}_\textrm{E}$ over $\textrm{TT}_\textrm{I}$ is established (Hofmann, 1997, Thm 3.2.5). Note that $\textrm{TT}_\textrm{I}$ has function extensionality and the UIP axiom, so it is different from what is usually called intensional Martin-Löf type theory (Nordstrom et al. 1990). Hofmann (1997, Ch 5.1, 5.3) also constructs setoid models of quotient types. The setoids are Prop-valued, or proof-irrelevant, in the sense that the truth values of equalities are in the type of propositions $\textrm{Prop}$ rather than in $\textrm{Set}$ , the type of small types, as in the Definition 3.1 we use below. The equality of morphisms and families in these setoid models are however definitional, which is an another difference.

The Minimalist Foundation of Maietti and Sambin (2005) is a two level type theory consisting of an extensional theory and a more fundamental theory, Minimal type theory, which is intensional. The extensional level is modeled (Maietti, 2009) into the minimal theory using a quotient construction. Moreover, equality of morphisms in this model is extensional as is ours. Further related constructions of extensional and quotients structures are in Maietti and Rosolini, (2012, 2013).

Some obstacles and opportunities for constructing convenient categories and universes of setoids inside intensional Martin-Löf type theory are demonstrated and discussed in Palmgren (2012, 2018b); Palmgren andWilander (2014); Wilander (2010, 2012).

3. Preliminaries

We recall some definitions and facts around setoids. Note that we use the propositions-as-types principle throughout. In particular, the notion of setoid uses this principle (in contrast to, for example, setoids of the standard library in the Coq system).

3.1 Setoids

Definition 3.1 A setoid $A=(|A|,=_A)$ is type A together with an equivalence relation $=_A$ , so that $x=_Ay$ is type. A setoid map, or extensional function $f: A \rightarrow B,$ is a pair $f=(|f|, \textrm{ext}_f)$ consisting of a function $|f| : |A| \rightarrow |B|$ and $\textrm{ext}_f$ a proof of extensionality, i.e.

$$(\forall x,y : |A|)[x =_A y \Rightarrow |f|(x) =_B |f|(y)].$$

Write $a : A$ for $a : |A|$ , and $f(x) = |f|(x)$ for a setoid A and a setoid map f.

For setoids A and B, the product setoid $A\times B$ is given by

$$|A\times B| =_\textrm{def} |A| \times |B|$$

and

$$(a,b) =_{A \times B} (c,d) \Longleftrightarrow_\textrm{def} a=_A c \land b =_B d.$$

For setoids A and B, the exponent setoid $[A \rightarrow B] = B^A $ is given by

$$|B^A| =_\textrm{def} (\Sigma f: |A| \rightarrow |B|)(\forall x,y: |A|)(x=_A y \Rightarrow f(x) =_B f(y))$$

and

$$(f,p) =_{B^A} (g,q) \Longleftrightarrow_\textrm{def} (\forall x: |A|)(f(x)=_B g(x)).$$

With type universes, we may introduce some distinctions of setoids which are useful and necessary to solve predicativity problems. Let $U_n$ , $n=0,1,2,\ldots$ denote the cumulative universes of a Martin-Löf type theory, à la Russell. (In Agda and Coq these are available as Set0, Set1, Set2, … resp Type0, Type1, Type2, ….) We recall a definition and some examples from Palmgren, (2018a ):

Definition 3.2 An (m, n)-setoid $A=(|A|,=_A)$ is a type $|A| \in U_m$ with an equivalence relation $=_A : |A| \rightarrow |A| \rightarrow U_n$ . An (n,n)-setoid will be called simply n-setoid. An $(n+1,n)$ -setoid is called an n-classoid.

As a justification for the term classoid, we note that there is a “replacement scheme”: if $f:A \rightarrow B$ is an extensional function from A, an m-setoid, to B an m-classoid, the image $\textrm{Im}(f)$ is an m-setoid. This is analogous to the replacement scheme in set theory.

Example 3.3 1. If $A \in U_n$ , then $\overline{A}=(A, \textrm{Id}_A(\cdot, \cdot))$ is an n-setoid.

2. The pair $\Omega_n = (U_n,\leftrightarrow)$ , where $\leftrightarrow$ is logical equivalence, is an n-classoid.

3. Aczel’s standard model V of CZF is built on the W-type over a universe $U_0$ with $|V| = W(U_0,T_0)$ and the equality $ =_V$ defined by bisimulation as function $|V| \rightarrow |V| \rightarrow U_0$ . Thus, $V=(|V|, =_V)$ forms a 0-classoid. Similarly, constructing $V_k$ from a universe $U_k$ , $T_k$ yields a k-classoid.

4. If A is an (m,n)-setoid and B is an (m’,n’)-setoid, then the exponential $[A \rightarrow B]$ is an $(\max(m,m',n,n'),\max(m,n'))$ -setoid. In particular, if A and B are both (m,n)-setoid, then $[A \to B]$ is an $(\max(m,n),\max(m,n))$ -setoid. Thus, (m,n)-setoids are also closed under exponents.

5. For an n-setoid A, the setoid of extensional propositional functions of level n

$$P_n(A) = [A \rightarrow \Omega_n]$$

is an n-classoid.

Subsetoids may be defined following Bishop cf. (Palmgren, 2005).

Definition 3.4 Let A be a setoid. A subsetoid of A is a setoid $\delta S$ together with an injective setoid map $\iota_S : \delta S \rightarrow A$ . An element $a : A$ is said to be a member of the subsetoid $S= (\delta S, \iota_S)$ if there is an $s:\delta S$ such that $a =_A \iota_S(s)$ . We then write $a \in S$ or $a \in_A S$ . (Note that s is unique.) If S and T are subsetoids of A, then we define

(4) \begin{equation} S \subseteq_A T \Longleftrightarrow_\textrm{def} (\forall x: A)[x \in_A S \Longrightarrow x \in_A T].\end{equation}

When A is clear from the context, we drop this subscript. Define also

$$S \equiv_A T \Longleftrightarrow_\textrm{def} S \subseteq_A T \land T \subseteq_A S.$$

Using the axiom of unique choice, it can be seen that

(5) \begin{equation} S \subseteq_A T \Longleftrightarrow (\exists f: [\delta S \rightarrow \delta T])[\iota_T \circ f = \iota_S].\end{equation}

Here, f is in fact injective and unique. Whenever $(\delta S,\iota_S) \equiv_A (\delta T,\iota_T),$ there is a unique isomorphism $\phi : \delta S \to \delta T$ such that $\iota_T \circ \phi = \iota_S$ .

Definition 3.5 Let A be a fixed (m,n)-setoid. Define $\textrm{Sub}^{m,n}_{k,\ell}(A)$ to be the type of all subsetoids $(S,\iota_S)$ such that S is a $(k,\ell)$ -setoid, and the type is equipped with the equivalence relation $\equiv_A$ . Each such subsetoid is given by the data

• $|S| \in U_k$ ,

• $=_S : |S| \rightarrow |S| \rightarrow U_\ell$ ,

• $|\iota_S| : |S| \rightarrow |A|$

such that $(\forall x, y : |S|)[x=_S y \leftrightarrow \iota_S(x) =_A \iota_S(y)].$

In this paper, we will be using only the cases $\textrm{Sub}^{m,m}_{m,m}(A)$ and $\textrm{Sub}^{m+1,m}_{m,m}(A)$ , i.e. when A is an m-setoid or an m-classoid, and we are collecting the m-subsetoids. However, the levels for the general cases can be analyzed as follows.

Remark 3.6 The data of the subsetoid are captured by a $\Sigma$ -construction in the universe of level $\max(k+1,\ell+1, m, n)$ . Two subsetoids $(S,\iota_S)$ and $(T,\iota_T)$ are equal, $(S,\iota_S) \equiv_A (T,\iota_T)$ , iff $(\exists f: [S \rightarrow T])[\iota_T \circ f = \iota_S]$ and $(\exists g: [T \rightarrow S])[\iota_S \circ g = \iota_T].$ Now, $[S \rightarrow T]$ has type level $\max(k,\ell)$ and $\iota_T \circ f = \iota_S$ has level $\max(k,n)$ . The level of the equivalence relation is thus $\max(k, \ell, n)$ .

Thus, $\textrm{Sub}^{m,n}_{k,\ell}(A)$ forms a $(\max(k+1,\ell+1, m, n), \max(k, \ell, n))$ -setoid.

In particular, $\textrm{Sub}^{m,m}_{m,m}(A)$ forms a $(m+1, m)$ -setoid, i.e. an m-classoid.

Further, $\textrm{Sub}^{m+1,m}_{m,m}(A)$ forms a $(m+1, m)$ -setoid, i.e. it is also an m-classoid.

For A an m-setoid or an m-classoid, we write $\textrm{Sub}(A)$ for $\textrm{Sub}^{m,m}_{m,m}(A)$ and $\textrm{Sub}^{m+1,m}_{m,m}(A),$ respectively. Thus, in either case $\textrm{Sub}(A)$ is an m-classoid.

Example 3.7 $\textrm{Sub}({\mathbb V})$ is a 0-classoid.

3.2 Families of setoids

Definition 3.8 Let A be a setoid. A proof-irrelevant setoid family consists of a family F(a) of setoids indexed by $a : A$ , with extensional transport functions $F(p) : F(a) \to F(b)$ for each proof $p : a =_A b$ , that are satisfying

• $F(p) =_\textrm{ext} F(q)$ for each pair of proofs $p, q : a =_A b$ (proof-irrelevance)

• $F(\textrm{r}_a) = \textrm{id}_{F(a)}$ where $\textrm{r}_a : a=_A a$ is the standard proof of reflexivity.

• $F(p \odot q) = F(p) \circ F(q)$ if $q : a =_A b$ and $p : b =_A c$ , and where $p \odot q : a =_A c$ , using the standard proof $\odot$ of transitivity.

We also write $p^{-1}: b =_A a$ for $a =_A b$ using the standard proof $(-)^{-1}$ of symmetry. Note that by functoriality and proof-irrelevance

$$F(p) \circ F(p^{-1}) = F(p^{-1}) \circ F(p) = \textrm{id}_{F(a)}$$

so each F(p) is an isomorphism.

Below, we will refer to a proof-irrelevant family of setoids as just a family of setoids, when there is no chance of confusion.

Example 3.9 The operation $\kappa$ of (2) extends to a family of setoids over the classoid ${\mathbb V}$ , for $p : \alpha =_V \beta$ we let $\kappa(p): [\kappa(\alpha) \rightarrow \kappa(\beta)]$ be given by

$$\kappa(p)(x)= \pi_1(p_1(x))$$

where

$$p= (p_1, p_2): ((\Pi x: A)(\Sigma y:B) f(x) =_V g(y)) \times ((\Pi y: B)(\Sigma x:A) f(x) =_V g(y))$$

assuming $\alpha= \textrm{sup}(A,f)$ and $\beta= \textrm{sup}(B,g)$ .

Functions into power setoids give families of setoids as can be expected:

Example 3.10 Let A and X be setoids. Let $F: A \rightarrow \textrm{Sub}(X)$ be an extensional function. Then, $F(x) = (\delta(F(x)), \iota_{F(x)})$ , with $\iota_{F(x)} : \delta F(x) \to X$ injective, and for $p: x =_A y$ , there is a unique isomorphism $\phi_p : \delta(F(x)) \rightarrow \delta(F(y))$ such that

(6) \begin{equation} \iota_{F(x)} = \iota_{F(y)} \phi_p.\end{equation}

Thus, we obtain a proof-irrelevant family $F^*$ of setoids over A by letting:

$$F^*(x) := \delta(F(x)) \qquad F^*(p) := \phi_p.$$

The conditions of the transport function are easy to check.

For a setoid map $f: A \to X,$ we say f is a global member of F if for all $x : A$ , $f(x) \in_X F(x)$ . Thus for every $x:A$ , there is a unique $s : \delta(F(x))$ such that

$$f(x) =_X \iota_{F(x)}(s).$$

Thus, there is a unique function $f^* : (\Pi x : |A|)|F^*(x)|$ such that

$$f(x) =_X \iota_{F(x)}(f^*(x)).$$

If $p: x=_A y$ , then $f(x) =_X f(y)$ , so indeed

$$\iota_{F(x)}(f^*(x)) =_X \iota_{F(y)}(f^*(y)).$$

By (6) we get

$$ \iota_{F(y)}(\phi_p(f^*(x))) =_X \iota_{F(y)}(f^*(y)).$$

Now since $\iota_{F(y)}$ is injective we have $ \phi_p(f^*(x)) =_{F^*(y)} f^*(y)$ , and thus,

$$F^*(p)(f^*(x)) =_{F^*(y)} f^*(y).$$

This leads to the following definition:

Definition 3.11 Let G be a setoid family over A. A global element of G is a family $g(x) : G(x)$ of elements indexed by $x:A$ , which is extensional in the sense that

$$(\forall p : x=_A y)[G(p)(g(x)) =_{G(y)} g(y)]$$

Note that if $g=(|g|, \textrm{ext}_g) : A \to B$ is an extensional function, and F is a family on B, the we can form a family by composition $F \circ g$ on A by defining

• $(F\circ g)(x) := F(|g|(x))$ for $x : A$

• $(F\circ g)(p) := F(\textrm{ext}_g(p))$ for $p : x =_A y$ and $x, y : A$

If f is a global element of F, then $f\circ g$ is a global element of $F\circ g$ .

Definition 3.12 For F a family on A, we can form the dependent sum $\Sigma(A,F)$ and the dependent product setoid $\Pi(A,F)$ as follows

$\Sigma(A,F) = ((\Sigma x: |A|)|F(x)|, \sim)$ where

$$(x,y) \sim (u,v) := (\exists p : x =_A u)[ F(p)(y) =_{B(u)} v ]$$

$\Pi(A,F) = (P, \sim)$ where

(7) \begin{align} P := (\Sigma f : (\Pi x: &|A|)|F(x)|)(\forall x, y : A)(\forall p : x=_A y)[F(p)(f(x)) =_{B(y)} f(y) ]\\&(f, e) \sim (g, e') := (\forall x: A)[f(x) =_{B(x)} g(y)].\end{align}

Note that $\Pi(A,F)$ consists of the global elements of F.

Next, we introduce an auxiliary notion. For a classoid X and a family H of setoids over X, we define a classoid of parameterizations

$$\textrm{Par}(X,H) =((\Sigma I : | X |)|[H(I) \rightarrow X]|, =_{\textrm{Par}(X,H)}).$$

where the equivalence relation $ (I, f) =_{\textrm{Par}(X,H)} (I',f')$ is defined as

$$(\exists p: I =_X I')(\forall x : H(I))f(x) =_X f'(H(p)(x))$$

This construction is used in (8) below.

4. Basic Types

From the type universe Set in Agda, we construct Aczel’s type of iterative sets V

which corresponds to the Agda recursive data type definition

(Note that V lives in the next type universe Set1 of Agda (Set is Set0).) This definition introduces two constants, $\texttt{V}$ a code for a type, and $\texttt{sup}$ an introduction constant. Explained in terms of LF (Section 10), one can say that Agda automatically generates the corresponding elimination constant and the associated computational equality.

Define operations to extract the index type A and the ath element f(a) from the set $\textrm{sup}(A,f)$ :

$$\# \textrm{sup}(A,f) = A \qquad \textrm{sup}(A,f) \blacktriangleright a = f(a).$$

The familiar set-theoretic construction $< a, b > \; = \{\{a\},\{a, b\}\}$ of ordered pairs is used.

The natural numbers in V may be constructed as the set

$$\textrm{natV}= \textrm{sup}(N, \textrm{nV})$$

where $\textrm{nV}(0) = \emptyset$ and $\textrm{nV}(s(m)) = \{\textrm{nV}(m)\}$ . Then, it can readily be shown that $\kappa(\textrm{natV})$ is isomorphic to the standard setoid ${\mathbb N}$ of natural numbers.

The set-theoretic version of the $\Sigma$ -construction is, for $a: V$ , $g : [\kappa(a) \rightarrow {\mathbb V}]$ ,

$$\textrm{sigmaV}(a,g) = \textrm{sup}((\Sigma y: \#(a)) \#(g(y)), \lambda u. )$$

or expressed in Agda code:

Here, $\texttt{pj1} \, \texttt{u}$ and $\texttt{pj2} \, \texttt{u}$ denote the first and second projection of the $\Sigma$ -type, respectively. Further, $\texttt{VV}$ is the classoid ${\mathbb V}$ , and $\texttt{setoidmap1}$ is the type of extensional maps. The operator $\cdot$ indicates application of such maps.

Viewing $\kappa$ as a family of setoids over ${\mathbb V,}$ we define

(8) \begin{equation} \textrm{sigma}{\mathbb V} = \lambda u.\textrm{sigmaV}(\pi_1(u), \pi_2(u)) : [\textrm{Par}({\mathbb V},\kappa) \rightarrow {\mathbb V}].\end{equation}

The set-theoretic $\Pi$ -construction is more involved. For $a : V$ and $g: [\kappa(a) \rightarrow {\mathbb V}],$ define

\begin{eqnarray*}\text{piV-iV}(a, g) &=& (\Sigma f : (\Pi x : \#(a))\#(g(a))) \\& & \quad (\forall x, y : \#(a))(\forall p : x =_{\kappa(a)} y) (\kappa \circ g)(p)(f(x)) =_{(\kappa \circ g)(y)} f(y) \\\text{piV-bV}(a, g) &=& \lambda h. \textrm{sup}(\#(a), (\lambda x. < a \blacktriangleright x , g(x) \blacktriangleright (\pi_1(h)(x)) >)) \\\text{piV}(a, g) &= & \textrm{sup}(\text{piV-iV}, \text{piV-bV}(a, g)) \\\end{eqnarray*}

The first type $\text{piV-iV}(a, g)$ singles out the extensional functions employing a $\Sigma$ -type just as in (7). The branching function piV-bV then transforms such an extensional function to its graph in terms of set-theoretic pairs. Similarly to the sigma-construction, we define:

$$\textrm{pi}{\mathbb V} = \lambda u.\textrm{piV}(\pi_1(u), \pi_2(u)) : [\textrm{Par}({\mathbb V},\kappa) \rightarrow {\mathbb V}]$$

Remark 4.1 The actual formalization uses the built-in $\Pi$ -type of Agda (x:A) -> B which may in contrast to the standard $\Pi$ -type of type theory satisfy the $\eta$ -rule. The Agda code is:

The interpretation of the extensional identity is as expected very simple: for $a:V$ and $x, y : \kappa(a)$ , let

$$\textrm{idV}(a, x, y) = \textrm{sup}((a \blacktriangleright x =_V a \blacktriangleright y), (\lambda u.a \blacktriangleright x)).$$

5. Universes

We use the type universe Set as a superuniverse (Palmgren, 1998). Agda’s data construct allows building universes via a so-called simultaneous inductive-recursive definition (Dybjer, 2000), such a definition has two parts, one inductive part which builds up the data part ( $\texttt{Uo}$ below) and a second part which defines a function ( $\texttt{To}$ below) by recursion on the data part. These parts may depend mutually on each other, as in the example below, where it is crucial.

To explain the above, we note that the universe

has the same closure rules as type universes à la Tarski in (Martin-Lof, 1984). In addition, it has constructors for lifting a given family A,(x)B into the universe

See Palmgren, (1998) for details.

Considering that the set universe V can be obtained by applying a W-type

to a type universe (Aczel 1982; Martin-Löf 1984), we get a method for constructing a hierarchy of Aczel universes. This gives us a set universe $\textrm{sV}(I, F)$ for each family of types I, F.

The elements of the small set universe $\textrm{sV}(I, F)$ can be embedded into V

and they form a set $\textrm{uV}(I,F)$ in V

We can think of $\textrm{uV}(I,F)$ as a constructive version of an inaccessible (Rathjen et al. 1998). Now, iterating the universe building operator

(here $I_0$ , $F_0$ is an empty family) we then obtain an infinite hierarchy of inaccessibles

$$V_k = \textrm{uV}(I_k, F_k)$$

in V such that $V_k \in V_{k+1}$ . Each is a transitive set so $V_k \subseteq V_{k+1} \subseteq V$ . This will be the basis for the interpretation of the hierarchy of universes in extensional type theory (Martin-Löf 1984).

6. Bracket and Quotient Types

The bracket type is a type construction which to any type A introduces a type [A] whose elements are all definitionally equal (Awodey and Bauer (2004)). The idea is that [A] is the proposition corresponding to A, and [A] is inhabited if and only if A is inhabited, but [A] does not distinguish the proof objects. These properties are expressed by introduction and elimination rules, and some further equalities. See Section 8.3.8 (where the notation $\textrm{Br}(A)$ is used for [A]).

A corresponding set-theoretic construction we use for the interpretation is the “set squasher.” If $\alpha= \textrm{sup}(A,f)$ is an arbitrary set, then its squashed version is

$$\textrm{Sq}(\alpha) =_\textrm{def} \textrm{sup}(A,\lambda x. \emptyset).$$

Clearly, all its elements must be equal (to $\emptyset$ ), and also $\textrm{Sq}(\alpha)$ has an element just in case $\alpha$ has an element.

Bracket types are one extreme form of quotient types. In fact, CZF and hence its models admit general quotient sets, see, for example, Aczel and Rathjen (2000/2001). Quotient rules for extensional type theory have been formulated by Hofmann (1997, Ch. 5.1.5), Maietti (2005) and for HoTT in The Univalent Foundations Program, (2013).

7. Interpretation

Now we fix the interpretation. Define the judgments on the left to have the meaning of those on the right.

(9) \begin{equation} \begin{array}{llll}&\Gamma \; \textrm{context} &\qquad & \Gamma : {\mathbb V} \\&\Gamma \Longrightarrow A \; \textrm{type} &\qquad & A : [\kappa(\Gamma) \rightarrow {\mathbb V}]\\&\Gamma \Longrightarrow A == B && A =_\textrm{ext} B : [\kappa(\Gamma) \rightarrow {\mathbb V}]\\&\Gamma \Longrightarrow a \; \textrm{raw} && a : [\kappa(\Gamma) \rightarrow {\mathbb V}] \\&\Gamma \Longrightarrow a :: A && \forall x : \kappa(\Gamma), a(x) \in_V A(x) \\& \Gamma \Longrightarrow a == b :: A && \forall x : \kappa(\Gamma), a(x)=_V b(x) \in_V A(x)\end{array}\end{equation}

Those on the right are judgments in Agda about setoids. As usual, we assume that judgments satisfy all their presuppositions.

Further, we introduce judgments for substitutions between contexts, and their corresponding interpretations

(10) \begin{equation} \begin{array}{llll}&f: \Delta \longrightarrow \Gamma &\qquad & f : [\kappa(\Delta) \rightarrow \kappa(\Gamma)] \\&f == g: \Delta \longrightarrow \Gamma &\qquad & f =_\textrm{ext} g : [\kappa(\Delta) \rightarrow \kappa(\Gamma)] \\\end{array}\end{equation}

The interpretation of application of substitutions to (raw) types and terms is given by composition

(11) \begin{equation} \begin{array}{llll}&a[f] &\qquad & a \circ f \\&A[f] &\qquad & A \circ f \\\end{array}\end{equation}

Composition of substitutions is interpreted as composition of maps

(12 \begin{equation} \begin{array}{llll}&f \frown g &\qquad & f \circ g\end{array}\end{equation}

Next, the operations for context extension $\rhd$ and the display map/left projection $\downarrow$ , last variable/right projection $\textrm{v}$ , and extension of substitutions $ \langle \; , \; \rangle$ are defined.

(13) \begin{equation} \begin{array}{llll}&\Gamma \rhd A &\qquad & \textrm{sigmaV}(\Gamma, A) \\& \downarrow(A) : \Gamma \rhd A \longrightarrow \Gamma & \qquad & \pi_1 : [\kappa(\textrm{sigmaV}(\Gamma, A)) \rightarrow \kappa(\Gamma)] \\& \Gamma \rhd A \Longrightarrow \textrm{v}_A \; \textrm{raw} & \qquad & \pi_2 : [\kappa(\textrm{sigmaV}(\Gamma, A)) \rightarrow {\mathbb V}] \\& \langle f , a \rangle_{A,p} : \Delta \longrightarrow \Gamma \rhd A& \qquad &\lambda u. < f(u) , \pi_1(p(u)) >\; : [\kappa(\Delta) \rightarrow \kappa(\textrm{sigmaV}(\Gamma, A)) ] \\\end{array}\end{equation}

here $p : (\Delta \Longrightarrow a :: A[f])$

Finally, we may introduce a judgment for equality of contexts which is interpreted as equality of sets

(14) \begin{equation} \begin{array}{llll}&\Delta == \Gamma &\qquad & \Delta =_V \Gamma \\\end{array}\end{equation}

Now by Example 3.9, each $p : \Delta =_V \Gamma$ gives an isomorphism

$$\phi_p =_\textrm{def} \kappa(p) : [\kappa(\Delta) \rightarrow \kappa(\Gamma)]$$

which is independent of p and functorial in p. Moreover, it has the property that

Some remarks about the notation to guide reading of the code. The interpretation will mainly use 0-setoids and 0-classoids, simply called setoids and classoids. Due to some limitations of Agda notation (no subscripts), we use the following notation for $a =_A a'$ and $b=_B b'$ , when A and B are, respectively, setoids and classoids

The underlying types $|A|$ and $|B|$ are denoted, respectively

When A, A’ are setoids, and B,B’ are classoids, we use the following notations for $|[A,A']|$ , $|[A,B]|$ and $|[B,B']|$

8. Interpreted Rules

The following is a list of the interpreted rules of the formalization (Section 9). The rule names refer to the Agda code.

We recall that the judgment forms are

$$\begin{array}{l@{\quad}l}\Gamma \; \text{context} & \Gamma \Longrightarrow A \; \text{type}\\\Gamma == \Delta & \Gamma \Longrightarrow A == B\\f: \Gamma \longrightarrow \Delta & \Gamma \Longrightarrow a :: A\\f == g : \Gamma \longrightarrow \Delta & \Gamma \Longrightarrow a==b :: A\end{array}$$

The following presupposition rules are valid in the model

8.1 Substitutions and general equality rules

8.2 Context extension and associated rules

Two derived rules:

8.3 Rules for particular type constructions

The general principle of Martin-Löf type theory is that each type construction comes with a formation rule, a finite number of introduction rules, one elimination rule, and computation rules. There may be additional equality rules in extended theories. Moreover, each constant has a congruence rule. If the theory is based on explicit substitution (as is the case here), there are also equality rules that state that substitutions commute with constants and abstractions.

8.3.1 $\Pi$ -rules

8.3.2 Id-rules

8.3.3 $\Sigma$ -rules

8.3.4 N-rules

Here, $\text{step-sub}(\Gamma): \Gamma \rhd \textrm{Nat}\longrightarrow \Gamma \rhd \textrm{Nat}$ is the straightforward substitution that applies the successor to the second argument.

8.3.5 $N_0$ -rules

8.3.6 $+$ -rules

Here, $\text{Sum-sub-lf}(A,B): \Gamma \rhd A \longrightarrow \Gamma \rhd \textrm{Sum}(A,B)$ and $\text{Sum-sub-rg}(A,B): \Gamma \rhd B \longrightarrow \Gamma \rhd \textrm{Sum}(A,B)$ are defined from $\textrm{lf}$ and $\textrm{rg}$ using $\langle,\rangle$ in the straightforward way.

8.3.7 Universe rules

For each $k \in {\mathbb N,}$

¹

8.3.8 Bracket type rules

The universes are closed under bracket types

8.4 Hidden arguments

In the actual verification of the rules in Agda (Section 9), there are implicit arguments that we have hidden in the above listing of rules. For instance,

looks as follows in Agda code:

9. Formalization in Agda

This paper describes the formalization available at²

https://github.com/peterlefanulumsdaine/palmgren-archive/tree/MSCS-2023/MLTT-and-setoids

The following files are part of the formalization we describe. Loading V-model-all-rules. agda in Agda verifies all relevant files. Agda version 2.5.2 has been used.

Basic definitions and results concerning setoids

Basic constructions and results concerning Aczel’s iterative sets

The setoid model of extensional Martin-Löf type theory

10. Comparing the Logical Framework and Agda

The Logical Framework (LF) is a dependently type lambda calculus which was designed to present dependent type theories in a compact form; see Nordström et al. (1990). There is one basic type former for dependent products (or dependent function space)

It has an introduction rule which gives the only abstraction construction for terms, together with an elimination rules which is application.

There are corresponding $\beta$ - and $\eta$ -rules. Usual syntactic conventions are used to reduce the number of parentheses $c(a_1) \cdots(a_n)$ is abbreviated as $c(a_1,\ldots,a_n)$ . If a type $\beta$ does not depend on x, $(x : \alpha) \beta$ is abbreviated as $\alpha \rightarrow \beta$ . LF has one basic dependent type which is a type universe $\textrm{Set}$ with a decoding function $\textrm{El}(\cdot)$ .

A type theory T can now be axiomatized by introducing a number of new constants $c_1,\ldots,c_m$ with types in contexts

and furthermore equations in contexts

In standard type theories, the constants are type formers, introduction and elimination constants, and the equations express the computation rules. We refer to Nordstrom et al. (1990), Ranta, (1995) for elaborations of Martin-Löf type theory in this form.

The interactive proof system Agda has similar basic constructions (cf. right column)

Agda has an infinite cumulative hierarchy of type universes $\textrm{Set} = \textrm{Set}0, \textrm{Set}1, \textrm{Set}2, \ldots $ with the rules

Note that there is no explicit decoding function $\textrm{El}$ . In fact, every type in the system belongs to some $\textrm{Set}N$ for some index N. Each universe $\textrm{Set}N$ is closed under inductive-recursive definitions, which includes record types (generalized $\Sigma$ types) and recursive data types, as well as inductive families.

11. Conclusion

In this paper, we have given a model of Martin-Löf extensional type theory with an infinite hierarchy of universes (cf. Martin-Lof, (1982)) inside Martin-Löf intensional type theory (cf. Nordstrom et al. (1990)). The model is completely formalized in Agda using setoid constructions. One may consider this as the first setoid model of the full extensional type theory in the intensional type theory.