It's a class with an array of integers in it with .length() == t - 1 and the same methods as Matrix.

In lean4, even without mathlib4, TCP/IP is way more code than a Rees algebra.

Math uses dense notation that is gigaoverloaded, and the disambiguating context was historically the leisure and proximity to have someone explain what the lexemes even mean.

lean4 is proving to be very revealing as an uncorruptible referee on a lot of things, including the relative difficulty of computer science and complex analysis.

  -- A Rees algebra over ℤ[t⁻¹] is this.
  -- That's it. That's the whole thing.
  structure ReesAlgebra where
    coeffs : Array Int   -- integers, indexed by grade
    -- grade k means the coefficient sits at t^k
    -- negative indices are the t⁻¹ part

  -- The "algebra" part: you can add them
  def ReesAlgebra.add (a b : ReesAlgebra) : ReesAlgebra :=
    ⟨a.coeffs.zipWith b.coeffs (· + ·)⟩

  -- And multiply them (convolution, same as polynomial multiplication)
  def ReesAlgebra.mul (a b : ReesAlgebra) : ReesAlgebra :=
    sorry -- it's Array.foldl over index pairs (i,j) summing into slot (i+j)
    -- exactly how you'd multiply polynomials in a job interview

  -- That's the entire mathematical content of
  -- "The Rees algebra is an algebra over Z[t^{-1}]"
  --
  -- Compare: a minimal TCP SYN handshake in Lean4 would be
  -- ~200 lines before you even get to retransmission.
  --
  -- The notation is the gate, not the math.

That's false. Z[n] in rings does not mean "an array of integers of length n", it means the subring generated by Z union with {n}, where n is an element of some other set. For example:

Z[i], the Gaussian integers, is the subring (of C) generated by Z union {i} where i is the imaginary unit in C, the complex numbers. The Gaussian integers correspond to the integer grid-points of the complex plane, if you want to visualize them.