English
Similarly for infimum-entourage bounds: the image under φ cannot increase the infimum-entourage compared to the source with the preimage relation.
Русский
Аналогично для инфимум-entourage: образ через φ не увеличивает инфимум-entourage по сравнению с исходной связью через предобраз.
LaTeX
$$$\\operatorname{coverEntropyInfEntourage}(T, \\phi(F), V) \\le \\operatorname{coverEntropyInfEntourage}(S, F, (\\phi \\times \\phi)^{-1}(V))$$$
Lean4
theorem coverEntropyInfEntourage_image_le (h : Semiconj φ S T) (F : Set X) (V : Set (Y × Y)) :
coverEntropyInfEntourage T (φ '' F) V ≤ coverEntropyInfEntourage S F ((map φ φ) ⁻¹' V) :=
expGrowthInf_monotone fun n ↦ ENat.toENNReal_mono (coverMincard_image_le h F V n)