← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (1449 chars)
No Metric.eventually_nhdsWithin_iff in this Mathlib. Use the chain: `eventually_nhdsWithin_iff` then `Metric.eventually_nhds_iff_ball`. So:
```lean
rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball] at hWtrivEq
obtain ⟨ε, hε, hWtrivEq⟩ := hWtrivEq
-- hWtrivEq : ∀ y ∈ ball (−2n) ε, y ∈ {−2n}ᶜ → logDeriv ζ y − mult/(y−(−2n)) = Wtriv y
```
Then in Fexpr_eventually_neg_two_mul: `rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]`; refine ⟨min 1 ε, lt_min (by norm_num) hε, fun s hsball hsne => ...⟩ — wait the goal after the two rewrites: ∃ ε > 0, ∀ y ∈ ball a ε, y ∈ {a}ᶜ → Fexpr y = W y — so refine ⟨min 1 ε, lt_min (by norm_num) hε, fun s hsball hsne => ?_⟩ with hsball : s ∈ ball (−2n) (min 1 ε), hsne : s ∈ {−2n}ᶜ. Then hs0 : s ≠ −2n := Set.mem_compl_singleton_iff.mp hsne ✓ (mem_compl_singleton_iff is .rfl — hsne IS s ≠ −2n definitionally ✓ use directly? hsne : s ∈ ({−2n}:Set ℂ)ᶜ — defeq to s ≠ −2n ✓ (the lemma says .rfl) — so `have hs0 : s ≠ -2 * (n:ℂ) := hsne` ✓ defeq ascription).
Same for Fexpr_eventually_zero: I used `rw [Metric.eventually_nhdsWithin_iff]` — WRONG name. Fix: `rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]`; refine ⟨1/2, by norm_num, fun s hsball hsne⟩: hs.1/hs.2 split becomes two args: hsball : s ∈ ball 0 (1/2), hsne : s ∈ {0}ᶜ — hs0 : s ≠ 0 := hsne ✓ defeq.
Let me now rewrite both witness blocks with the corrected eventually handling. Let me redo the edits.
No Metric version — must chain `eventually_nhdsWithin_iff` + `Metric.eventually_nhds_iff_ball`. Fixing both witness blocks.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean",
"old_string": " \u00b7 -- punctured agreement on ball(0, 1/2)\n rw [Metric.eventually_nhdsWithin_iff]\n refine \u27e81 / 2, by norm_num, fun s hs => ?_\u27e9\n have hs0 : s \u2260 0 := hs.2\n have hsball : s \u2208 Metric.ball 0 (1 / 2) := hs.1",
"new_string": " \u00b7 -- punctured agreement on ball(0, 1/2)\n rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]\n refine \u27e81 / 2, by norm_num, fun s hsball hsne => ?_\u27e9\n have hs0 : s \u2260 0 := hsne"
}
raw content_json (database row)
[{"type":"thinking","thinking":"No Metric.eventually_nhdsWithin_iff in this Mathlib. Use the chain: `eventually_nhdsWithin_iff` then `Metric.eventually_nhds_iff_ball`. So:\n\n```lean\n rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball] at hWtrivEq\n obtain ⟨ε, hε, hWtrivEq⟩ := hWtrivEq\n -- hWtrivEq : ∀ y ∈ ball (−2n) ε, y ∈ {−2n}ᶜ → logDeriv ζ y − mult/(y−(−2n)) = Wtriv y\n```\n\nThen in Fexpr_eventually_neg_two_mul: `rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]`; refine ⟨min 1 ε, lt_min (by norm_num) hε, fun s hsball hsne => ...⟩ — wait the goal after the two rewrites: ∃ ε > 0, ∀ y ∈ ball a ε, y ∈ {a}ᶜ → Fexpr y = W y — so refine ⟨min 1 ε, lt_min (by norm_num) hε, fun s hsball hsne => ?_⟩ with hsball : s ∈ ball (−2n) (min 1 ε), hsne : s ∈ {−2n}ᶜ. Then hs0 : s ≠ −2n := Set.mem_compl_singleton_iff.mp hsne ✓ (mem_compl_singleton_iff is .rfl — hsne IS s ≠ −2n definitionally ✓ use directly? hsne : s ∈ ({−2n}:Set ℂ)ᶜ — defeq to s ≠ −2n ✓ (the lemma says .rfl) — so `have hs0 : s ≠ -2 * (n:ℂ) := hsne` ✓ defeq ascription).\n\nSame for Fexpr_eventually_zero: I used `rw [Metric.eventually_nhdsWithin_iff]` — WRONG name. Fix: `rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]`; refine ⟨1/2, by norm_num, fun s hsball hsne⟩: hs.1/hs.2 split becomes two args: hsball : s ∈ ball 0 (1/2), hsne : s ∈ {0}ᶜ — hs0 : s ≠ 0 := hsne ✓ defeq.\n\nLet me now rewrite both witness blocks with the corrected eventually handling. Let me redo the edits.","signature":""},{"type":"text","text":"No Metric version — must chain `eventually_nhdsWithin_iff` + `Metric.eventually_nhds_iff_ball`. Fixing both witness blocks."},{"type":"tool_use","id":"Edit_139","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":" · -- punctured agreement on ball(0, 1/2)\n rw [Metric.eventually_nhdsWithin_iff]\n refine ⟨1 / 2, by norm_num, fun s hs => ?_⟩\n have hs0 : s ≠ 0 := hs.2\n have hsball : s ∈ Metric.ball 0 (1 / 2) := hs.1","new_string":" · -- punctured agreement on ball(0, 1/2)\n rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff_ball]\n refine ⟨1 / 2, by norm_num, fun s hsball hsne => ?_⟩\n have hs0 : s ≠ 0 := hsne"},"extra_content":null}]