- Recap for Product/Sum Types
- Recursive Types in a Partial Language
- Bridging Recursive Types and Programs

# Recap for Product/Sum Types

Brief recap for product and sum to have a consensus on notations.

## Product Types

### Introductions

### Eliminations

## Sum Types

### Introduction

### Eliminations

## Unit

# Recursive Types in a Partial Language

## Examples

### Natural Numbers

We can write

where $\cong$ means “type isomorphism” which means to write functions back and forth that compose to the identity.

From left to right:

From right to left:

It works because anything is observationally equivalent to $<>$ at type $unit$:

### Lists

Any list is either the empty list or a cons with the type and another list.

From left to right:

## Construction

What we’ve got here is that: certain types can be characterize as solutions to equations of the form:

$\mathbb{N}$ solves $t=unit+t$

$\tau list$ solves $t=unit+(\tau\times t)$

which is a/the solution to $t\cong \tau(t)$. i.e.

## Introductions

## Eliminations

## Dynamics

## Examples using Recursive Types

# Bridging Recursive Types and Programs

In this language, even though there is no loops, no recursive functions in the terms,
we can still get general recursion just from self-referencial types. Meaning that
**self-referential types will give us self-referential codes**. How to do?

- Define a type of self-referential programs, and get $fix(x.e)$ from it
- Define the type of self-referential programs from $rec(t.\tau)$

## Self Types

The following thing is only used as a bridge between $fix$ and $rec$ such that we can define both $fix$ and $rec$ from it. It’s only to help the constructions.

In the following rules, $x$ could be interpreted as `this`

in a normal programming language.

$\tau self$ represents a self-referential computation of a $\tau$

### Introductions

### Eliminations

### Dynamics

## Construct Recursive Programs

Now we have $\tau self$, the above two steps can be rephrased as:

- We want to get a recursive computation of any type, i.e. $fix(x.e):\tau$, from self types $\tau self$
- We want to get self types $\tau self$ from recursive types $rec(x.e)$

### From Recursive Computation to Self Types

We have $\tau self$, and we want:

Solution:

### From Self Types to Recursive Types

We want to solve:

Solution: