```
lemma "(A ? B) ? (B ? A)"
apply (rule impI)
apply (erule conjE)
apply (rule conjI)
apply assumption+
done
```

Can you briefly explain what ISABELLE is used for?

ISABELLE is used for formal verification. It's a proof assistant tool that helps in the development of formal proofs by providing a framework for creating and managing such proofs.

How would you install ISABELLE on a Linux machine?

You would first download the ISABELLE distribution, then extract the archive and set the 'ISABELLE_HOME' environment variable to the location of the extracted archive. After that, you would add 'ISABELLE_HOME/bin' to your PATH.

What are some of the main features of ISABELLE?

ISABELLE has a number of features including support for higher-order logic, a customizable proof engine, and an integrated development environment.

How would you specify a simple theorem in ISABELLE?

You would use the 'theorem' keyword followed by the name of the theorem and its specification. For example, 'theorem add_comm: "a + b = b + a"'.

What is a proof in the context of ISABELLE?

A proof in ISABELLE is a formal demonstration that given certain assumptions, a particular statement is true. It is constructed using a sequence of inference rules.

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Does the candidate have a strong understanding of ISABELLE?

Has the candidate demonstrated problem-solving skills?

Is the candidate able to communicate effectively?

Has the candidate shown experience with similar projects?

Describe the difference between ISABELLE/HOL and ISABELLE/ZF.

ISABELLE/HOL is based on higher-order logic, which allows for reasoning about functions and sets in a simple and intuitive way. ISABELLE/ZF, on the other hand, is based on Zermelo-Fraenkel set theory, which is more expressive but also more complex.

What is the role of the Isar language in ISABELLE?

Isar is a language for structured proofs in ISABELLE. It provides a high-level language for writing proofs that are easy to read and write.

Can you explain how tactics work in ISABELLE?

Tactics in ISABELLE are functions that take a goal and return a sequence of subgoals. They are used to guide the proof process by breaking down complex goals into simpler ones.

How would you prove a theorem in ISABELLE using the 'apply' method?

You would first state the theorem using the 'theorem' keyword, then use the 'apply' method to apply tactics to the goal. For example, 'apply (rule add_comm)' would apply the commutativity rule to the current goal.

What are some of the advantages of using a proof assistant like ISABELLE?

Proof assistants like ISABELLE provide a rigorous framework for developing formal proofs, which can increase confidence in the correctness of the results. They also provide tools for managing complex proofs and automating routine tasks.

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What does this simple Isabelle code do?

```
lemma "(A ? B) ? (B ? A)"
apply (rule impI)
apply (erule conjE)
apply (rule conjI)
apply assumption+
done
```

This Isabelle code is proving the lemma that if A and B are true, then B and A are true. It's essentially proving the commutativity of logical 'and' operator.

What does this Isabelle code snippet do?

```
datatype 'a list = Nil | Cons 'a 'a list
fun append :: "'a list ? 'a list ? 'a list" where
"append Nil ys = ys" |
"append (Cons x xs) ys = Cons x (append xs ys)"
```

This code defines a list datatype and an append function for the list. The append function takes two lists and appends the second list to the first one. The function uses pattern matching for two cases: when the first list is Nil, and when the first list is a Cons.

What will this Isabelle code do?

```
fun rev :: "'a list ? 'a list" where
"rev Nil = Nil" |
"rev (Cons x xs) = append (rev xs) (Cons x Nil)"
```

This Isabelle code defines a function rev that reverses a list. It uses pattern matching for two cases: when the list is Nil, and when the list is a Cons. In the latter case, it recursively calls rev on the tail of the list, and appends the head to the result.

What does this Isabelle code snippet do?

```
datatype 'a tree = Leaf | Node "'a tree" 'a "'a tree"
fun mirror :: "'a tree ? 'a tree" where
"mirror Leaf = Leaf" |
"mirror (Node l x r) = Node (mirror r) x (mirror l)"
```

This code defines a binary tree datatype and a mirror function for the tree. The mirror function takes a tree and returns a new tree that is the mirror image of the original one. It uses pattern matching for two cases: when the tree is a Leaf, and when the tree is a Node.

How would you use ISABELLE to verify a complex software system?

You would first formalize the specifications of the system in ISABELLE, then construct a proof that the system meets these specifications. This might involve proving properties about the system's behavior, or showing that certain error conditions cannot occur.

Describe the difference between a 'proof' and a 'lemma' in ISABELLE.

In ISABELLE, a 'proof' is a formal demonstration that a particular statement is true, while a 'lemma' is a subsidiary or intermediate theorem in a proof. Lemmas are used to break down complex proofs into smaller, more manageable parts.

How would you debug a proof in ISABELLE that isn't going as expected?

ISABELLE provides several tools for debugging proofs, including the 'proof' command, which shows the current goal and the assumptions, and the 'oops' command, which discards the current proof attempt. You can also use the 'apply' command with different tactics to see how they affect the goal.

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