exists_sum_fiber_le_of_maps_to_of_sum_le_nsmul

English

IsFractionalPolymorphismFor ω for a VCSP template Γ holds if every function in Γ admits ω.

Русский

IsFractionalPolymorphismFor ω для Γ означает, что каждая функция из Γ допускает ω.

LaTeX

$$$\\forall f ∈ Γ, f.snd.AdmitsFractional ω$$$

Lean4

/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
version: if the total weight of a finite set of pigeons is less than or equal to `n • b`, and they
are sorted into `n > 0` pigeonholes, then for some pigeonhole, the total weight of the pigeons in
this pigeonhole is less than or equal to `b`. -/
theorem exists_sum_fiber_le_of_maps_to_of_sum_le_nsmul (hf : ∀ a ∈ s, f a ∈ t) (ht : t.Nonempty)
    (hb : ∑ x ∈ s, w x ≤ #t • b) : ∃ y ∈ t, ∑ x ∈ s with f x = y, w x ≤ b :=
  exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum (M := Mᵒᵈ) hf ht hb

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