import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct

namespace OAI

section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical

namespace CriticalPercolation.BernoulliProduct
variable {A E : Type*}

lemma law_map_preimage (q : E → unitInterval) (i : A → E) (hi : Injective i) :
    (law q).map (fun ω : Set E => i ⁻¹' ω)=law (q∘i) := by
  have hind : iIndepFun (fun a (ω : Set E) => i a∈ω) (law q) :=
    iIndepFun.precomp hi (independent_coordinates q)
  have hh := hind.map_fun_eq_infinitePi_map (fun a => measurable_set_mem (i a))
  change (law q).map (fun ω : Set E => i ⁻¹' ω)=_ at hh
  rw [hh]
  rw [law_eq_infinitePi (q∘i)]
  apply congrArg Measure.infinitePi
  funext a
  rw [law_eq_infinitePi]
  exact Measure.infinitePi_map_eval _ (i a)

end CriticalPercolation.BernoulliProduct
end
end
end

end OAI