\begin{definition}[$\eta$]\label{eta-def-multi}\lean{multiRefPackage}\leanok
We set $\eta := \frac{1}{32m^3}$.
\end{definition}
/-- A structure that packages all the fixed information in the main argument. See https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Problem.20when.20instances.20are.20inside.20a.20structure for more discussion of the design choices here. -/
structure multiRefPackage (G Ω₀ : Type*) [MeasureableFinGroup G] [MeasureSpace Ω₀] where
  /-- The torsion index of the group we are considering. -/
  (m : ℕ)
  (hm : m ≥ 2)
  (htorsion : ∀ x : G, m • x = 0)
  (hprob : IsProbabilityMeasure (ℙ : Measure Ω₀))
  /-- The random variable -/
  (X₀ : Ω₀ → G)
  (hmeas : Measurable X₀)
  /-- A small constant. The argument will only work for suitably small `η`. -/
  (η : ℝ)
  (hη : 0 < η)
  (hη': η ≤ 1)

/-- If $(X_i)_{1 \leq i \leq m}$ is a tuple, we define its $\tau$-functional
$$ \tau[ (X_i)_{1 \leq i \leq m}] := D[(X_i)_{1 \leq i \leq m}] + \eta \sum_{i=1}^m d[X_i; X^0].$$
-/
noncomputable def multiTau {G Ω₀ : Type*} [MeasureableFinGroup G] [MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type*) (hΩ : ∀ i, MeasureSpace (Ω i)) (X : ∀ i, Ω i → G) : ℝ :=
  D[X; hΩ] + p.η * ∑ i, d[ X i # p.X₀ ]

-- I can't figure out how to make a τ notation due to the dependent types in the arguments. But perhaps we don't need one. Also it may be better to define multiTau in terms of probability measures on G, rather than G-valued random variables, again to avoid dependent type issues.

-- had to force objects to lie in a fixed universe `u` here to avoid a compilation error