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I'm relatively new to type theory and dependent programming. I've been studying the calculus of constructions (CoC) and other pure type systems. I'm particularly interested in using it as a proof-preserving intermediate representation for a compiler system.

I understand that (co-)recursive types are representable, computationally, using $\Pi$ as the only type constructor. I've read, though, that they cannot be used to build proofs by induction (forgive me, I can't find where now!), e.g., that I couldn't prove that $0\neq 1$ in plain CoC (even though $\texttt{Nat}$ is typeable as $\Pi(\mathbb{N}:*).\Pi(S:\mathbb{N}\rightarrow\mathbb{N}).\Pi(Z:\mathbb{N}).\mathbb{N}$).

I assume this is why they built the calculus of inductive constructions (CIC). Is this correct? But why? I couldn't find any material explaining why such proofs cannot be represented without using (co-)inductive types as primitives. If this is not true, then why add them as primitives in CIC?

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I'm not an expert, but I'll share what I understood so far with an example.

Let's consider the boolean type in CoC, using its standard encoding: $$ \begin{array}{l} \mathbb{B} = \Pi_{\tau:*} \tau \to \tau \to \tau \\ \mathsf{tt} = \lambda \tau:*,x:\tau,y:\tau.\ x \\ \mathsf{ff} = \lambda \tau:*,x:\tau,y:\tau.\ y \end{array} $$ We might expect to be able to prove $$ \Pi_{b : \mathbb{B}} b={\sf tt} \lor b = {\sf ff} \qquad(*) $$ Indeed, this quickly follows from the dependent elimination/induction principle we have e.g. in CiC $$ \mathbb{B}_{ind} : \Pi_{P:\mathbb{B}\to*} P({\sf tt}) \rightarrow P({\sf ff}) \rightarrow \Pi_{b:\mathbb{B}} P(b) $$

However, we can not really expect (*) to hold in all models of CoC! Intuitively, a value in $\mathbb{B}$ should roughly be a family of functions $\{f_\tau\}_{\tau}$ assigning to each type $\tau$ a value in the interpretation of $\tau\to\tau\to\tau$. But this does not force $f_\tau$ to be one between the values of $\sf tt,ff$. We could have e.g. (informally) $$ f_{\mathbb{N}}(n)(m) = n+m $$

To be sure that the values of $\sf tt,ff$ are the only possible values, we need to restrict ourselves to parametric models. Indeed (I think) the property $(*)$ can be proven from the free theorem associated to the polytype $\mathbb{B}$.

However, as far as I understand, CoC does not rule out ad-hoc models, where parametricity does not hold. In some of those, $(*)$ is simply false. By soundness, in the presence of a countermodel, we conclude that $(*)$ is not inhabited in CoC. Consequently, there's no $\mathbb{B}_{ind}$ term in CoC, either.

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  • $\begingroup$ I'm not sure I follow. For example, given an expression as $(\lambda(Nat:*).(...)) (\Pi(\mathbb{N}:*).\Pi(S:\mathbb{N}\rightarrow\mathbb{N}).\Pi(Z:\mathbb{N}).\mathbb{N})$, there are many ways that I could make constructors for Nat, but, ultimately, wouldn't them all be built up from either $S$ or $Z$? $\endgroup$ – paulotorrens Jan 6 '17 at 18:01
  • $\begingroup$ @paulotorrens Inside the logic, yes, I believe that these are the only options. But in a model of CoC (an ad-hoc model), there might be values of $Nat$ which can not be defined through $S,Z$. Think of a natural value $n$ such that $n(T)(S_T)(Z_T) = Z_T$ for all types $T$ except for $T=\mathbb{B}$ where instead $n(\mathbb{B})(S)(Z) = S(Z)$. The value $n$ behaves as "zero" on most types, and as "one" for booleans. We can't write $\lambda T:*. {\sf if}\ T=\mathbb{B}\ {\sf then}\ \ldots$ in CoC to define that value $n$, but in an ad-hoc model that value might be present nevertheless. $\endgroup$ – chi Jan 6 '17 at 22:04
  • $\begingroup$ @paulotorrens Maybe you can understand the issue more easily if you think about $\Pi_{T:*} T\to T$. This type is only inhabited by (polymorphic) identity, and in parametric models that's indeed the only possible value for terms of that type. But, in an ad-hoc model, we might define a value $v(T)(x) = x$ for all types $T$ but for $T=\mathbb{N}$ where $v(\mathbb{N})(x) = x+1$. $\endgroup$ – chi Jan 6 '17 at 22:08

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