```
id : ? {A : Set} ? A ? A
id x = x
```

How would you define Agda?

Agda is a dependently typed functional programming language that has been designed to be a language for mathematics, software verification, and general-purpose programming.

What are the main features of Agda?

Agda is a dependently typed language, which means that types can depend on values. It also supports interactive theorem proving, inductive and coinductive types, and has a module system.

Describe the difference between Agda and other functional programming languages.

The main difference is that Agda is a dependently typed language, which means that types can depend on values. This allows for more expressive types and more powerful type checking. Agda also supports interactive theorem proving, which is not common in other functional programming languages.

How would you define a dependent type in Agda?

A dependent type in Agda is a type that depends on a value. This allows for more expressive types and more powerful type checking.

What are inductive and coinductive types in Agda?

Inductive types in Agda are types that are defined by a finite set of constructors. Coinductive types, on the other hand, are types that are defined by a potentially infinite set of observations.

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Does the candidate have a strong understanding of Agda?

Can the candidate demonstrate problem-solving skills?

Is the candidate able to communicate effectively?

Does the candidate have experience with other functional programming languages?

How would you use interactive theorem proving in Agda?

Interactive theorem proving in Agda is done by writing a program that represents the proof of the theorem. The Agda type checker then verifies the correctness of the proof.

Describe the difference between dependent types and regular types in Agda.

The main difference is that dependent types can depend on values, while regular types cannot. This allows for more expressive types and more powerful type checking in Agda.

How would you define a module in Agda?

A module in Agda is a way to group related definitions. It can contain types, functions, and other modules.

What are the benefits of using Agda for software verification?

The main benefit of using Agda for software verification is that it allows for formal verification of the correctness of the software. This is done by writing a program that represents the proof of the correctness of the software, which is then verified by the Agda type checker.

How would you use coinductive types in Agda?

Coinductive types in Agda are used to represent potentially infinite data structures. They are defined by a potentially infinite set of observations, and can be used to model things like streams and infinite lists.

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What does this simple Agda function do?

```
id : ? {A : Set} ? A ? A
id x = x
```

This is an identity function. It takes an argument of any type A and returns the same value.

What does the following Agda code do?

```
_+_ : N ? N ? N
zero + n = n
(suc m) + n = suc (m + n)
```

This code defines a function for addition on natural numbers. It is defined recursively: the sum of zero and any number n is n, and the sum of the successor of m (suc m) and n is the successor of the sum of m and n.

What does this Agda code do?

```
filter : ? {A : Set} ? (A ? Bool) ? List A ? List A
filter p [] = []
filter p (x ? xs) with p x
... | true = x ? filter p xs
... | false = filter p xs
```

This code defines a filter function for lists. The function takes a predicate and a list, and returns a new list containing only the elements of the original list that satisfy the predicate.

What does this Agda code do?

```
mutual
data _|_ (m n : N) : Set where
zero|n : zero | n
suc|n : ? {m'} ? suc m' | n ? m | suc n
_/_ : N ? N ? N
m / n with m |? n
... | yes p = zero
... | no ¬p with suc (m / n)
... | q = suc q
```

This code defines a division function for natural numbers. It uses a mutual block to define both a relation that represents divisibility and the division function itself. The division function uses the divisibility relation to decide whether to return zero or the successor of the result of a recursive call.

Describe the difference between inductive and coinductive types in Agda.

The main difference is that inductive types are defined by a finite set of constructors, while coinductive types are defined by a potentially infinite set of observations. This makes inductive types suitable for representing finite data structures, and coinductive types suitable for representing potentially infinite data structures.

How would you use dependent types for software verification in Agda?

Dependent types can be used for software verification in Agda by encoding the properties of the software in the types. The Agda type checker then verifies that the software satisfies these properties.

What are the challenges of using Agda for general-purpose programming?

One of the main challenges of using Agda for general-purpose programming is that it requires a different mindset compared to traditional programming languages. In particular, programming in Agda often involves proving properties about the program, which can be more time-consuming and difficult than traditional programming.

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