What is induction-induction?

Question

What is induction-induction?

The resources I found are:

• the HoTT book, at the end of chapter 5.7.
• nLab's article
• a paper called Inductive-inductive definitions
• this blog post also mentions inductive-inductive types

The first two references are too brief for me, and the latter two are too technical. Can anyone explain it in layman's term? It would be better if there's Agda code.

Answer 1

Supplemental 2016-10-03: I mixed up induction-induction and induction-recursion (not the first time I did that!). My apologies for the mess. I updated the answer to cover both.

I find the explanations in the Forsberg & Setzer's paper A finite axiomatisation of inductive-inductive definitions illuminating.

Induction-recursion

An inductive-recursive definition is one in which we define a type $A$ and a type family $B : A \to \mathsf{Type}$ simultaneously in a special way:

1. $A$ is defined as an inductive type.
2. $B$ is defined by recursion on $A$.
3. Crucially, the definition of $A$ may use $B$.

Without the third requirement, we could first define $A$ and then separately $B$.

Here is a baby example. Define $A$ inductively to have the following constructors:

• $a : A$
• $\ell : \big(\sum_{x : A} B(x)\big) \to A$

The type family $B$ is defined by

• $B(a) = \mathtt{bool}$
• $B(\ell(x, f)) = \mathtt{nat}$.

So, what is in $A$? First of all we have an element $$a : A.$$ Because of that, there is a type $B(a)$ which is defined to be $\mathtt{bool}$. Therefore, we can form two new elements $$\ell(a, \mathtt{false})$$ and $$\ell(a, \mathtt{true})$$ in $A$. Now we have $B(\ell(a, \mathtt{false})) = B(\ell(a, \mathtt{true})) = \mathtt{nat}$, so we can also form for every $n : \mathtt{nat}$ the elements $$\ell(\ell(a, \mathtt{false}), n) : A$$ and $$\ell(\ell(a, \mathtt{true}), n) : A$$ We can keep going like this. The next stage will be that since $$B(\ell(\ell(a, \mathtt{true}), n)) = \mathtt{nat}$$ there is for every $m : \mathtt{nat}$ the element $$\ell(\ell(\ell(a, \mathtt{true}), n), m) : A$$ and the element $$\ell(\ell(\ell(a, \mathtt{false}), n), m) : A$$ We can keep going. A little bit of thinking reveals that $A$ is more or less two copies of lists of natural numbers, sharing a common empty list. I will leave it as an exercise to figure out why.

Induction-induction

An inductive-inductive definition also defines a type $A$ and simultaneously a type family $B : A \to \mathsf{Type}$:

1. $A$ is defined inductively
2. $B$ is defined inductively, and it may refer to $A$
3. Crucially, $A$ may refer to $B$.

It is important to understand the difference between induction-recursion and induction-induction. In induction-recursion we define $B$ by providing equations of the form $$B(\mathsf{c}(\ldots)) = \cdots$$ where $\mathsf{c}(\ldots)$ is a constructor for $A$. In an inductive-inductive definition we define $B$ by providing constructors for forming the elements of $B$.

Let us reformulate our previous example as induction-induction. First we have the inductively given tpye $A$:

• $a : A$
• $\ell : \big(\sum_{x : A} B(x)\big) \to A$

The type family $B$ is defined by the following constructors:

• $\mathsf{Tru} : B(a)$
• $\mathsf{Fal} : B(a)$
• if $x : A$ and $y : B(x)$ then $\mathsf{Zer} : B(\ell(x,y))$
• if $x : A$ and $y : B(x)$ and $z : B(\ell(x,y))$ then $\mathsf{Suc}(z) : B(\ell(x,y))$.

As you can see, we gave rules for generating the elements of $B$ which amount to saying that $B(a)$ is (isomoprhic to) the booleans, and $B(\ell(x,y))$ is (isomorphic to) the natural numbers.