← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (1198 chars)
`ContinuousAt.eventually_ne (hg1 : ContinuousAt g x) (hg2 : g x ≠ y)` — so it's the general form with {y} implicit! The call `.eventually_ne (by rw [riemannZeta_zero]; norm_num)` leaves y as a metavariable — the by-block's goal is `riemannZeta 0 ≠ ?y` — rw [riemannZeta_zero] gives `¬(−1/2 = ?y)` and norm_num can't close with the metavariable. In LiResidue it worked because the RESULT type was pinned by the ascribed statement `zetaA_ne_zero_near_one : ∀ᶠ w in 𝓝 1, zetaA w ≠ 0` — the ascribed type fixed y := 0. Here the eventually_ne result feeds `Metric.eventually_nhds_iff_ball.mp` directly — no ascription — so y is free. Fix: pin y := 0 by ascribing the result:
```lean
have hz0 : ∀ᶠ w in 𝓝 (0 : ℂ), riemannZeta w ≠ 0 :=
(analyticAt_riemannZeta (by norm_num : (0 : ℂ) ≠ 1)).continuousAt.eventually_ne
(by rw [riemannZeta_zero]; norm_num)
obtain ⟨ε₁, hε₁, hball₁⟩ := Metric.eventually_nhds_iff_ball.mp hz0
```
The ascription pins y = 0 ✓ then the by-block's goal is riemannZeta 0 ≠ 0 ✓ norm_num closes after rw ✓.
Same pattern was used in §3's NF witness: `(hh.continuousAt.eventually_ne hh0)` — hh0 : h ρ₀ ≠ 0 ✓ pinned by hh0 ✓ fine.
Now fix the three sites.
`eventually_ne` has `{y}` implicit — the result type must be pinned by ascription. Fixing that plus the `simp; ring` and the EventuallyEq unfold-shows.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean",
"old_string": " -- s = 0: the \u03b6-continuity ball \u2229 ball(0, 1/2)\n obtain \u27e8\u03b5\u2081, h\u03b5\u2081, hball\u2081\u27e9 := Metric.eventually_nhds_iff_ball.mp\n ((analyticAt_riemannZeta (by norm_num : (0 : \u2102) \u2260 1)).continuousAt.eventually_ne\n (by rw [riemannZeta_zero]; norm_num))",
"new_string": " -- s = 0: the \u03b6-continuity ball \u2229 ball(0, 1/2)\n have hz0 : \u2200\u1da0 w in \ud835\udcdd (0 : \u2102), riemannZeta w \u2260 0 :=\n (analyticAt_riemannZeta (by norm_num : (0 : \u2102) \u2260 1)).continuousAt.eventually_ne\n (by rw [riemannZeta_zero]; norm_num)\n obtain \u27e8\u03b5\u2081, h\u03b5\u2081, hball\u2081\u27e9 := Metric.eventually_nhds_iff_ball.mp hz0"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`ContinuousAt.eventually_ne (hg1 : ContinuousAt g x) (hg2 : g x ≠ y)` — so it's the general form with {y} implicit! The call `.eventually_ne (by rw [riemannZeta_zero]; norm_num)` leaves y as a metavariable — the by-block's goal is `riemannZeta 0 ≠ ?y` — rw [riemannZeta_zero] gives `¬(−1/2 = ?y)` and norm_num can't close with the metavariable. In LiResidue it worked because the RESULT type was pinned by the ascribed statement `zetaA_ne_zero_near_one : ∀ᶠ w in 𝓝 1, zetaA w ≠ 0` — the ascribed type fixed y := 0. Here the eventually_ne result feeds `Metric.eventually_nhds_iff_ball.mp` directly — no ascription — so y is free. Fix: pin y := 0 by ascribing the result:\n```lean\n have hz0 : ∀ᶠ w in 𝓝 (0 : ℂ), riemannZeta w ≠ 0 :=\n (analyticAt_riemannZeta (by norm_num : (0 : ℂ) ≠ 1)).continuousAt.eventually_ne\n (by rw [riemannZeta_zero]; norm_num)\n obtain ⟨ε₁, hε₁, hball₁⟩ := Metric.eventually_nhds_iff_ball.mp hz0\n```\nThe ascription pins y = 0 ✓ then the by-block's goal is riemannZeta 0 ≠ 0 ✓ norm_num closes after rw ✓.\n\nSame pattern was used in §3's NF witness: `(hh.continuousAt.eventually_ne hh0)` — hh0 : h ρ₀ ≠ 0 ✓ pinned by hh0 ✓ fine.\n\nNow fix the three sites.","signature":""},{"type":"text","text":"`eventually_ne` has `{y}` implicit — the result type must be pinned by ascription. Fixing that plus the `simp; ring` and the EventuallyEq unfold-shows."},{"type":"tool_use","id":"Edit_209","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":" -- s = 0: the ζ-continuity ball ∩ ball(0, 1/2)\n obtain ⟨ε₁, hε₁, hball₁⟩ := Metric.eventually_nhds_iff_ball.mp\n ((analyticAt_riemannZeta (by norm_num : (0 : ℂ) ≠ 1)).continuousAt.eventually_ne\n (by rw [riemannZeta_zero]; norm_num))","new_string":" -- s = 0: the ζ-continuity ball ∩ ball(0, 1/2)\n have hz0 : ∀ᶠ w in 𝓝 (0 : ℂ), riemannZeta w ≠ 0 :=\n (analyticAt_riemannZeta (by norm_num : (0 : ℂ) ≠ 1)).continuousAt.eventually_ne\n (by rw [riemannZeta_zero]; norm_num)\n obtain ⟨ε₁, hε₁, hball₁⟩ := Metric.eventually_nhds_iff_ball.mp hz0"},"extra_content":null}]