geomSum_ofColex_strictMono — Mathlib

English

There exists an initSeg for 𝒜 with card r equal to s.card and 𝒜 = initSeg s.

Русский

Существует initSeg для 𝒜 так, что card r равно card s и 𝒜 = initSeg s.

LaTeX

$$∀ α, IsInitSeg 𝒜 r → 𝒜 ≠ ∅ → ∃ s, |s| = r ∧ 𝒜 = initSeg s$$

Lean4

theorem geomSum_ofColex_strictMono (hn : 2 ≤ n) : StrictMono fun s ↦ ∑ k ∈ ofColex s, n ^ k :=
  by
  intro s t hst
  rw [lt_iff_exists_forall_lt] at hst
  obtain ⟨a, hat, has, ha⟩ := hst
  rw [← sum_sdiff_lt_sum_sdiff]
  exact (Nat.geomSum_lt hn <| by simpa).trans_le <| single_le_sum (fun _ _ ↦ by cutsat) <| mem_sdiff.2 ⟨hat, has⟩

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