Coq Theorem Prover

My Journey to Understanding Proofs better by Learning the Model of a Theorem Prover: Coq

Plan of Attack

  1. Understand a proof with natural numbers with Coq.
  2. Understand a proof with vectors.
  3. Understand a proof with Qubits.
  4. Going further with software foundations?

More basics with Natural Numbers

As for Vectors

I'll be using "Mathematical Components":

The point: To be able to do exercises from "Quantum Mechanics: The Theoretical Minimum"

Exercise 1.1: a) Using the axioms for inner products, prove {⟨A| + ⟨B|} |C⟩ = ⟨A|C⟩ + ⟨B|C⟩.

First example of a vector proof. For this simple case I'll likely find many examples off the shelf.

b) Prove ⟨A|A⟩ is a real number.

I think this is defined by the complex conjugate of A_dagger dot A. Since the complex conjugate just negates the i component.

The dot product of two vectors a = [a1, a2, …, an] and b = [b1, b2, …, bn] is defined as:[1]

Exercise 1.2: Show that the inner product defined by Eq. 1.2 satisfies all the axioms of inner products.

Exercise 2.1: Prove that the vector |r in Eq. 2.5 is orthogonal to vector |l in Eq. 2.6.