exists_lt_card_fiber_of_nsmul_lt_card

English

There exist y ∈ β such that the fiber over y has size strictly between 0 and a given bound when the total weight or total count surpasses the bound.

Русский

Существует y ∈ β such that размер фибера над y строго между нулём и данным пределом, при условии превысания общего веса/числа пределов.

LaTeX

$$$\\exists y,\\; 0 < \\#\\{x \\in s \\mid f(x) = y\\} \\le b$$$

Lean4

/-- The strong pigeonhole principle for finitely many pigeons and pigeonholes. There is a pigeonhole
with at least as many pigeons as the ceiling of the average number of pigeons across all
pigeonholes. -/
theorem exists_lt_card_fiber_of_nsmul_lt_card (hb : card β • b < card α) : ∃ y : β, b < #{x | f x = y} :=
  let ⟨y, _, h⟩ := exists_lt_card_fiber_of_nsmul_lt_card_of_maps_to (fun _ _ => mem_univ _) hb
  ⟨y, h⟩

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