coverEntropyEntourage_empty — Mathlib

English

For any nonempty F, the entropy over the universal entourage is zero: coverEntropyInfEntourage T F univ = 0.

Русский

Для любого непустого F энтропия по универсуальной связи нулевая: coverEntropyInfEntourage T F univ = 0.

LaTeX

$$$coverEntropyInfEntourage(T,F,\univ)=0$$$

Lean4

@[simp]
theorem coverEntropyEntourage_empty {T : X → X} {U : Set (X × X)} : coverEntropyEntourage T ∅ U = ⊥ :=
  by
  simp only [coverEntropyEntourage, coverMincard_empty]
  rw [ENat.toENNReal_zero, ← Pi.zero_def, expGrowthSup_zero]

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