le_collapse_of_mem — Mathlib

English

If s is not containing a, and f is nonnegative, and t equals s, then f(t) ≤ collapse 𝒜 a f s provided t ∈ 𝒜.

Русский

Если s не содержит a и f неотрицательно, и t = s, то f(t) ≤ collapse 𝒜 a f s при условии t ∈ 𝒜.

LaTeX

$$f(t) ≤ collapse(𝒜,a,f,s) when t=s ∈ 𝒜 and a ∉ s.$$

Lean4

theorem le_collapse_of_mem (ha : a ∉ s) (hf : 0 ≤ f) (hts : t = s) (ht : t ∈ 𝒜) : f t ≤ collapse 𝒜 a f s :=
  by
  subst hts
  rw [collapse_eq ha, if_pos ht]
  split_ifs
  · exact le_add_of_nonneg_right <| hf _
  · rw [add_zero]

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