[{"type":"thinking","thinking":"`AnalyticAt.comp_of_eq h1 h2 (by show ... = ...; ring)` — the pattern: `AnalyticAt.comp_of_eq (hf : AnalyticAt ℂ f c') (hg : AnalyticAt ℂ g z) (h : g z = c')` — the third arg is the point-equality proof. Let me check its exact signature... from usage: h1 : AnalyticAt at the target, h2 : the inner, then `by show ...; ring` proves the inner-value = outer-point. So:\n\n```lean\n  exact ((analyticAt_riemannZeta_of_ne_one hne).comp_of_eq haff rfl).div analyticAt_const (...)\n```\nThe point-equality: (fun w => discCenter t + 5/2*w) w = discCenter t + 5/2*w — rfl ✓ (beta). So `AnalyticAt.comp_of_eq hζ haff rfl`.\n\nLet me check the signature quickly — `AnalyticAt.comp_of_eq` in Mathlib: probably `AnalyticAt.comp_of_eq (hf : AnalyticAt 𝕜 f y) (hg : AnalyticAt 𝕜 g x) (h : g x = y) : AnalyticAt 𝕜 (f ∘ g) x`? Then the result is about f ∘ g — need to unfold ∘: the goal is AnalyticAt (fun w => riemannZeta (discCenter t + 5/2*w)) w — and (riemannZeta ∘ fun w => discCenter t + 5/2*w) — defeq ✓ or via `Function.comp_def`... Then the .div: (…).div analyticAt_const (proof ζ(discCenter t) ≠ 0): the div produces AnalyticAt ((riemannZeta ∘ affine) / const) w — and scaledZeta t w = ζ(...)/ζ(...) — the function equality: scaledZeta t w = (ζ∘aff) w / ζ(discCenter t) ✓ defeq-ish. The target of scaledZeta_analytic: AnalyticOnNhd ℂ (scaledZeta t) (closedBall 0 1) — intro w hw — AnalyticAt (scaledZeta t) w — scaledZeta t = fun w => ζ(...)/ζ(...) — the final exact needs the forms to match: exact (the div result) — the div result's function: (riemannZeta ∘ fun w => discCenter t + 5/2*w) / fun _ => riemannZeta (discCenter t)?? — AnalyticAt.div hf hg : AnalyticAt (f / g) — the Pi div — vs scaledZeta's fun w => ζ(aff w)/ζ(s₀) — defeq via Pi.div_apply ✓ exact should handle.\n\nLet me fix these two spots.","signature":""},{"type":"text","text":"`AnalyticAt.comp_of_eq` is the pattern (with the point-equality proof). Fixing both spots."},{"type":"tool_use","id":"Edit_358","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","old_string":"      have h2 : |t| ≤ |t + (5 / 2) * w.im| + |(5 / 2) * w.im| := by\n        calc |t| = |(t + (5 / 2) * w.im) - ((5 / 2) * w.im)| := by ring_nf\n          _ ≤ |t + (5 / 2) * w.im| + |(5 / 2) * w.im| := abs_sub_le _ _","new_string":"      have h2 : |t| ≤ |t + (5 / 2) * w.im| + |(5 / 2) * w.im| := by\n        have h3 := abs_add_le (t + (5 / 2) * w.im) (-((5 / 2) * w.im))\n        rw [abs_neg, add_neg_cancel_right] at h3\n        exact h3"},"extra_content":null}]