AlexKontorovich/PrimeNumberTheoremAnd
Source indexedlemma · leanprover/lean4:v4.32.0
CH2.LadderParams.upperRectangle_subset_Rpos
PrimeNumberTheoremAnd.IEANTN.CH2.CH2 · PrimeNumberTheoremAnd/IEANTN/CH2/CH2.lean:341 to 356
Source documentation
The rectangle used in lemma_5_1_a is contained in Rpos.
Exact Lean statement
lemma LadderParams.upperRectangle_subset_Rpos (l : LadderParams) (n : ℕ) :
Rectangle ((l.σ n : ℂ) + (l.δ : ℂ) * Complex.I) (1 + (l.T : ℂ) * Complex.I) ⊆ l.RposComplete declaration
Lean source
Full Lean sourceLean 4
lemma LadderParams.upperRectangle_subset_Rpos (l : LadderParams) (n : ℕ) : Rectangle ((l.σ n : ℂ) + (l.δ : ℂ) * Complex.I) (1 + (l.T : ℂ) * Complex.I) ⊆ l.Rpos := by intro z hz have hδ_le_T : l.δ ≤ l.T := by have hδ := l.hδ.2 have hT : 0 < l.T := l.hT linarith rcases (mem_Rect (z := ((l.σ n : ℂ) + (l.δ : ℂ) * Complex.I)) (w := (1 + (l.T : ℂ) * Complex.I)) (p := z) (zRe_lt_wRe := by simpa using l.hσ n) (zIm_lt_wIm := by simpa using hδ_le_T)).1 hz with ⟨hz_re_left, hz_re_right, hz_im_bot, hz_im_top⟩ simp only [LadderParams.Rpos, Set.mem_setOf_eq, Set.mem_Icc] exact ⟨by simpa using hz_re_right, ⟨by simpa using hz_im_bot, by simpa using hz_im_top⟩⟩