2.1 A simple dependency: length-indexed vectors

In our language, definitions such as $(\!+\!)$ are opaque by default – they are not unfolded automatically. To illustrate the need to selectively unfold $(\!+\!)$ , consider the indexed inductive type of length-indexed vectors with the following constructors:

\begin{align*} \begin{array}{l} {[\hspace {0.1pt}]}{} : \mathsf{vec}\,\mathsf{ze}\,A\\ (\!\mathbin {::}\!) : A\to \mathsf{vec}\,n\,A\to \mathsf{vec}\,(\mathsf{su}\,n)\,A \end{array} \end{align*}

Suppose we attempt to define the append operation on vectors by dependent pattern matching on the first vector. Our goals would be as follows:

As it stands, the goals above are in normal form and cannot be proved; however, we may indicate that the definition of $(\!+\!)$ should be unfolded within the definition of $(\!\oplus\!)$ by adding the following top-level $\mathbf{unfolds}$ annotation:

With our new declaration, the goals simplify:

The first goal is solved with $v$ itself; for the second goal, we begin by applying the $\mathsf{vcons}$ constructor:

The remaining goal is just our induction hypothesis $u\oplus v$ . All in all, we have:

\begin{align*} \begin{array}{l} (\!\oplus\!)\,{\mathbf{unfolds}}\,(\!+\!)\\(\!\oplus\!) : \mathsf{vec}\,m\,A\to \mathsf{vec}\,n\,A\to \mathsf{vec}\,(m+n)\,A\\ {[\hspace {0.1pt}]}{} \oplus v = v\\ (a \mathbin {::}{} u) \oplus v = a \mathbin {::}{} (u\oplus v) \end{array} \end{align*}

2.2 Transitive unfolding

Now suppose we want to prove that $\mathsf{map}$ distributes over $(\!\oplus\!)$ . In doing so, we will certainly need to unfold $\mathsf{map}$ , but it turns out this will not be enough:

\begin{align*} \begin{array}{l} \mathsf{map} : (A\to B) \to \mathsf{vec}\,n\,A \to \mathsf{vec}\,n\,B\\ \mathsf{map}\,f\,{[\hspace {0.1pt}]}{} = {[\hspace {0.1pt}]}{}\\ \mathsf{map}\,f\,(a \mathbin {::}{} u) = f\,a \mathbin {::}{} \mathsf{map}\,f\,u \end{array} \end{align*}

To make further progress, we must also unfold $(\!\oplus\!)$ :

In our language, unfolding $(\!\oplus\!)$ has the side effect of also unfolding $(\!+\!)$ : in other words, unfolding is transitive. To see why this is the case, observe that the unfolding of $(a \mathbin {::}{} u) \oplus v : \mathsf{vec}\,({\mathsf{su}\,m} + n)\,A$ , namely $a \mathbin {::}{} (u\oplus v) : \mathsf{vec}\,(\mathsf{su}\,(m + n))\,A$ , would otherwise not be well-typed. From an implementation perspective, one can think of the transitivity of unfolding as necessary for subject reduction. Having unfolded $\mathsf{map}$ , $(\!\oplus\!)$ , and thus $(\!+\!)$ , we complete our definition:

\begin{align*} \begin{array}{l} \mathsf{cong} : (f:A\to B)\to a\equiv a^{\prime }\to f\,a\equiv f\,a^{\prime }\\ \mathsf{cong}\,f\,\mathsf{refl} = \mathsf{refl} \end{array} \end{align*}

\begin{align*} \begin{array}{l} \mathsf{map\text{-}{\oplus }}\,{\mathbf{unfolds}}\,\mathsf{map}; (\!\oplus\!)\\ \mathsf{map\text{-}{\oplus }} : (f:A\to B)\,(u:\mathsf{vec}\,m\,A)\,(v:\mathsf{vec}\,n\,A) \to \mathsf{map}\,f\,(u\oplus v) \equiv \mathsf{map}\,f\,u \oplus \mathsf{map}\,f\,v\\ \mathsf{map\text{-}{\oplus }}\,f\,{[\hspace {0.1pt}]}{}\,v = \mathsf{refl}\\ \mathsf{map\text{-}{\oplus }}\,f\,(a \mathbin {::}{} u)\,v = \mathsf{cong}\,(f\,a \mathbin {::}{})\,(\mathsf{map\text{-}{\oplus }}\,f\,u\,v) \end{array} \end{align*}

2.3 Recovering unconditionally transparent/opaque definitions

There are also times when we intend a given definition to be a fully transparent abbreviation, in the sense of being unfolded automatically whenever possible. We indicate this with an $\mathbf{abbreviation}$ declaration:

Then the following lemma can be defined without any explicit unfolding:

\begin{align*} \begin{array}{l} \mathsf{abbrv\text{-}{}example} : \mathsf{singleton}\,5 \equiv (5 \mathbin {::}{} {[\hspace {0.1pt}]}{})\\ \mathsf{abbrv\text{-}{}example} = \mathsf{refl} \end{array} \end{align*}

The meaning of the $\mathbf{abbreviation}$ keyword must account for unfolding constraints. For instance, what would it mean to make $\mathsf{map\text{-}{\oplus }}$ an abbreviation?

\begin{align*} \begin{array}{l} {\mathbf{abbreviation}}\,\mathsf{map\text{-}{\oplus }}\\ \mathsf{map\text{-}{\oplus }}\,{\mathbf{unfolds}}\,\mathsf{map};\; (\!\oplus\!)\\ \dots \end{array} \end{align*}

We cannot unfold $\mathsf{map\text{-}{\oplus }}$ in all contexts, because its definition is only well-typed when $\mathsf{map}$ and $(\!\oplus\!)$ are unfolded. The meaning of this declaration must, therefore, be that $\mathsf{map\text{-}{\oplus }}$ shall be unfolded just as soon as $\mathsf{map}$ and $(\!\oplus\!)$ are unfolded. In other words, ${\mathbf{abbreviation}}\,\vartheta$ followed by $\vartheta \,{\mathbf{unfolds}}\,\kappa _1;\ldots ;\;\kappa _n$ means that unfolding $\vartheta$ is synonymous with unfolding all of $\kappa _1;\ldots ;\;\kappa _n$ .

Conversely, we may intend a given definition never to unfold, which we may indicate by a corresponding $\mathbf{abstract}$ declaration. Because definitions in our system do not automatically unfold, the force of ${\mathbf{abstract}}\,\vartheta$ is simply to prohibit users from including $\vartheta$ in any subsequent $\mathbf{unfolds}$ annotations.

2.4 Unfolding within the type

The effect of a $\vartheta \,{\mathbf{unfolds}}\,\kappa _1;\ldots ;\;\kappa _n$ declaration is to make $\kappa _1;\ldots \kappa _n$ unfold within the definition of $\vartheta$ , but still not within its type; it will happen, however, that a type might not be expressible without some unfolding. First, we will show how to accommodate this situation using only features we have introduced so far, and then in Section 2.5, we will devise a more general and ergonomic solution.

Consider the left-unit law for $(\!\oplus\!)$ : in order to state that a vector $u$ is equal to the vector ${[\hspace {0.1pt}]}{}\oplus u$ , we must contend with their differing types $\mathsf{vec}\,n\,A$ and $\mathsf{vec}\,(\mathsf{ze}+n)\,A$ , respectively. One approach is to rewrite along the left-unit law for $\mathbb{N}$ ; indeed, to state the right-unit law for $(\!\oplus\!)$ , one must rewrite along the right-unit law for $\mathbb{N}$ . But here, because $(\!+\!)$ computes on its first argument, $\mathsf{vec}\,n\,A$ and $\mathsf{vec}\,(\mathsf{ze}+n)\,A$ would be definitionally equal types if we could unfold $(\!+\!)$ .

In order to formulate the left-unit law for $(\!\oplus\!)$ , we start by defining its type as an abbreviation that unfolds $(\!+\!)$ :

\begin{align*} \begin{array}{l} {\mathbf{abbreviation}}\,\oplus \mathsf {\text{-}left\text{-}unit\text{-}type}\\ \oplus \mathsf {\text{-}left\text{-}unit\text{-}type}\,{\mathbf{unfolds}}\,(\!+\!)\\ \oplus \mathsf {\text{-}left\text{-}unit\text{-}type} : \mathsf{vec}\,n\,A\to \mathsf{Type}\\ \oplus \mathsf {\text{-}left\text{-}unit\text{-}type}\,u = {[\hspace {0.1pt}]}{}\oplus u \equiv u \end{array} \end{align*}

Now we may state the intended lemma using the type defined above:

Clearly, we must unfold $(\!+\!)$ and thus $\oplus \mathsf {\text{-}left\text{-}unit\text{-}type}$ to simplify our goal:

We complete the proof by unfolding $(\!\oplus\!)$ itself, which transitively unfolds $(\!+\!)$ :

\begin{align*} \begin{array}{l} \oplus \mathsf {\text{-}left\text{-}unit}\,{\mathbf{unfolds}}\,(\!\oplus\!)\\ \oplus \mathsf {\text{-}left\text{-}unit} : (u : \mathsf{vec}\,n\,A) \to \oplus \mathsf {\text{-}left\text{-}unit\text{-}type}\,u\\ \oplus \mathsf {\text{-}left\text{-}unit}\,u = \mathsf{refl} \end{array} \end{align*}

2.5 Unfolding within subexpressions

We have just demonstrated how to unfold definitions within the type of a declaration by defining that type as an additional declaration; using the same technique, we can introduce unfoldings within any subexpression by hoisting that subexpression to a top-level definition with its own unfolding constraint.

Unfolding within the type, revisited. Rather than repeating the somewhat verbose pattern of Section 2.4, we abstract it as a new language feature that is easily eliminated by elaboration. In particular, we introduce a new expression former ${\mathbf{unfold}}\,\kappa \,{\mathbf{in}}\,M$ that can be placed in any expression context. Let us replay the example from Section 2.4, but using $\mathbf{unfold}$ rather than an auxiliary definition:

The type ${\mathbf{unfold}}\,(\!+\!)\,{\mathbf{in}}\,\quad\oplus u \equiv u$ is in normal form; the only way to simplify it is to unfold $(\!+\!)$ . We could do this with another inline $\mathbf{unfold}$ expression (see $\oplus \mathsf {\text{-}left\text{-}unit'}$ below), but here we will use a top-level declaration:

By virtue of the above, the $\mathbf{unfold}$ expression in our hole has computed away and we are left with as $(\!\oplus\!)$ is still abstract in this scope. To make progress, we strengthen the declaration to unfold $(\!\oplus\!)$ in addition to $(\!+\!)$ :

The meaning of the code above is exactly as described in Section 2.4: the $\mathbf{unfold}$ scope is elaborated to a new top-level $\mathbf{abbreviation}$ that unfolds $(\!+\!)$ .

Expression-level vs. top-level unfolding. We noted in our definition of $\oplus \mathsf {\text{-}left\text{-}unit}$ above that we could have replaced the top-level ${\mathbf{unfolds}}\,(\!\oplus\!)$ directive of $\oplus \mathsf {\text{-}left\text{-}unit}$ with the new expression-level ${\mathbf{unfold}}\,(\!\oplus\!)\,{\mathbf{in}}$ as follows:

\begin{align*} \begin{array}{l} \oplus \mathsf {\text{-}left\text{-}unit'} : (u : \mathsf{vec}\,n\,A) \to {\mathbf{unfold}}\,(\!+\!)\,{\mathbf{in}}\,{[\hspace {0.1pt}]}{}\oplus u\equiv u\\ \oplus \mathsf {\text{-}left\text{-}unit'}\,u = {\mathbf{unfold}}\,(\!\oplus\!)\,{\mathbf{in}}\,\mathsf{refl} \end{array} \end{align*}

The resulting definition of $\oplus \mathsf {\text{-}left\text{-}unit'}$ has slightly different behavior than $\oplus \mathsf {\text{-}left\text{-}unit}$ above: whereas unfolding $\oplus \mathsf {\text{-}left\text{-}unit}$ causes $(\!\oplus\!)$ to unfold transitively, we can unfold $\oplus \mathsf {\text{-}left\text{-}unit'}$ without unfolding $(\!\oplus\!)$ – at the cost of ${\mathbf{unfold}}\,(\!\oplus\!)$ expressions appearing in our goal. This more granular behavior may be desirable in some cases, and it is a strength of our language and its elaborative semantics that the programmer can manipulate unfolding in such a fine-grained manner.

For completeness, we show the elaborated version of $\oplus \mathsf {\text{-}left\text{-}unit'}$ resulting from eliminating expression-level unfolding from the definition. We defer a systematic discussion of this transformation till Section 4.

\begin{align*} \begin{array}{l} {\mathbf{abbreviation}}\,\oplus\mathsf{\text{-}left\text{-}unit'\text{-}type}\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}type}\,{\mathbf{unfolds}}\,(\!+\!)\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}type} : \mathsf{vec}\,n\,A\to \mathsf{Type}\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}type}\,u = {[\hspace {0.1pt}]}{}\oplus u \equiv u \end{array} \end{align*}

\begin{align*} \begin{array}{l} {\mathbf{abbreviation}}\,\oplus \mathsf {\text{-}left\text{-}unit'\text{-}body}\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}body}\,{\mathbf{unfolds}}\,(\!\oplus\!)\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}body} : (u : \mathsf{vec}\,n\,A)\to \oplus \mathsf {\text{-}left\text{-}unit'\text{-}type}\,u\\ \oplus \mathsf {\text{-}left\text{-}unit'\text{-}body}\,u = \mathsf{refl} \end{array} \end{align*}

\begin{align*} \begin{array}{l} \oplus \mathsf {\text{-}left\text{-}unit'} : (u : \mathsf{vec}\,n\,A)\to \oplus \mathsf {\text{-}left\text{-}unit'\text{-}type}\,u\\ \oplus \mathsf {\text{-}left\text{-}unit'}\,u = \oplus \mathsf {\text{-}left\text{-}unit'\text{-}body}\,u \end{array} \end{align*}

In our experience, expression-level unfolding seems more commonly useful for end users than top-level unfolding; on the other hand, the clearest semantics for expression-level unfolding are stated in terms of top-level unfolding. Because one of our goals is to provide an account of unfolding that admits a reliable and precise mental model for programmers, it is desirable to include both top-level and expression-level unfolding in the surface language.

3.1 A core calculus with proposition symbols

Our core calculus parameterizes intensional MLTT (Martin-Löf Reference Martin-Löf, Rose and Shepherdson1975) by a bounded meet semilattice of proposition symbols $p\in \mathbb{P}$ and adjoins to the type theory a new form of context extension and two new type formers $\{p\}\, A$ and $\lbrace A\vert p \hookrightarrow M \rbrace$ involving proposition symbols:

The bounded meet semilattice structure on $\mathbb{P}$ closes proposition symbols under conjunction $\land$ and the true proposition $\top$ , thereby partially ordering $\mathbb{P}$ by entailment $p \leq q$ (“ $p$ entails $q$ ”) satisfying the usual principles of propositional logic. We say $p$ is true if $\top$ entails $p$ ; the context extension $\Gamma ,p$ hypothesizes that $p$ is true.

The type $\{p\}\, A$ is the dependent product “ $\{\_ : p\}\to A$ ,” that is, $\{p\}\, A$ is well-formed when $A$ is a type under the hypothesis that $p$ is true, and $f : \{p\}\, A$ when, given that $p$ is true, we may conclude $f : A$ . The extension type $\lbrace A\vert p \hookrightarrow a_p\rbrace$ is well-formed when $A$ is a type and $a_p : \{p\}\, A$ ; its elements $a : \lbrace A\vert p \hookrightarrow a_p\rbrace$ are terms $a : A$ satisfying the side condition that when $p$ is true, we have $a = a_p : A$ . We provide inference rules for the core calculus, including these connectives, in Section 4.1.

3.2 Elaborating controlled unfolding to our core calculus

Our surface language extends a generic surface language for dependent type theory with a new expression former $\mathbf{unfold}$ and several new declaration forms: $\vartheta \,{\mathbf{unfolds}}\,\kappa _1;\dots ;\;\kappa _n$ for controlled unfolding, ${\mathbf{abbreviation}}\,\vartheta$ for transparent definitions, and ${\mathbf{abstract}}\,\vartheta$ for opaque definitions. Elaboration transforms these surface-language declarations into core-language signatures, that is, sequences of declarations over our core calculus of MLTT with proposition symbols.

Our signatures include the following declaration forms:

• • ${\mathbf{prop}}\,p \leq q$ introduces a fresh proposition symbol $p$ such that $p$ entails $q\in \mathbb{P}$ ;

• • ${\mathbf{prop}}\,p = q$ defines the proposition symbol $p$ to be an abbreviation for $q\in \mathbb{P}$ ;

• • ${\mathbf{const}}\,\vartheta : A$ introduces a constant $\vartheta$ of type $A$ .

We now revisit our examples from Section 2, illustrating how they are elaborated into our core calculus:

Plain definitions

Recall our unadorned definition of $(\!+\!)$ from Section 2:

\begin{align*} \begin{array}{l} (\!+\!) : \mathbb{N}\to \mathbb{N}\to \mathbb{N}\\ \mathsf{ze} + n = n\\ {\mathsf{su}\,m} + n = \mathsf{su}\,(m + n) \end{array} \end{align*}

We elaborate $(\!+\!)$ into a sequence of declarations: first, we introduce a new proposition symbol $\Upsilon _{+}$ corresponding to the proposition that “ $(\!+\!)$ unfolds.” Next, we introduce a new definition $\delta _{+}:\mathbb{N}\to \mathbb{N}\to \mathbb{N}$ satisfying the defining clauses of $(\!+\!)$ above, under the (trivial) assumption of $\top$ ; finally, we introduce a new constant $(\!+\!)$ involving the extension type of $\delta _{+}$ along $\Upsilon _{+}$ :

\begin{align*} \begin{array}{l} {\mathbf{prop}}\,\Upsilon _{+}\leq \top \\ \\[-8pt] \delta _{+} : \{\top \}\,(m\,n : \mathbb{N})\to \mathbb{N}\\ \delta _{+}\,\mathsf{ze}\,n = n\\ \delta _{+}\,(\mathsf{su}\,m)\,n = \mathsf{su}\,(\delta _{+}\,m\,n)\\ \\[-8pt] {\mathbf{const}}\,(\!+\!) : \lbrace \mathbb{N}\to \mathbb{N}\to \mathbb{N} \vert \Upsilon _{+} \hookrightarrow \delta _{+}\rbrace \end{array} \end{align*}

Top-level unfolding

To understand why we have elaborated $(\!+\!)$ in this way, let us examine how to elaborate top-level unfolding declarations (Section 2.1):

\begin{align*} \begin{array}{l} (\!\oplus\!)\,{\mathbf{unfolds}}\,(\!+\!)\\ (\!\oplus\!) : \mathsf{vec}\,m\,A\to \mathsf{vec}\,n\,A\to \mathsf{vec}\,(m+n)\,A\\ {[\hspace {0.1pt}]}{} \oplus v = v\\ (a \mathbin {::}{} u) \oplus v = a \mathbin {::}{} (u\oplus v) \end{array} \end{align*}

To elaborate $(\!\oplus\!)\,{\mathbf{unfolds}}\,(\!+\!)$ , we define the proposition symbol $\Upsilon _{\oplus }$ to entail $\Upsilon _{+}$ , capturing the idea that unfolding $(\!\oplus\!)$ always causes $(\!+\!)$ to unfold; in order to cause $(\!+\!)$ to unfold in the body of $(\!\oplus\!)$ , we assume $\Upsilon _{+}$ in the definition of $\delta _{\oplus }$ . In full, we elaborate the definition of $(\!\oplus\!)$ as follows:

\begin{align*} \begin{array}{l} {\mathbf{prop}}\,\Upsilon _{\oplus }\leq \Upsilon _{+}\\ \\[-5pt] \delta _{\oplus } : \{\Upsilon _{+}\} \, (u:\mathsf{vec}\,m\,A)\,(v:\mathsf{vec}\,n\,A)\to \mathsf{vec}\,(m+n)\,A\\ \delta _{\oplus }\,{[\hspace {0.1pt}]}{}\,v = v\\ \delta _{\oplus }\,(a \mathbin {::}{} u)\,v = a \mathbin {::}{} (\delta _{\oplus }\,u\,v)\\ \\[-5pt] {\mathbf{const}}\,(\!\oplus\!) : \lbrace \mathsf{vec}\,m\,A\to \mathsf{vec}\,n\,A\to \mathsf{vec}\,(m+n)\,A \vert \Upsilon _{\oplus } \hookrightarrow \delta _{\oplus }\rbrace \end{array} \end{align*}

Observe that the definition of $\delta _{\oplus }$ is well-typed because $\Upsilon _{+}$ is true in its scope: thus, the extension type of $(\!+\!)$ causes $\mathsf{ze}+n$ to be definitionally equal to $\delta _{+}\,\mathsf{ze}\,n$ , which in turn is defined to be $n$ . The constraint $\Upsilon _{\oplus }\hookrightarrow \delta _{\oplus }$ is well-typed because $\Upsilon _{\oplus }$ entails $\Upsilon _{+}$ .

If a definition $\vartheta$ unfolds multiple definitions $\kappa _1;\dots ;\;\kappa _n$ , we define $\Upsilon _{\vartheta }$ to entail (and define $\delta _{\vartheta }$ to assume) the conjunction $\Upsilon _{\kappa _1}\land \dots \land \Upsilon _{\kappa _n}$ ; if a definition $\vartheta$ unfolds no definitions, then $\Upsilon _{\vartheta }$ entails (and $\delta _{\vartheta }$ assumes) $\top$ , as in our $(\!+\!)$ example.

Abbreviations

To elaborate the combination of the declarations ${\mathbf{abbreviation}}\,\vartheta$ and $\vartheta \,{\mathbf{unfolds}}\,\kappa _1;\dots ;\;\kappa _n$ , we define $\Upsilon _{\vartheta }$ to equal the conjunction $\Upsilon _{\kappa _1}\land \dots \land \Upsilon _{\kappa _n}$ . For example, consider the following code from Section 2.3:

\begin{align*} \begin{array}{l} {\mathbf{abbreviation}}\,\mathsf{map\text{-}{\oplus }}\\ \mathsf{map\text{-}{\oplus }}\,{\mathbf{unfolds}}\,\mathsf{map};\; (\!\oplus\!)\\ \mathsf{map\text{-}{\oplus }} : (f:A\to B)\,(u:\mathsf{vec}\,m\,A)\,(v:\mathsf{vec}\,n\,A) \to \mathsf{map}\,f\,(u\oplus v) \equiv \mathsf{map}\,f\,u \oplus \mathsf{map}\,f\,v\\ \mathsf{map\text{-}{\oplus }}\,f\,{[\hspace {0.1pt}]}{}\,v = \mathsf{refl}\\ \mathsf{map\text{-}{\oplus }}\,f\,(a \mathbin {::}{} u)\,v = \mathsf{cong}\,((f\,a) \mathbin {::}{}-)\,(\mathsf{map\text{-}{\oplus }}\,f\,u\,v) \end{array} \end{align*}

Let us write $\mathfrak{C}$ for the following type:

\begin{align*} & (f:A\to B)\,(u:\mathsf{vec}\,m\,A)\,(v:\mathsf{vec}\,n\,A) \to \mathsf{map}\,f\,(u\oplus v) \equiv \mathsf{map}\,f\,u \oplus \mathsf{map}\,f\,v \end{align*}

The above example is then elaborated as follows:

\begin{align*} \begin{array}{l} {\mathbf{prop}}\,\Upsilon _{\mathsf{map\text{-}{\oplus }}} = \Upsilon _{\mathsf{map}}\land \Upsilon _{\oplus }\\ \\[-5pt] \delta _{\mathsf{map\text{-}{\oplus }}} : \{\Upsilon _{\mathsf{map}}\land \Upsilon _{\oplus }\}\,\mathfrak{C}\\ \delta _{\mathsf{map\text{-}{\oplus }}}\,f\,{[\hspace {0.1pt}]}{}\,v = \mathsf{refl}\\ \delta _{\mathsf{map\text{-}{\oplus }}}\,f\,(a \mathbin {::}{} u)\,v = \mathsf{cong}\,((f\,a) \mathbin {::}{} -)\,(\delta _{\mathsf{map\text{-}{\oplus }}}\,f\,u\,v)\\ \\[-5pt] {\mathbf{const}}\, \mathsf{map\text{-}{\oplus }} : \lbrace \mathfrak{C} \mid \Upsilon _{\mathsf{map\text{-}{\oplus }}} \hookrightarrow \delta _{\mathsf{map\text{-}{\oplus }}} \rbrace \end{array} \end{align*}

Expression-level unfolding

The elaboration of the expression-level unfolding construct ${\mathbf{unfold}}\,\kappa \,{\mathbf{in}}\,M$ to our core calculus factors through the elaboration of expression-level unfolding to top-level unfolding as described in Section 2.5; we return to this in Section 4.3.

4.1 The core calculus $\mathbf {TT}_{\mathbb{P}}$

Our core calculus $\mathbf{TT}_{\mathbb{P}}$ is intensional MLTT (Martin-Löf Reference Martin-Löf, Rose and Shepherdson1975) with dependent sums and products, a Tarski universe, etc., extended with (1) a collection of proof-irrelevant proposition symbols, (2) dependent products over propositions, and (3) extension types for those propositions (Riehl and Shulman Reference Riehl and Shulman2017).

In fact, $\mathbf{TT}_{\mathbb{P}}$ is actually a family of type theories parameterized by a bounded meet semilattice $(\mathbb{P},\top ,\land )$ whose underlying set $\mathbb{P}$ is the set of proposition symbols of $\mathbf{TT}_{\mathbb{P}}$ ; the semilattice structure on $\mathbb{P}$ axiomatizes the conjunctive fragment of propositional logic with $\land$ as conjunction, $\top$ as the true proposition, and $\leq$ as entailment (where $p\leq q$ is defined as $p\land q = p$ ), subject to the usual logical principles such as $p\land q \leq p$ and $p\land q \leq q$ and $p \leq \top$ .

The language $\mathbf{TT}_{\mathbb{P}}$ augments ordinary MLTT with a new judgment $\Gamma \vdash p\,\textit{true}$ (for $p\in \mathbb{P}$ ) and the corresponding context extension $\Gamma ,p$ (for $p\in \mathbb{P}$ ). The judgment $\Gamma \vdash p\,\textit{true}$ states that the proposition $p$ is true in context $\Gamma$ , that is, the conjunction of the propositional hypotheses in $\Gamma$ entails $p$ while $\Gamma ,p$ extends $\Gamma$ with the hypothesis that $p$ is true.

\begin{align*} & \frac { \Gamma \ \textit{ctx} \qquad p\in \mathbb{P} }{ \Gamma ,p\ \textit{ctx} } \quad\qquad \frac { p\in \mathbb{P} }{ \Gamma , p \vdash p\,\textit{true} } \quad\qquad \frac { \Gamma ,p \vdash {\mathscr{J}} \qquad \Gamma \vdash p\,\textit{true} }{ \Gamma \vdash \mathscr{J} } \\[5pt] & \frac { }{ \Gamma \vdash \top \,\textit{true} } \quad\qquad \frac { \Gamma \vdash p\,\textit{true} \qquad \Gamma \vdash q\,\textit{true} }{ \Gamma \vdash p\land q\,\textit{true} } \quad\qquad \frac { \Gamma \vdash p\,\textit{true} \qquad p\leq q }{ \Gamma \vdash q\,\textit{true} } \end{align*}

The dependent product $\{p\}\,A$ is defined as an ordinary dependent product:

\begin{align*} &\frac { {\Gamma }, p \vdash A\ \textit{type} }{ {\Gamma } \vdash \{p\}\,A\ \textit{type} } \quad\qquad \frac { \Gamma , p \vdash\, M:{A} }{ \Gamma \vdash \langle p \rangle \,M : \{p\}\,A } \quad\qquad \frac { \Gamma \vdash M : \{p\}\,A \qquad \Gamma \vdash p\,\textit{true} }{ \Gamma \vdash M \mathbin {@} p : A }\\[5pt] &\quad\qquad \frac { \Gamma ,p \vdash\, M:A \qquad \Gamma \vdash p\, true}{\Gamma\vdash(\langle p \rangle M) \mathbin {@} p = M : A} \qquad\qquad \frac{\Gamma\vdash M:{p}A}{\Gamma\vdash M =\langle p \rangle(M \mathbin {@} p):\{p\}A} \end{align*}

The remaining feature of $\mathbf{TT}_{\mathbb{P}}$ is the extension type $\lbrace A\vert p \hookrightarrow a_{p} \rbrace$ . Given a proposition $p\in \mathbb{P}$ and an element $a_{p}$ of $A$ under the hypothesis $p$ , the elements of $\lbrace A\vert p \hookrightarrow a_{p}\rbrace$ correspond to elements of $A$ that equal $a_{p}$ when $p$ holds.

\begin{align*} &\frac { {\Gamma } \vdash A\ \textit{type} \qquad \Gamma ,p \vdash a_{p}:{A} }{ {\Gamma } \vdash \lbrace A\vert p \hookrightarrow a_{p}\rbrace \ \textit{type} } \quad \frac { \begin{array}{l} \Gamma \vdash a : A\\ \Gamma , p \vdash a_{p} : A \\ {\Gamma }, p \vdash a =a_{p}:A \end{array}}{ \Gamma \vdash \mathsf{in}_p\,a : \lbrace A\vert p \hookrightarrow a_{p} \rbrace } \qquad \frac { \Gamma \vdash a : \lbrace A\vert p \hookrightarrow a_{p} \rbrace }{ \Gamma \vdash \mathsf{out}_p\,a : A } \\[8pt] &\qquad\qquad \frac { \Gamma \vdash a : A }{ {\Gamma } \vdash \mathsf{out}_p(\mathsf{in}_p\,a) = a : A } \qquad \frac { \Gamma \vdash a : \lbrace A\vert p \hookrightarrow a_{p} \rbrace }{ {\Gamma } \vdash \mathsf{in}_p(\mathsf{out}_p\,a) = a : \lbrace A\vert p \hookrightarrow a_{p} \rbrace } \\[8pt] &\qquad\qquad\qquad \qquad\qquad \frac { \Gamma \vdash p\,\textit{true} \qquad \Gamma \vdash a : \lbrace A\vert p \hookrightarrow a_{p} \rbrace }{ {\Gamma } \vdash \mathsf{out}_p\,a = a_{p}:{A} } \end{align*}

4.2 Signatures over $\mathbf{{TT}}_{\mathbb{P}}$

Our elaboration procedure takes as input a sequence of surface-language definitions and outputs a well-formed signature, a list of declarations over $\mathbf{TT}_{\mathbb{P}}$ .

A signature is well-formed precisely when each declaration in $\Sigma$ is well-formed relative to the earlier declarations in $\Sigma$ . Our well-formedness judgment $\vdash \Sigma \,\textit{sig} \longrightarrow \mathbb{P}, \Gamma$ computes from $\Sigma$ the $\mathbf{TT}_{\mathbb{P}}$ context $\Gamma$ and proposition semilattice $\mathbb{P}$ specified by $\Sigma$ ’s $\mathbf{const}$ and $\mathbf{prop}$ declarations, respectively.

The rules for signature well-formedness are standard except for the ${\mathbf{prop}}\,p \le q$ and ${\mathbf{prop}}\,p = q$ declarations, which extend $\mathbb{P}$ with a new element $p$ satisfying $p \le q$ or $p = q$ , respectively. Recalling that our core calculus $\mathbf{TT}_{\mathbb{P}}$ is really a family of type theories parameterized by a semilattice $\mathbb{P}$ , these declarations shift us between type theories, for example, from $\mathbf{TT}_{\mathbb{P}}$ to TT $_{\mathbb{Q}}$ , where $\mathbb{Q} = \mathbb{P}[p \le q]$ is the minimal semilattice containing $\mathbb{P}$ and an element $p$ satisfying $p \le q$ . This shifting between theories is justified by Remark 5.

\begin{align*} &\qquad\quad \frac { }{ \vdash \epsilon \,\textit{sig} \longrightarrow \{\top \}, \cdot } \qquad\qquad \frac { \vdash \Sigma \,\textit{sig} \longrightarrow \mathbb{P}, \Gamma \qquad \Gamma \vdash _{\mathbf{TT}_{\mathbb{P}}} A\,\textit{type} }{ \vdash (\Sigma ,\ {\mathbf{const}}\, x : A)\,\textit{sig} \longrightarrow \mathbb{P}, (\Gamma , x : A) }\\[5pt] &\frac { \vdash \Sigma \,\textit{sig} \longrightarrow \mathbb{P}, \Gamma \qquad q\in \mathbb{P} }{ \vdash (\Sigma ,\ {\mathbf{prop}}\, p \le q) \,\textit{sig} \longrightarrow \mathbb{P}[p \le q] , \Gamma \qquad \vdash (\Sigma ,\ {\mathbf{prop}}\, p = q) \,\textit{sig} \longrightarrow \mathbb{P}[p = q], \Gamma } \end{align*}

4.3 Bidirectional elaboration

We adopt a bidirectional elaboration algorithm which mirrors bidirectional type-checking algorithms (Coquand Reference Coquand1996; Pierce and Turner Reference Pierce and Turner2000). The top-level elaboration judgment $\Sigma \vdash \vec {S} \rightsquigarrow \Sigma ^{\prime }$ takes as input the current well-formed signature $\Sigma$ and a list of surface-level definitions $\vec {S}$ and outputs a new well-formed signature $\Sigma ^{\prime }$ .

We define $\Sigma \vdash \vec {S} \rightsquigarrow \Sigma ^{\prime }$ in terms of three auxiliary judgments for elaborating surface-language types and terms; in the bidirectional style, we divide term elaboration into a checking judgment $\Sigma ;\;\Gamma \vdash {\texttt {e}} \Leftarrow A \rightsquigarrow \Sigma ^{\prime },M$ taking a core type as input and a synthesis judgment $\Sigma ;\;\Gamma \vdash {\texttt {e}} \Rightarrow A \rightsquigarrow \Sigma ^{\prime },M$ producing a core type as output. All three judgments take as input a signature $\Sigma$ and a context (telescope) over $\Sigma$ and output a new signature along with a core type or term.

We represent a surface-level definition $S$ as a tuple:

\begin{equation*} ({\mathbf{def}}\,\vartheta :{\texttt {A}},\textit {abbrv?},\textit {abstr?},[\kappa _1,\ldots \kappa _n],{\texttt {e}}) \end{equation*}

In this expression, $\vartheta$ is the name of the definiendum, $\texttt {A}$ is the surface-level type of the definition, abbrv? and abstr? are flags governing whether $\vartheta$ is an $\mathbf{abbreviation}$ (resp., is $\mathbf{abstract}$ ), $[\kappa _1,\ldots ,\kappa _n]$ are the names of the definitions that $\vartheta$ unfolds, and $\texttt {e}$ is the surface-level definiens.

The elaboration judgment elaborates each surface definition in sequence:

The rules for term and type elaboration are largely standard: for instance, we elaborate a surface-dependent product to a core-dependent product by recursively elaborating the first and second components. We single out two cases below: the boundary between checking and synthesis, and the expression-level $\mathbf{unfold}$ .

\begin{align*} \frac {\begin{array}{l} \Sigma ;\;\Gamma \vdash {\texttt {e}} \Rightarrow A \rightsquigarrow \Sigma _1;\;M \\ \Sigma _1;\;\Gamma \vdash \mathsf{conv}\,A\,B\end{array} }{ \Sigma ;\;\Gamma \vdash {\texttt {e}} \Leftarrow B \rightsquigarrow \Sigma _1;\;M } \quad \frac {\begin{array}{l} \Sigma ;\;\Gamma ,\Upsilon _{\vartheta } \vdash {\texttt {e}} \Leftarrow A \rightsquigarrow \Sigma _1;\;M \qquad {\mathbf{let}}\,\chi := \textit {gensym}\,() \\ {\mathbf{let}}\,\Sigma _2 := \Sigma _1, {\mathbf{const}}\,\chi :\prod _{\Gamma }{\lbrace A\vert \Upsilon _{\vartheta } \hookrightarrow M\rbrace }\end{array} }{ \Sigma ;\;\Gamma \vdash {\mathbf{unfold}}\,\vartheta \,{\mathbf{in}}\,{\texttt {e}} \Leftarrow A \rightsquigarrow \Sigma _2;\; \mathsf{out}_{\Upsilon _{\vartheta }}\,\chi [\Gamma ] } \end{align*}

The first rule states that a term synthesizing a type $A$ can be checked against a type $B$ provided that $A$ and $B$ are definitionally equal; in order to implement this rule algorithmically, we need definitional equality to be decidable. Additionally, our (omitted) type-directed elaboration rules are only well-defined if type constructors are injective up to definitional equality, for example, $A \to B = C \to D$ if and only if $A = C$ and $B = D$ .

Elaborating expression-level unfolding requires the ability to hoist a type to the top level by iterating dependent products over its context, an operation notated $\prod _{\Gamma }$ above. Because $\Gamma$ can hypothesize (the truth of) propositions, this operation relies crucially on the presence of dependent products $\{p\}\,A$ .

6.1 Type theories as categories with representable maps

While any number of logical frameworks are available (generalized algebraic theories (Cartmell Reference Cartmell1978), essentially algebraic theories (Freyd Reference Freyd1972), locally cartesian closed categories (Gratzer and Sterling Reference Gratzer and Sterling2020), etc.), Uemura’s categories with representable maps (Uemura, Reference Uemura2021, Reference Uemura2023) are particularly attractive because they express exactly the binding and dependency structure needed for type theory: a second-order version of generalized algebraic theories.

Uemura’s logical framework axiomatizes the category of judgments of $\mathbf{TT}_{\mathbb{P}}$ as a particular category with representable maps $\mathbb{T}$ . The finite limit structure of $\mathbb{T}$ encodes substitution as well as equality judgments, while the class of representable maps carves out those judgments that may be hypothesized. Uemura (Reference Uemura2023) develops a syntactic method for presenting a CwR as a signature within a variant of extensional type theory, which he has rephrased in terms of second-order generalized algebraic theories in his doctoral dissertation (Uemura Reference Uemura2021). Although we will use the type-theoretic presentation for convenience, the difference between these two accounts is only superficial.

Each judgment of $\mathbf{TT}_{\mathbb{P}}$ is rendered as a (dependent) sort, while operators are modeled by elements of the given sorts. In order to record whether a given judgment may be hypothesized, the sorts of the type theory are stratified by meta-sorts $\star \subseteq \Box$ where $A : \star$ signifies that $A$ is a representable sort (i.e., a context-former) and can be hypothesized, whereas $B : \Box$ cannot parameterize a framework-level dependent product.

We will often refer to a category with representable maps $\mathbb{T}$ as a type theory; indeed, as the category of judgments of a given type theory, $\mathbb{T}$ is a suitable invariant replacement for it.

Proposition 10 describes the universal property of a type theory generated by a given signature in a logical framework. Type theories qua CwRs thus give rise to a form of functorial semantics in which algebras (interpretations) arrange into a groupoid of CwR functors $\operatorname{hom}{\mathbf{CwR}}{\mathbb{T}}{\mathscr{E}}$ .

This is an appropriate setting for studying the syntax of type theory, but it is somewhat inappropriate for studying the semantics of type theory – in which one expects models to correspond to structured CwFs (Dybjer Reference Dybjer, Berardi and Coppo1996) or natural models (Awodey Reference Awodey2018), which themselves arrange into a (2,1)-category. The second notion of functorial semantics, developed by Uemura in his doctoral dissertation (Uemura Reference Uemura2021), is a generalization of the theory of CwFs and pseudo-morphisms between them (Clairambault and Dybjer Reference Clairambault and Dybjer2014; Newstead Reference Newstead2018).

Note that we may always regard a presheaf category $\mathbf{Pr}\,{\mathscr{C}}$ as a CwR with the representable maps being representable natural transformations, that is, families of presheaves whose fibers at representables are representable (Awodey Reference Awodey2018).

Models are arranged into a (2,1)-category ${\mathbf{Mod}}\,\mathbb{T}$ (see Appendix A). Essentially, a morphism of models ${\mathbf{M}}\to {\mathbf{N}}$ is given by a functor ${\alpha _\diamond }:{\mathbf{M}_\diamond }\to {\mathbf{N}_\diamond }$ together with a natural transformation ${\mathbf{M}}\to {\alpha ^{\ast}\mathbf{N}}\in \operatorname{hom}_{\mathbf{CwR}}({\mathbb{T}},{\mathbf{Pr}\,{\mathbf{M}_\diamond }})$ that preserves context extensions up to isomorphism; an isomorphism between morphisms of models is a natural isomorphism between the underlying functors satisfying an additional property.

For each CwR $\mathbb{T}$ , Uemura has shown the following theorem:

6.2 Encoding $\mathbf{TT}_{\mathbb{P}}$ in the logical framework

We begin by defining the signature for a category with representable maps $\mathbb{T}_0$ containing exactly the bare judgmental structure of $\mathbf{TT}_{\mathbb{P}}$ , namely the propositions and the judgments for types and terms. In our signature, we make liberal use of the Agda-style notation for implicit arguments. As always, $p$ ranges over $\mathbb{P}$ :

\begin{align*} \begin{array}{l} \langle p \rangle :\star \\\mathsf{tp} \square \\ \mathsf{tm} : \mathsf{tp} \Longrightarrow \star \\ \_:\{u,v:\langle p \rangle \}\Longrightarrow u = v\\ \_:\{\_ : \langle \bigwedge _{i\lt n}p_i\rangle \}\Longrightarrow \langle p_k\rangle \\ \_:\{\_:\langle p_i\rangle ,\dots \}\Longrightarrow \langle {\bigwedge _{i\lt n} p_i}\rangle \end{array} \end{align*}

Note that already this signature encodes the necessary theory of propositions for $\mathbf{TT}_{\mathbb{P}}$ . For instance, if $p \le q$ in $\mathbb{P}$ , then a combination of the final two implications in the signature implies $\langle p \rangle \to \langle q \rangle$ . We next extend the above to include the type formers of $\mathbf{TT}_{\mathbb{P}}$ , writing $\mathbb{T}$ for the CwR generated by the full signature.

For instance, the following constants specify the rules of extension types given in Section 4.1:

\begin{align*} \begin{array}{l} \mathsf{ext}p : (A : \mathsf{tp}\,(a : \{p\}\, \mathsf{tm}{A})\Longrightarrow \mathsf{tp}\\ \mathsf{in}_p : (A : \mathsf{tp}\,(a : \{p\}\, \mathsf{tm}{A})\,(u : \mathsf{tm}{A})\,\{\_ : \{p\}\, u=a\} \Longrightarrow \mathsf{tm}(\mathsf{ext}p\,A\,a)\\ \mathsf{out}_p : (A:\mathsf{tp}\,(a: \{p\}\,\mathsf{tm}{A})\,(u:\mathsf{tm}(\mathsf{ext}p\,A\,a))\Longrightarrow \mathsf{tm}{A}\\ \_ : (A:\mathsf{tp}\,(a:\{p\}\,\mathsf{tm}{A})\,(u:\mathsf{tm}(\mathsf{ext}p\,A\,a))\,\{\_:\langle p \rangle \} \Longrightarrow \mathsf{out}_p\,A\,a\,u = a\\ \_ : (A:\mathsf{tp}\,(a:\{p\}\,\mathsf{tm}{A})\,(u:\mathsf{tm}{A})\,\{\_:\{p\}\, u = a\}\Longrightarrow \mathsf{out}_p\,A\,a\,(\mathsf{in}_p\,A\,a\,u) = u\\ \_ : (A:\mathsf{tp}\,(a:\{p\}\,\mathsf{tm}{A})\,(u:\mathsf{tm}(\mathsf{ext}p\,A\,a))\Longrightarrow \mathsf{in}_p\,A\,a\,(\mathsf{out}_p\,A\,a\,u) = u \end{array} \end{align*}

The full list of non-standard constants is specified in Figure 1. Once the signature is complete, we obtain from Uemura’s framework a category with representable maps $\mathbb{T}$ together with a bi-initial model $\mathbf{I}$ .

Figure 1.

The non-standard aspects of the LF signature for $\textbf{TT}_{\mathbb{P}}$ .

6.3 The atomic figure shape and its universal property

For each context $\Gamma$ and type ${\Gamma }, \Gamma \vdash A\ \textit{type}$ , it is possible to axiomatize the normal forms of type $A$ ; unfortunately, this assignment of sets of normal forms does not immediately extend to a presheaf on the category of contexts $\mathbf{I}_\diamond$ , precisely because normal forms are not a priori closed under substitution. In fact, closing normal forms under substitution is the purpose of normalization, so we are not able to assume it beforehand.

Normal forms are, however, closed under substitutions of variables for variables (often called structural renamings), and in our case we shall be able to close them additionally under the “phase transitions” ${\Gamma ,\langle p \rangle }\to {\Gamma ,\langle q \rangle }$ when $\Gamma , p \vdash q\,\textit{true}$ is derivable. We shall refer to these substitutions as atomic substitutions, and we wish to organize them into a category.

It is possible to inductively define a category of “atomic contexts” whose objects are those of $\mathbf{I}_\diamond$ and whose morphisms are atomic substitutions, but this construction obscures a beautiful and simple (2,1)-categorical universal property first exposed by Bocquet et al. (Reference Bocquet, Kaposi and Sattler2021) that leads to a more modular proof. To explicate this universal property, first note that the theory $\mathbb{T}_0$ axiomatizes exactly the structure of variables and phase transitions, and that the initial model $\mathbf{I}$ of $\mathbb{T}$ is, by restriction along ${\mathbb{T}_0}\to {\mathbb{T}}$ , also a model of $\mathbb{T}_0$ .

Atomic substitution model over $\mathbf{M}$ arrange themselves into a (2,1)-category, a full subcategory of ${\mathbf{Mod}}\,\mathbb{T}_0\downarrow \mathbf{M}$ . The following result is due to Bocquet et al. (Reference Bocquet, Kaposi and Sattler2021).

When $(\boldsymbol{\mathbf{A}}, {\alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{I}})$ is the bi-initial atomic substitution model over $\mathbf{I}$ as in Proposition 16, we shall refer to an object $\Gamma \in \boldsymbol{\mathbf{A}}$ as an atomic context and a morphism ${\gamma }:{\Delta }\to {\Gamma }$ in $\boldsymbol{\mathbf{A}}$ as an atomic substitution. We shall assume without loss of generality that $\boldsymbol{\mathbf{A}}(\mathsf{tp}) = \alpha ^{\ast}\mathbf{I}(\mathsf{tp})$ so that the component $\alpha _{\mathsf{tp}}$ is the identity map.

6.4 Computability spaces by gluing along the atomic figure shape

We shall use the bi-initial atomic substitution model over $\mathbf{I}$ as a figure shape in the sense of Sterling (Reference Sterling2021, §4.3) to instantiate STC. Here, we transition into the 2-category of Grothendieck topoi, geometric morphisms, and geometric transformations, guided by a phase distinction between “object-space” and “meta-space” (Sterling Reference Sterling2021);Footnote ⁴ object-space refers to the object language embodied in the model $\mathbf{I}$ , whereas meta-space refers to the metalanguage embodied in the model $\boldsymbol{\mathbf{A}}$ . Later on, we will construct a glued topos in which we may speak of constructs that have extent in both object-space and meta-space. We follow Vickers (Reference Vickers, Aiello, Pratt-Hartmann and Van Benthem2007) and Anel and Joyal (Reference Anel, Joyal, Anel and Catren2021) in emphasizing the distinction between a topos $\boldsymbol{\mathsf{x}}$ and the category of sheaves $\mathbf{Sh}\,{\boldsymbol{\mathsf{x}}}$ presenting it:

That ${\alpha }:{\boldsymbol{\mathsf{A}}}\to {\boldsymbol{\mathsf{I}}}$ is essential means that its inverse image ${\alpha ^{\ast}}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}$ has a left adjoint ${\alpha _!}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}$ ; from the point of view of presheaves, this is precisely the Yoneda extension of ${\alpha _\diamond }:{\boldsymbol{\mathbf{A}}_\diamond }\to {\mathbf{I}_\diamond }$ as depicted below:

Following Sterling (Reference Sterling2025), we shall refer to a sheaf on $\boldsymbol{\mathsf{G}}$ as a computability space. A computability space $X\in \mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ is then identified with a family ${\pi _X}:{{\boldsymbol{i}}^{\ast}X}\to {\alpha^{\ast} {\boldsymbol{j}}^{\ast}X}$ in $\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}$ . Because the assignment $\pi _X$ is natural in computability spaces $X$ , it corresponds to a 2-cell ${\pi }:{{\boldsymbol{j}}\circ \alpha }\to {{\boldsymbol{i}}}$ in the 2-category of Grothendieck topoi. The universal property of $\boldsymbol{\mathsf{G}}$ is then expressed by the fact that ${\pi }:{\boldsymbol{{j}}\circ \alpha }\to {{\boldsymbol {i}}}$ is a co-comma cell in the 2-category of Grothendieck topoi:

6.4.1 Reflection of object and meta-space

By definition, the inverse image functors ${j}^{\ast},{i}^{\ast}$ have fully faithful right adjoints ${j}_{\ast}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ and ${i}_{\ast}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ ,respectively. These are computed as follows:

\begin{align*} {j}_{\ast}E &= (E, {1_{\alpha ^{\ast}E}}:{\alpha ^{\ast}E}\to {\alpha ^{\ast}E})\\ {i}_{\ast}A &= (\mathbf {1}{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}, {!_{A}}:{A}\to {{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\cong \alpha ^{\ast}{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}}) \end{align*}

Thus, the adjunctions ${j}^{\ast}\dashv {j}_{\ast}$ and ${i}^{\ast}\dashv {i}_{\ast}$ exhibit $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ and $\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}$ as reflective subcategories of $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ :

1. (1) The essential image of the reflective embedding ${j}_{\ast}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ is spanned by computability spaces $X$ for which ${\pi _X}:{{i}^{\ast}X}\to {\alpha ^{\ast} {j}^{\ast}X}$ is an isomorphism, that is, such that $\pi _X$ is terminal in the slice $\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}\downarrow \alpha ^{\ast}{j}^{\ast}X$ .

2. (2) The essential of image of the reflective embedding ${{i}}_{\ast}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ is spanned by computability spaces $X$ such that ${j}^{\ast}X$ is terminal in $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ .

Definition 21 (Vocabulary for reflective subcategories). When a computability space lies in the essential image of ${{j}_{\ast}}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ , we shall refer to it as lying in object-space. Likewise, when a computabiltiy space lies in the essential image of ${{i}_{\ast}}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ , we shall say that it lies in meta-space.

6.4.2 Coreflection of object- and meta-space

Both the open and closed immersions are essential morphisms of topoi, in the sense that we have additional (necessarily fully faithful) left adjoints ${j}_!\dashv {j}^{\ast}\colon \mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}\to \mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ and ${i}_!\dashv {i}^{\ast}\colon \mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}\to \mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}$ that are computed as follows:

\begin{align*} {j}_!E &= (E, {!_{\alpha ^{\ast}E}}:{{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}}\to {\alpha ^{\ast}E}) \\ {i}_!A &= (\alpha _!A, {\eta _A}:{A}\to {\alpha ^{\ast}\alpha _!A}) \end{align*}

Thus, $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ and $\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}$ are not only reflective in $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ – they are also coreflective.

6.5 The language of STC

As $\boldsymbol{\mathsf{I}}$ and $\boldsymbol{\mathsf{A}}$ are both subtopoi of $\boldsymbol{\mathsf{G}}$ , their reflections (Section 6.4.1) can be expressed in the internal language of $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ by means of a pair of complementary lex idempotent monads ( $\large \circ$ , $\bullet$ ). The internal language of $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ is presented by the $\large \circ$ -modal or object-space types and $\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}$ is presented by $\bullet$ -modal or meta-space types. Because they form an open/closed partition, these modal subuniverses admit a particularly simple formulation:

As a presheaf topos, $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ inherits a hierarchy of cumulative universes $\mathscr{U}_{i}$ , each of which supports the strict gluing or (mixed-phase) refinement type (Gratzer et al. Reference Gratzer, Shulman and Sterling2022): a version of the dependent sum of a family of meta-space types indexed in an object-space type $A$ that additionally restricts within object-space to exactly $A$ :

\begin{equation*} \frac { A : \{\mathsf{obj}\}\, \mathscr{U}_{i} \qquad B : (\{\mathsf{obj}\}\, A) \to \mathscr{U}_{i} \qquad \{\mathsf{obj}\} (a : A) \to (B\,a \cong {}) }{(x : A) \ltimes B\,x : \lbrace \mathscr{U}_{i}\vert \mathsf{obj} \hookrightarrow A\rbrace \qquad \mathsf{gl} : \lbrace ((x : \{\mathsf{obj}\}\, A)\times B\,x) \cong (x : A) \ltimes B\,x \vert \mathsf{obj} \hookrightarrow \pi _{1}\rbrace } \end{equation*}

Notation 28. We write $[\mathsf{obj} \hookrightarrow a \vert b]$ for $\mathsf{gl}\,(a,b)$ and $\mathsf{ungl}\,x$ for $\pi _{2}(\mathsf{gl}^{-1} x)$ . When constructing particularly complex inhabitants of $(x : A) \ltimes B\,x$ , we will avail ourselves of copattern matching notation and write the following instead of $c = [\mathsf{obj} \hookrightarrow a \vert b]$ :

\begin{align*} \begin{array}{l} \mathsf{obj} \hookrightarrow c = a\\ \mathsf{ungl}\,c = b \end{array} \end{align*}

Both $\large \circ$ and $\bullet$ induce reflective subuniverses ${\mathscr{U}_{\large \circ }^i,\mathscr{U}_{\bullet }^i}\hookrightarrow {\mathscr{U}_{i}}$ spanned by modal types, and these universes are themselves modal. Following Sterling (Reference Sterling2021), we use strict gluing to choose these universes with additional strict properties:

\begin{equation*} \mathscr{U}_{\large \circ }^i : \lbrace \mathscr{U}_{i+1}\vert \mathsf{obj} \hookrightarrow \mathscr{U}_{i}\rbrace \qquad \mathscr{U}_{\bullet }^i : {\lbrace \mathscr{U}_{i+1}\vert \mathsf{obj} \hookrightarrow \rbrace } \end{equation*}

Furthermore, the inclusion ${\mathscr{U}_{\large \circ }^i}\hookrightarrow {\mathscr{U}_{i}}$ restricts to the identity under $\mathsf{obj}$ . With the modal universes to hand, we may choose ${\large \circ } : \mathscr{U}_{i} \to \mathscr{U}_{i}$ and $\bullet : \mathscr{U}_{i} \to \mathscr{U}_{i}$ to factor through $\mathscr{U}_{\large \circ }^i$ and $\mathscr{U}_\bullet ^i$ , respectively. Henceforth, we will suppress the inclusions ${\mathscr{U}_{\large \circ }^i,\mathscr{U}_{\bullet }^i}\hookrightarrow {\mathscr{U}_{i}}$ and write, for example, ${\large \circ } : \mathscr{U}_{i} \to \mathscr{U}_{\large \circ }^i$ for the reflections.

The interpretation of the $\mathbf{TT}_{\mathbb{P}}$ signature within $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ internalizes into $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ as a sequence of constants valued in the subuniverse $\mathscr{U}_{\large \circ }^0$ ; for instance, we have:

\begin{align*} \begin{array}{l} \mathsf{tp} :\mathscr{U}_{\large \circ }^0\\ \mathsf{tm} :\mathsf{tp} \mathscr{U}_{\large \circ }^0\\ \langle p \rangle :\Omega _{\textit {dec}}\ \ \text{(for $p\in \mathbb{P}$)}\\ \mathsf{ext}p : (A : \mathsf{tp})\to (a : \{\langle p \rangle \}\, \mathsf{tm}{A}) \to \mathsf{tp}\\ \mathsf{in}_p : (A : \mathsf{tp})\,(a : \{\langle p \rangle \}\, \mathsf{tm}{A}) \to \lbrace \mathsf{tm}A\vert \langle p \rangle \hookrightarrow a\rbrace \cong \mathsf{tm}(\mathsf{ext}p\,A\,a) \end{array} \end{align*}

Following Remark 27, we package the pair $(\mathsf{in}_p,\mathsf{out}_p)$ as a single isomorphism $\mathsf{in}_p$ .

The presheaf of terms in the model $\boldsymbol{\mathbf{A}}$ internalizes as a meta-space type of variables which by virtue of the structure map ${\boldsymbol{\mathbf{A}}}\to {\mathbf{I}}$ can be indexed over the object-space collection of terms. We realize this synthetically as follows:

\begin{align*} \mathsf{var} : (A:\mathsf{tp}) \to \lbrace \mathscr{U}_{}\vert \mathsf{obj} \hookrightarrow \mathsf{tm}A\rbrace \end{align*}

We refer to extensional type theory extended with these constants and modalities as the language of STC.

6.6 Normal and neutral forms

Internally to STC, we now specify the normal and neutral forms of terms, and the normal forms of types. Following Sterling and Angiuli (Reference Sterling and Angiuli2021), we index the type of neutral forms by a frontier of instability, a proposition at which the neutral form is no longer meaningful. Our construction proceeds in two steps. First, we define a series of indexed quotient-inductive definitions (Kaposi et al. Reference Kaposi, Kovács and Altenkirch2019) specifying the meta-space components of normal and neutral forms:

\begin{align*} \begin{array}{l} {\mathsf{nf}}_\bullet : (A : \mathsf{tp}) \to \mathsf{tm}A \to \mathscr{U}_{\bullet }^0\\ \mathsf{ne}_\bullet : (A : \mathsf{tp}) \to \Omega _{\textit {dec}}\to \mathsf{tm}A \to \mathscr{U}_{\bullet }^0\\ \mathsf{nftp}_\bullet : \mathsf{tp} \to \mathscr{U}_{\bullet }^0 \end{array} \end{align*}

Next, we use the strict gluing connective to define the types of normals, neutrals, and normal types such that they lie strictly over $\mathsf {tm}$ and $\mathsf {tp}$ :

\begin{align*} \begin{array}{l} {\mathsf{nf}}\,A = (a : \mathsf{tm}A) \ltimes {\mathsf{nf}}_\bullet \,A\,a\\ \mathsf{ne}_\phi \,A = (a : \mathsf{tm}A) \ltimes \mathsf{ne}_\bullet \,A\,\phi \,a\\ \mathsf{nftp} = (A : \mathsf{tp}) \ltimes \mathsf{nftp}_\bullet \,A \end{array} \end{align*}

We illustrate a representative fragment of the inductive definitions in Figure 2.

The induction principles for ${\mathsf{nf}}_\bullet ,\mathsf{ne}_\bullet$ and $\mathsf{nftp}_\bullet$ play no role in the main development, which works with any algebra for these constants. These induction principles, however, are needed in order to prove Theorem 48 and deduce the decidability of definitional equality and the injectivity of type constructors. These same considerations motivate our choice to index $\mathsf{ne}_\bullet$ over $\Omega _{\textit {dec}}$ rather than $\Omega$ .

Figure 2.

Selected rules from the definition of $\mathsf{nf}$ , $\mathsf{ne}$ , and $\mathsf{nftp}$ .

6.7 A glued normalization algebra

We can now construct a new $\mathbf{TT}_{\mathbb{P}}$ -algebra internally to $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ , satisfying the constraint that each of its constituents restricts under $\mathsf{obj}$ to the corresponding constant from the $\mathbf{TT}_{\mathbb{P}}$ -algebra inherited from $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ . We shall refer to this as the normalization algebra. For instance, we must define types representing object types and terms:

\begin{equation*} \mathsf {tp}^{\ast} : \lbrace \mathscr{U}_{2}\vert \mathsf{obj} \hookrightarrow \mathsf {tp}\rbrace \qquad \mathsf {tm}^{\ast} : \lbrace \mathsf {tp}^{\ast} \to \mathscr{U}_{1}\vert \mathsf{obj} \hookrightarrow \mathsf {tm}\rbrace \end{equation*}

The meta-space component of the computability structure of types is given as a dependent record below:

\begin{align*} \begin{array}{l} \mathbf{record}\,\mathsf{tp}_\bullet \,(A:\mathsf{tp}):{\mathscr{U}_{2}}\; \mathbf{where}\\ \quad \mathsf{code} : \mathsf{nftp}_\bullet \,A\\ \quad \mathsf{tm}_\bullet : \mathsf{tm}A \to \mathscr{U}_{\bullet }^1\\ \quad \mathsf{reflect} : (a : \mathsf{tm}A)\,(\phi : \Omega _{\textit {dec}})\,(e : \mathsf{ne}_\bullet \,A\,\phi \,a) \to (a_\phi : \{\phi \}\,\mathsf{tm}_\bullet \,a) \to \lbrace {\mathsf{tm}_\bullet \,a}\vert {\phi \hookrightarrow a_\phi }\rbrace \\ \quad \mathsf{reify} : (a : \mathsf{tm}A) \to \mathsf{tm}_\bullet \,a \to {\mathsf{nf}}_\bullet \,A\,a \end{array} \end{align*}

The $\mathsf{tm}_\bullet$ field classifies the meta-space component of a given element; the $\mathsf{reflect}$ and $\mathsf{reify}$ fields generalize the familiar operations of normalization by evaluation, subject to Sterling and Angiuli’s stabilization yoga (Sterling and Angiuli Reference Sterling and Angiuli2021). We finally define both $\mathsf{tp}$ and $\mathsf{tm}$ using strict gluing to achieve the correct boundary:

\begin{align*} \begin{array}{l} \mathsf{tp}^{\ast} = (A : \mathsf{tp}) \ltimes \mathsf{tp}\bullet \,A\\ \mathsf{tm}^{\ast}\,A = (a : \mathsf{tm}A) \ltimes (\mathsf{ungl}\,A).\mathsf{tm}_\bullet \,a \end{array} \end{align*}

We must also define $\langle p \rangle ^{\ast} : \Omega _{\textit {dec}}$ for each $p \in \mathbb{P}$ subject to the condition that $\mathsf{obj}$ implies $\langle p \rangle ^{\ast} = \langle p \rangle$ . As there is no normalization data associated with these propositions, we define $\langle p \rangle ^{\ast} = \langle p \rangle$ which clearly satisfies the boundary condition. It remains to show that $(\mathsf{tp}^{\ast},\mathsf{tm}^{\ast})$ are closed under all the connectives of $\mathbf{TT}_{\mathbb{P}}$ . We show two representative cases: extension types and the universe.

6.7.1 Extension types

Fixing $A : \mathsf{tp}^{\ast}$ , $p : \mathbb{P}$ , $a : \{\langle p \rangle \}\,\mathsf{tm}^{\ast}\,A$ , we must construct the following pair of constants:

\begin{align*} \begin{array}{l} \mathsf{ext}^{\ast}_p\,A\,a : \lbrace \mathsf{tp}^{\ast} \vert \mathsf{obj} \hookrightarrow \mathsf{ext}^{\ast}p\,A\,a\rbrace \\ \mathsf{in}^{\ast}_p\,A\,a : \lbrace \lbrace \mathsf{tm}^{\ast}\,A \vert \langle p \rangle \hookrightarrow a\rbrace \cong \mathsf{tm}^{\ast}\,(\mathsf{ext}^{\ast}_p\,A\,a) \vert \mathsf{obj} \hookrightarrow \mathsf{in}_p\,A\,a\rbrace \end{array} \end{align*}

Recalling the definition of $\mathsf{tp}$ as a strict gluing type, we observe that the boundary condition on $\mathsf{ext}_p$ already fully constrains the first component:

In the above, we have used Notation 28 for constructing elements of a strict gluing type.

We define the second component as follows, using copattern matching notation:

In the clauses of $\mathsf{reify}$ and $\mathsf{reflect}$ , we were allowed to assume that the argument was of the form $\eta _\bullet x$ where $\eta _\bullet$ is the unit of the modality $\bullet$ : $\eta _\bullet : A \to \bullet A$ . This is because we are mapping into meta-space types and so this “pattern-matching” amounts to the bind operation of the monad $\bullet$ .

Remark 32. Stabilized neutrals are crucial to the definition of $(\mathsf{ext}^{\ast}_p\,A\,a) \hspace {.1ex} . \hspace {.1ex} \mathsf{reflect}$ above: without them, we could not ensure that reflecting lies within the specified subtype of $A \hspace {.1ex} . \hspace {.1ex} \mathsf{tm}_\bullet$ .

The definition of $\mathsf{in}_p$ is now straightforward:

\begin{align*} \mathsf{in}^{\ast}_p\,A\,a\,x = [{\mathsf{obj} \hookrightarrow \mathsf{in}_p\,A\,a\,x}\vert {\mathsf{ungl}\,x}] \end{align*}

We leave the routine verification of the various boundary conditions to the reader; nearly all of them follow immediately from the properties of strict gluing.

Figure 3.

The normalization structure on the universe.

6.7.2 The universe

We now turn to the construction of the universe in the normalization algebra; it is here that the complexity of unstable neutrals becomes evident. Once again the boundary conditions on $\mathsf{uni}$ force part of its definition:

The second component of $\mathsf{uni}^{\ast}$ is complex, and we present its definition in Figure 3. The inclusion of $\mathsf{el\text{-}{}code}$ in $\mathsf{uni}_\bullet$ is necessary in order to define $\mathsf{el}^{\ast}$ :

\begin{align*} \begin{array}{l} \mathsf{obj} \hookrightarrow \mathsf{el}^{\ast}\,A = \mathsf{el}A\\ (\mathsf{el}^{\ast}\,(\eta _\bullet A)) \hspace {.1ex} . \hspace {.1ex} \mathsf{code} = A \hspace {.1ex} . \hspace {.1ex} \mathsf{el\text{-}{}code}\\ (\mathsf{el}^{\ast}\,(\eta _\bullet A)) \hspace {.1ex} . \hspace {.1ex} \mathsf{tm}_\bullet = A \hspace {.1ex} . \hspace {.1ex} \mathsf{tm}_\bullet \\ (\mathsf{el}^{\ast}\,(\eta _\bullet A)) \hspace {.1ex} . \hspace {.1ex} \mathsf{reflect} = A \hspace {.1ex} . \hspace {.1ex} \mathsf{reflect}\\ (\mathsf{el}^{\ast}\,(\eta _\bullet A)) \hspace {.1ex} . \hspace {.1ex} \mathsf{reify} = A \hspace {.1ex} . \hspace {.1ex} \mathsf{reify} \end{array} \end{align*}

Finally, we must show that $\mathsf{uni}$ is closed under all small type formers and that $\mathsf{el}$ preserves them. This flows from the cumulativity of universes in $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ ; to close $\mathsf{uni}$ under, for example, products, we essentially "redo" the construction of products in $\mathsf{tp}$ by altering its predicate to be valued in $\mathscr{U}_{0}$ rather than $\mathscr{U}_{1}$ .

6.8 The normalization algorithm

Having constructed the normalization algebra in $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ (Section 6.7), we can now define the actual normalization function using an argument based on those presented by Fiore (Reference Fiore2002, Reference Fiore2022, §II.2) and Sterling (Reference Sterling2025, §3.3), making use of the inserter model of atomic substitutions introduced by Bocquet et al. (Reference Bocquet, Kaposi and Sattler2021). As our results are constructive, our normalization function corresponds to an actual normalization by evaluation algorithm – whose executable computational content Fiore (Reference Fiore2022) has demonstrated explicitly in the simply typed case.

6.8.1 Stripping of atomic contexts

We first must establish an intermediate result: that the functor ${\boldsymbol{\mathbf{A}}\langle - \rangle }:{\mathbb{P}}\to {\boldsymbol{\mathbf{A}}_\diamond }$ sending each $p\in \mathbb{P}$ to the associated unary atomic context is fully faithful and has a left adjoint, that is, that $\mathbb{P}$ is reflective in $\boldsymbol{\mathbf{A}}_\diamond$ . The left adjoint allows an atomic context to be “stripped” of anything that induces variables, leaving only propositional assumptions. This result is ultimately used in Lemma 40 to exhibit an isomorphism $\boldsymbol{\mathbf{A}}\langle p \rangle \cong \alpha ^{\ast}\mathbf{I}\langle p \rangle$ .

While it is straightforward to imagine how such a reflection can be defined by “induction on atomic contexts and repeated weakening,” we have not given an inductive specification of $\boldsymbol{\mathbf{A}}_{\diamond }$ and instead opted to specify it through its universal property. Accordingly, we define this stripping map using an model to which we may apply the universal property of $\boldsymbol{\mathbf{A}}$ . Fundamentally, however, the resulting constructions are the same, but our insistence on using only these universal properties enables us to avoid fixing a particular and explicit construction of atomic contexts.

Construction 33 (The stripping model). We consider a model $\mathbf{P}$ of $\mathbb{T}$ in which we set $\mathbf{P}_\diamond = \mathbb{P}$ , $\mathbf{P}(\mathsf{tp}) = \mathbf{P}(\mathsf{tm}) = \mathbf{1}_{{\mathbf{Pr}\,{\mathbb{P}}}}$ , and $\mathbf{P}\langle p \rangle = {}_{\mathsf {Y}{\mathbb{P}}}{p}$ . All the remaining constructs of the model are trivial by virtue of these definitions. From the universal property of $\mathbf{I}$ as the bi-initial $\mathbb{T}$ -model, we obtain a unique homomorphism of models ${I_{\mathbf{P}}}:{\mathbf{I}}\to {\mathbf{P}}$ whose contextual component is a product preserving functor ${I_{\mathbf{P}}^\diamond }:{\mathbf{I}_\diamond }\to {\mathbb{P}}$ .

Lemma 34. The component ${\boldsymbol{\mathbf{A}}\langle - \rangle }:{\mathbb{P}}\to {\boldsymbol{\mathbf{A}}_\diamond }$ of the bi-initial atomic substitution model over $\mathbf{I}$ is full and faithful.

Proof. Any functor out of a poset is necessarily faithful. To see that ${\boldsymbol{\mathbf{A}}\langle - \rangle }:{\mathbb{P}}\to {\boldsymbol{\mathbf{A}}_\diamond }$ is full, we fix a morphism ${\boldsymbol{\mathbf{A}}\langle p \rangle }\to {\boldsymbol{\mathbf{A}}\langle q \rangle }$ in $\boldsymbol{\mathbf{A}}_\diamond$ ; as this morphism is necessarily unique, it suffices to show that $p\leq q$ in $\mathbb{P}$ . We consider the image of ${\boldsymbol{\mathbf{A}}\langle p \rangle }\to {\boldsymbol{\mathbf{A}}\langle q \rangle }$ under the contextual component of the composite homomorphism ${I_{\mathbf{P}}\circ \alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{P}}$ of $\mathbb{T}_0$ -models, which gives precisely the desired inequality $p\leq q$ , recalling that each $\langle r \rangle$ is is representable in $\mathbb{T}_0$ and thus preserved by homomorphisms of models.

Lemma 35. The functor between categories of contexts induced by ${I_{\mathbf{P}}\circ \alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{P}}$ is left adjoint to the embedding ${\boldsymbol{\mathbf{A}}\langle - \rangle }:{\mathbb{P}}\hookrightarrow {\boldsymbol{\mathbf{A}}_\diamond }$ .

Proof. The counit in $\mathbb{P}$ is given by the identity inequality, as each $\langle p \rangle$ is representable in $\mathbb{T}_0$ and thus preserved by homomorphisms. For the unit, we must construct a (necessarily unique) arrow ${\Gamma }\to {\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond \Gamma )}}\rangle }$ in $\boldsymbol{\mathbf{A}}_\diamond$ for each atomic context $\Gamma$ .

For this, we consider a new atomic substitution model $\mathbf{E}$ over $\boldsymbol{\mathbf{A}}$ whose category of contexts $\mathbf{E}_\diamond$ is the following inserter object (Lack Reference Lack2009, Section 6.5) in $\mathsf{Cat}$ :

Equivalently, $\mathbf{E}_\diamond$ is the full subcategory of $\boldsymbol{\mathbf{A}}_\diamond$ spanned by atomic contexts $\Gamma$ for which there exists an arrow ${\Gamma }\to {\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond \Gamma )}}\rangle }$ ; as the codomain is subterminal, such arrows are necessarily unique. We define all the constructs of $\mathbb{T}_0$ in $\mathbf{E}$ as in $\boldsymbol{\mathbf{A}}$ , and it remains only to check that $\mathbf{E}$ has a terminal object and is closed under context comprehension and phase comprehension.

1. (1) For the terminal object, we see that $\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond {})}}\rangle$ is already terminal.

2. (2) For the context comprehension, we fix $\Gamma \in \mathbf{E}_\diamond$ and $A\in \boldsymbol{\mathbf{A}}(\mathsf{tp}(\Gamma )$ , and we must check that there exists a map ${\Gamma .A}\to {\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma .A))}}\rangle }$ . As $\alpha \circ I_{\mathbf{P}}$ is a homomorphism of models, it preserves context comprehensions; unraveling definitions, we ultimately have $I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma .A))=I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma ))$ and so we are done.

3. (3) For phase comprehension, we fix $\Gamma \in \mathbf{E}_\diamond$ and $p\in \mathbb{P}$ to check that there exists an arrow ${\Gamma .\boldsymbol{\mathbf{A}}\langle p \rangle }\to {\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma .\boldsymbol{\mathbf{A}}{(\langle p \rangle )}))}}\rangle }$ . But we have $I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma .\boldsymbol{\mathbf{A}}{(\langle p \rangle )})) = I_{\mathbf{P}}^\diamond (\alpha _\diamond (\Gamma ))\land p$ , so we may use the projection ${\Gamma .\boldsymbol{\mathbf{A}}\langle p \rangle }\to {\boldsymbol{\mathbf{A}}\langle p \rangle }$ .

We evidently have a homomorphism of $\mathbb{T}_0$ -models ${\eta }:{\mathbf{E}}\to {\boldsymbol{\mathbf{A}}}$ that exhibits $\mathbf{E}$ as a atomic substitution model over $\boldsymbol{\mathbf{A}}$ . Postcomposing with the structure map ${\alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{I}}$ , we can view $\mathbf{E}$ as a atomic substitution model over $\mathbf{I}$ . Thus, by the universal property of $\boldsymbol{\mathbf{A}}$ , we have a universal section ${J_{\mathbf{E}}}:{\boldsymbol{\mathbf{A}}}\to {\mathbf{E}}$ to ${\eta }:{\mathbf{E}}\to {\boldsymbol{\mathbf{A}}}$ . This shows that every atomic context $\Gamma \in \boldsymbol{\mathbf{A}}_\diamond$ can be equipped with an arrow ${\Gamma }\to {\boldsymbol{\mathbf{A}}\langle {{I_{\mathbf{P}}^\diamond (\alpha _\diamond \Gamma )}}\rangle }$ . Assembling all these arrows together, we have the unit of the adjunction $I_{\mathbf{P}}^\diamond \circ \alpha _\diamond \dashv \boldsymbol{\mathbf{A}}\langle {-}\rangle$ .

The force of Lemma 35 is to show that $\mathbb{P}$ is a reflective subcategory of $\boldsymbol{\mathbf{A}}_\diamond$ .

6.8.2 Computability spaces of atomic and computable substitutions

We will consider two computability spaces induced by an atomic context $\Gamma$ : the computability space $[\![\Gamma ]\!]$ of “computable substitutions into $\Gamma$ ” and the computability space $(\!|{\Gamma }|\!)$ of “atomic substitutions into $\Gamma$ .”

Construction 36 (The computability space of computable substitutions). The computability space $[\![\Gamma ]\!]$ of computable substitutions into an atomic context $\Gamma$ is defined in terms of the interpretation of $\mathbb{T}$ into $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ as follows, sending each atomic context to the computability space determined by the algebra structure:

Construction 37 (The computability space of atomic substitutions). We define an embedding ${(\!|-|\!)}:{\boldsymbol{\mathbf{A}}_\diamond }\hookrightarrow {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ sending $\Gamma \in \boldsymbol{\mathbf{A}}_\diamond$ to the computability space $(\!|\Gamma |\!)$ with ${j}^{\ast}(\!|{\Gamma }|\!) = \mathsf {Y}{\mathbf{I}_\diamond }{\alpha _\diamond \Gamma }$ and ${i}^{\ast}(\!|{\Gamma }|\!) = \mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma$ , such that ${\pi _{(\!|\Gamma |\!)}}\to {\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }{\alpha ^{\ast}\mathsf {Y}{\mathbf{I}_\diamond }\alpha _\diamond \Gamma }$ is defined on generalized elements by the functorial action of ${\alpha _\diamond }:{\boldsymbol{\mathbf{A}}_\diamond }\to {\mathbf{I}_\diamond }$ as follows:

\begin{equation*} \pi _{(\!|\Gamma |\!)}^\Delta ({\gamma }:{\Delta }\to {\Gamma }) = {\alpha _\diamond {\gamma }}:{\alpha _\diamond \Delta }\to {\alpha _\diamond \Gamma } \end{equation*}

Miraculously, the coreflective embedding ${{i}_!}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{A}}}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ sends $\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma$ to precisely the computability space $(\!|\Gamma |\!)$ , up to isomorphism:

\begin{align*} {{i}}_!\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma &= (\alpha _!\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma , {}:{\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }\to {\alpha ^{\ast}\alpha _!\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }) \\ &\cong (\mathsf {Y}{\mathbf{I}_\diamond }\alpha _\diamond \Gamma , {}:{\mathsf {Y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }\to {\alpha ^{\ast}\mathsf {Y}{\mathbf{I}_\diamond }\alpha _\diamond \Gamma }) \\ &= (\!|\Gamma |\!) \end{align*}

Note that the functors $[\![-]\!],(\!|-|\!)$ lift into the slice $\mathsf{Cat}\downarrow \mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ in the following sense:

Notation 38. Let $\Gamma$ be an atomic context, and let ${A}: {(\!|\Gamma |\!)}\to {\mathsf{tp}}$ be an object-space type, which we may regard as a morphism ${\alpha \Gamma }\to {\mathsf{tp}}$ in $\mathbb{T}$ . We shall write $[\![A]\!]\colon [\![\Gamma ]\!]\to \mathsf{tp}^{\ast}$ for the image of $A$ under the interpretation functor $I_{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ .

Lemma 39. Let $\Gamma$ be an atomic context, and let ${A}:{(\!|\Gamma |\!)}\to {\mathsf{tp}}$ be an object-space type (which we may regard as a morphism ${\alpha _\diamond \Gamma }\to {\mathsf{tp}}$ in $\mathbb{T}$ ). Then we have the following cartesian squares:

Stated in the internal language, we have canonical isomorphisms $(\!|\Gamma .A|\!)\cong \sum _{\gamma :(\!|\Gamma |\!)}\mathsf{var}\,(A\gamma )$ and $[\![\Gamma .A]\!] \cong \sum _{\gamma :[\![\Gamma ]\!]}\mathsf{tm}\,([\![A]\!]\gamma )$ .

Proof. The latter is the image of a pullback square in $\mathbb{T}$ under $I_{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ , which is finitely continuous. The former can be seen by means of an explicit computation.

We now come to an important result relating the interpretation of $\langle p \rangle$ in $\mathbf{I}$ to the interpretation of the same in $\boldsymbol{\mathbf{A}}$ . Lemma 40 is the raison d’être for the stripping model $\mathbf{P}$ in Section 6.8.1.

Lemma 40. We have a (necessarily unique) isomorphism $\boldsymbol{\mathbf{A}}\langle p \rangle \cong \alpha ^{\ast}\mathbf{I}\langle p \rangle$ .

Proof. As both presheaves are subterminal, it is enough to see that one is inhabited if and only if the other is.

1. (1) We may transpose a map ${}_{{\mathsf {y}{\boldsymbol{\mathbf{A}}_\diamond }} \Gamma }\to {\alpha ^{\ast}\mathbf{I}\langle p \rangle }$ to get ${\alpha _\diamond \Gamma }\to {\mathbf{I}\langle p \rangle }$ ; applying the functorial action of ${I_{\mathbf{P}}^\diamond }\to {\mathbf{I}_\diamond }{\mathbb{P}}$ , we have $I_{\mathbf{P}}^\diamond \alpha _\diamond \Gamma \leq I_{\mathbf{P}}^\diamond \mathbf{I}\langle p \rangle =p$ and by adjoint transpose with Lemma 35, we have ${\Gamma }\to {\boldsymbol{\mathbf{A}}\langle p \rangle }$ .

2. (2) Conversely, given an arrow ${\Gamma }\to {\boldsymbol{\mathbf{A}}\langle p \rangle }$ we may apply the functorial action of ${\alpha _\diamond }:{\boldsymbol{\mathbf{A}}_\diamond }\to {\mathbf{I}_\diamond }$ to obtain an arrow ${\alpha _\diamond \Gamma }\to {\alpha _\diamond \boldsymbol{\mathbf{A}}\langle p \rangle }$ . As $\langle p \rangle$ is representable in $\mathbb{T}_0$ , it is preserved by morphisms of models like ${\alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{I}}$ ; thus, $\alpha _\diamond \boldsymbol{\mathbf{A}}\langle p \rangle \cong \mathbf{I}\langle p \rangle$ and so we have ${\alpha _\diamond \Gamma }\to {\mathbf{I}\langle p \rangle }$ , which we may transpose to obtain ${}_{\mathsf {y}{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }\to {\alpha ^{\ast}\mathbf{I}\langle p \rangle }$ .

6.8.3 Hydration of atomic substitutions

A critical point in concrete normalization by evaluation algorithms is to “reflect” a vector of variables as an environment of (computable) values against which the computability interpretation of an open term can be executed; in a concrete setting, this operation is defined by recursion on the atomic contexts. The same process, which we shall refer to here as the hydration of atomic substitutions, plays an equally important role in semantic proofs of normalization in the guise of a certain hydration map ${\nearrow }:\to {(\!|{-}|\!)}{[\![-]\!]}$ in $\mathsf{Cat}\downarrow \mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ that we shall need to construct.

Just as in the definition of the stripping map, we are confronted by the fact that we have defined the atomic contexts only by means of a universal property (the bi-initial atomic substitution model over $\mathbf{I}$ ), so we do not immediately have anything concrete to do recursion on. The innovation of Bocquet et al. (Reference Bocquet, Kaposi and Sattler2021) was to find the correct categorical induction motive that explains the usual recursive argument purely in terms of the universal property of the bi-initial atomic substitution model (likewise due to op. cit.). In what follows, we adapt their ideas to our setting and show how to construct the desired hydration map.

We begin by defining a new model $\mathbf{H}$ of $\mathbb{T}_0$ that we shall refer to as the hydration model. The category of contexts $\mathbf{H}_\diamond$ is defined to be the underlying category of the inserter object for ${(\!|-|\!),[\![-]\!]}:{\boldsymbol{\mathbf{A}}_\diamond }\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ in $\mathsf{Cat}\downarrow \mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ as depicted below:

Explicitly, an object of $\mathbf{H}_{\diamond}$ is a pair $(\Gamma ,h_\Gamma )$ of an object $\Gamma \in {\mathbf{A}}_\diamond$ together with an arrow ${h_\Gamma }:{(\!|\Gamma |\!)}\to {[\![\Gamma ]\!]}$ whose image under ${{\mathbf {j}}^{\ast}}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}$ is the identity map on $\mathsf {Y}{\mathbf{I}_\diamond }\alpha _\diamond \Gamma$ . An arrow from $(\Delta ,h_\Delta )$ to $(\Gamma ,h_\Gamma )$ is given by an arrow ${\gamma }:{\Delta }\to {\Gamma }$ in ${\mathbf{A}}_\diamond$ making the following square commute:

Clearly, $\mathbf{H}_\diamond$ has a terminal object because $[\![\mathbf{1}_{\boldsymbol{\mathbf{A}}_\diamond }]\!]$ is terminal. Anticipating that $\psi _\diamond$ should lift to a morphism of models exhibiting $\mathbf{H}$ as a atomic substitution model over $\boldsymbol{\mathbf{A}}$ , we define $\mathbf{H}\langle p \rangle = \psi _\diamond ^{\ast}\boldsymbol{\mathbf{A}}\langle p \rangle$ and $\mathbf{H}(\mathsf{tp}) = \psi _\diamond ^{\ast}\boldsymbol{\mathbf{A}}(\mathsf{tp})$ . In order to define $\mathbf{H}(\mathsf{tm})$ , it will be easiest to first define context comprehensions.

Construction 41 (Context comprehensions in $\mathbf{H}_\diamond$ ). Given a context $(\Gamma ,h_\Gamma )$ in $\mathbf{H}_\diamond$ and a type ${A}:{\mathsf {Y}_{\boldsymbol{\mathbf{A}}_\diamond }\Gamma }\to {\boldsymbol{\mathbf{A}}(\mathsf{tp})}$ , we can lift the context comprehension $\Gamma .A\in \boldsymbol{\mathbf{A}}_\diamond$ into $\mathbf{H}_\diamond$ by finding a suitable map ${h_{\Gamma .A}}:{(\!|\Gamma .A|\!)}\to {[\![\Gamma .A]\!]}$ whose image in $\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ is the identity. Recalling Lemma 39, we may equivalently construct (from the internal point of view) the following map:

The projection ${p_A}:{\Gamma .A}\to {\Gamma }$ tracks a morphism ${(\Gamma .A,h_{\Gamma .A})}\to {(\Gamma ,h_\Gamma )}$ , by definition of $h_{\Gamma .A}$ .

Construction 42 (Phase comprehensions in $\mathbf{H}_\diamond$ ). Given $\Gamma \in \mathbf{H}_\diamond$ and $p\in \mathbb{P}$ , we must exhibit a map ${h_{\Gamma .\boldsymbol{\mathbf{A}}\langle p \rangle }}:{(\!|\Gamma .\boldsymbol{\mathbf{A}}\langle p \rangle |\!)}\to {[\![\Gamma .\boldsymbol{\mathbf{A}}\langle p \rangle ]\!]}$ . Using Lemma 39, we construct this by combining the assumed map ${h_\Gamma }:{(\!|\Gamma |\!)}\to {[\![\Gamma ]\!]}$ with the (necessarily unique) map ${(\!|\boldsymbol{\mathbf{A}}\langle p \rangle |\!)}\to {[\![\boldsymbol{\mathbf{A}}\langle p \rangle ]\!]}$ obtained from the identity map on $\boldsymbol{\mathbf{A}}\langle p \rangle$ in the following way:

1. (1) First, we observe that $[\![\boldsymbol{\mathbf{A}}\langle p \rangle ]\!] \cong {\mathbf {j}}_{\ast}\mathbf{I}\langle p \rangle$ as follows:

   \begin{align*} [\![\boldsymbol{\mathbf{A}}\langle p \rangle ]\!] &= I_{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}\alpha _\diamond \boldsymbol{\mathbf{A}}\langle p \rangle &&\text{by definition} \\ &\cong I_{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}\mathbf{I}\langle p \rangle &&\text{$\alpha $ is a homomorphism} \\ &\cong {\mathbf {j}}_{\ast}\mathbf{I}\langle p \rangle &&\text{by definition} \end{align*}

2. (2) Then we proceed by adjoint calisthenics:

   

Construction 43 (The presheaf of terms). We define $\mathbf{H}(\mathsf{tm})\in \mathbf{Pr}\,{\mathbf{H}_\diamond }\downarrow \mathbf{H}(\mathsf{tp})$ to send $A\in \mathbf{H}(\mathsf{tp})(\Gamma ,h_\Gamma )$ to the set of sections of the projection ${p_A}:{(\Gamma .A,h_{\Gamma .A})}\to {(\Gamma ,h_\Gamma )}$ in $\mathbf{H}_\diamond$ .

Construction 44 (The hydration model). With the definitions that we have given, the projection functor ${\psi _\diamond }:{\mathbf{H}_\diamond }\to {\boldsymbol{\mathbf{A}}_\diamond }$ easily extends to a morphism of models, exhibiting $\mathbf{H}$ as a atomic substitution model over $\boldsymbol{\mathbf{A}}$ .

Construction 45 (The hydration map). As we may compose ${\psi }:{\mathbf{H}}\to {\boldsymbol{\mathbf{A}}}$ with the structure map ${\alpha }:{\boldsymbol{\mathbf{A}}}\to {\mathbf{I}}$ , we can view $\mathbf{H}$ as a atomic substitution model over $\mathbf{I}$ as well, and thus by the universal property of $\boldsymbol{\mathbf{A}}$ we have a universal section ${\boldsymbol{\mathbf{A}}}\to {\mathbf{H}}$ to ${\psi }:{\mathbf{H}}\to {\boldsymbol{\mathbf{A}}}$ . Unraveling the section ${\boldsymbol{\mathbf{A}}}\to {\mathbf{H}}$ , we obtain precisely a natural assignment of hydration map component ${h_\Gamma }:{(\!|\Gamma |\!)}\to {[\![\Gamma ]\!]}$ from which we may assemble a single hydration map ${\nearrow }:{(\!|-|\!)}\to {[\![-]\!]}$ in $\mathsf{Cat}\downarrow \mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}$ .

6.8.4 Normalization and decidability

We can now show how to compute the normal form of a type $\Gamma \vdash A\ \textit{type}$ , which we regard as an arrow ${A}:{\alpha _\diamond \Gamma }\to {\mathsf{tp}}$ in $\mathbb{T}$ . Then we may apply the functorial action of the evaluation functor ${I_{\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}}:{\mathbb{T}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ to obtain ${[\![A]\!]}:{[\![\Gamma ]\!]}\to {\mathsf{tp}^{\ast}}$ and then postcompose with the projection of normal forms to obtain ${[\![A]\!] \hspace {.1ex} . \hspace {.1ex} \mathsf{code}}:{[\![\Gamma ]\!]}\to {\mathsf{nftp}}$ . Unraveling the meaning of such a map in the gluing category $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ , we see that this amounts not to a normal form for $A$ but instead to an assignment of normal forms of $A$ to computability witnesses for all the variables in the context $\Gamma$ . It is precisely this gap that hydration fills:

\begin{equation*} \mathsf{norm}_{\mathsf{tp}}(\Gamma \vdash A\ \textit{type}) = (\!|\Gamma |\!)\xrightarrow {\nearrow _\Gamma }[\![\Gamma ]\!]\xrightarrow {[\![A]\!]}\mathsf{tp}^{\ast}\xrightarrow {- \hspace {.1ex} . \hspace {.1ex} \mathsf{code}}\mathsf{nftp} \end{equation*}

The entire map above restricts within object-space, by construction, to the original type $A$ , or (to be more precise) its image in $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ under ${{j}_{\ast}}:{\mathbf{Sh}\,{\boldsymbol{\mathsf{I}}}}\to {\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}}$ . The meta-space component of such a morphism ${(\!|\Gamma |\!)}\to {\mathsf{nftp}}$ is precisely a normal form for $A$ .

Theorem 46. Normalization is sound and complete:

1. (1) Soundness. If $\mathsf{norm}_{\mathsf{tp}}( \Gamma \vdash A\ \textit{type}) = \mathsf{norm}_{\mathsf{tp}}( \Gamma \vdash B\ \textit{type})$ , then ${\Gamma } \vdash A = {B}$ .

2. (2) Completeness. If ${\Gamma } \vdash A = B$ , then $\mathsf{norm}_{\mathsf{tp}}(\Gamma \vdash A\ \textit{type}) = \mathsf{norm}_{\mathsf{tp}}(\Gamma \vdash B\ \textit{type})$ .

Proof. Completeness holds by definition, as the normalization function is defined on the denizens of the syntactic CwR rather than on raw terms. Soundness follows from the fact that $\mathsf{norm}_{\mathsf{tp}}(\Gamma \vdash A\ \textit{type})$ restricts in object-space to $A$ itself.

In the same way, we can construct a normalization function for terms and prove that it is sound and complete (though we do not do so here). Of course, deciding equality for terms is practically important only insofar as it arises in the context of deciding equality for the types that mention them.

Definition 47. An object $X \in \mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ has levelwise decidable equality when for each $\Gamma \in \boldsymbol{\mathbf{A}}_{\diamond }$ , the set $({i}^{\ast}X)\Gamma$ has decidable equality where $i:A\hookrightarrow {\boldsymbol{\mathsf{G}}}$ is as in Definition 19.

Theorem 48. Viewed as objects of $\mathbf{Sh}\,{\boldsymbol{\mathsf{G}}}$ , the following have levelwise decidable equality:

\begin{equation*} \mathsf{nftp} \qquad (A : \mathsf{tp}) \times {\mathsf{nf}}\,A \qquad (A : \mathsf{tp}) \times (\phi :\Omega _{\textit {dec}}) \times \mathsf{ne}_\phi {A} \end{equation*}

From this, we obtain our main results concerning $\mathbf{TT}_{\mathbb{P}}$ :

Corollary 49. Definitional equality in $\mathbf{TT}_{\mathbb{P}}$ is decidable.

7.1 Library-level features

Various library-level idioms for abstract definitions are used in practice such as SSReflect’s lock idiom. While such approaches are flexible and compatible with existing proof assistants, they are often cumbersome in practice. For instance, lock relies on various tactics with subtle behavior, which makes it difficult to use locking idioms in pure Gallina code.

7.2 Language-level features

Many proof assistants include a feature like Agda’s abstract blocks which marks a definition as completely opaque to the remainder of the development. In Remark 1, we explained how to recover Agda’s abstract definitions using controlled unfolding. Moreover, as controlled unfolding does not require a user to decide up front whether a definition can be unfolded, it gives a more realistic and flexible discipline for abstraction in a proof assistant. In practice, however, abstract is often used for performance reasons instead of merely for controlling abstraction; unfolding large or complex definitions can significantly slow down type checking and unification. While we have not discussed performance considerations for controlled unfolding, the same optimizations apply to our mechanism for definitions that are never unfolded. In total, controlled unfolding strictly generalizes abstract blocks.

Recently, Kovács ( Reference Kovács2023, Reference Kovács2024) has proposed a glued evaluation technique to both improve the pretty-printing of goals and more efficiently handle unfolding during conversion testing. Roughly, the proof assistant’s kernel may choose to unfold any definition but avoids doing so whenever possible for efficiency and strives to never show unfolded goals to the user. Both glued evaluation and controlled unfolding relate to the unfolding of definitions, and they are largely orthogonal and complementary. In particular, glued evaluation does not require user intervention, unlike controlled unfolding, but it does not actually preclude any unfolding from taking place. Thus, glued evaluation does not impact the well-formedness of a program and can be used as a “drop-in” technique for improving performance and usability. However, for the same reasons, glued evaluation cannot be used to enforce modularity and independence in the same way as controlled unfolding. Ideally, a proof assistant would support both glued evaluation and controlled unfolding: the more advanced evaluation algorithm would improve baseline performance and controlled unfolding would facilitate users enforcing stronger abstraction boundaries within their programs and assisting the kernel by manually designating certain definitions as opaque.

Program verifiers such as VeriFast and Chalice include similar unfolding mechanisms to cope specifically with recursive definitions ( Jacobs et al. Reference Jacobs, Vogels and Piessens2015; Summers and Drossopoulou Reference Summers, Drossopoulou and Castagna2013). Like our mechanism, these features allow users fine-grained control over how definitions are unfolded. However, these verifiers work only within simply typed theories and thus avoid the substantial complexity of dependency. Moreover, these mechanisms manage a different problem than controlled unfolding; they allow a user to unfold recursive definitions step-by-step, while controlled unfolding is used to control when each definition can be fully inlined.

7.3 Translucent ascription in module systems

Thus far we have focused on proof assistants, but similar considerations arise for ML-style module systems ( Dreyer et al. Reference Dreyer, Crary and Harper2003; Harper and Stone Reference Harper, Stone, Plotkin, Stirling and Tofte2000; Milner et al. Reference Milner, Tofte, Harper and MacQueen1997; Sterling and Harper Reference Sterling and Harper2021). The default opacity for definitions in module systems is the same as in controlled unfolding and opposite to proof assistants: types are abstract unless marked otherwise. The treatment of translucent type declarations in module systems (Harper and Stone Reference Harper, Stone, Plotkin, Stirling and Tofte2000) relies on singleton kinds (Aspinall Reference Aspinall, Pacholski and Tiuryn1995; Stone and Harper Reference Stone and Harper2006), which are the special case of extension types whose boundary proposition is $\top$ . Generalizing from compiletime kinds to mixed compiletime–runtime module signatures, Sterling and Harper have pointed out that transparent ascriptions are best handled by an extension type whose boundary proposition represents the compiletime phase itself (Sterling and Harper Reference Sterling and Harper2021). Thus, the translucency of compiletime module components can be seen as a particular controlled unfolding policy in the sense of this paper.

7.4 Controlled unfolding in Agda

Inspired by our implementation of controlled unfolding in , Amélia Liao and Jesper Cockx have implemented a version of this mechanism called opaque within Agda 2.6.4 (Liao and Cockx Reference Liao and Cockx2022). However, rather than using extension types, their Agda implementation simulates the necessary behaviors by instrumenting conversion checking – a workaround made possible by the very restricted ways in which our elaboration procedure relies on extension types. This demonstrates that controlled unfolding can be adapted to proof assistants like Coq whose core calculi do not presently support extension types.

At the time of writing, Agda’s opaque declarations are new enough that only two major Agda libraries, the 1Lab (The 1Lab Development Team 2022) and the Cubical Agda library (The Agda Community 2023), use them extensively; the Agda standard library may adopt opaque declarations in a future major revision (The Agda Development Team 2023). As of publication, 65 modules in the 1Lab use opaque and 19 use unfolding, in addition to over 100 using abstract blocks; in the Cubical Agda library, 27 modules use opaque, 6 use unfolding, and 35 use abstract.