17. Formal Verification and Proof Assistants

Formal Verification and Proof Assistants

Formal verification is the process of proving the correctness of software or hardware systems using mathematical methods.

It involves specifying the desired properties of a system and then proving that these properties hold under all possible circumstances.

Proof assistants are software tools designed to assist in the development and verification of mathematical proofs.

They provide a formal language for expressing mathematical statements and proofs, along with mechanisms for checking the validity of these proofs.

Key features of proof assistants include:

• Interactive proof development: Users can interactively construct proofs step by step, guided by the system's feedback.
• Formal specification languages: Proof assistants support formal languages for specifying properties and requirements of systems.
• Automated reasoning: They often include libraries of tactics or combinators to automate common proof tasks.
• Proof reuse and sharing: Proofs developed in a proof assistant can be reused, shared, and checked by others.

Coq is one of the most widely used proof assistants, known for its expressive type system and powerful proof automation features.

Let's demonstrate a simple proof in Coq:

Theorem plus_comm : forall n m : nat, n + m = m + n.
Proof.
  intros n m.
  induction n as [| n' IHn'].
  - simpl. rewrite <- plus_n_O. reflexivity.
  - simpl. rewrite <- IHn'. rewrite <- plus_n_Sm. reflexivity.
Qed.
  

In this lab activity, we prove the commutativity of addition over natural numbers using Coq.

We start by introducing the universal quantifiers using the `intros` tactic.

We then perform induction on the first argument `n` using the `induction` tactic.

In the base case, we simplify the expression and apply the `plus_n_O` lemma.

In the inductive case, we simplify the expression, use the induction hypothesis `IHn'`, and apply the `plus_n_Sm` lemma.