← Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)
thinking (2834 chars)
Fixing remaining issues:
1. Line 61 (hbd): the hδ application — `hδ (by rwa [dist_eq_norm, sub_zero] exact hz.1)` — the structure: hδ needs `z ∈ {1}ᶜ` and `dist z 1 < δ` — my by-block produced the wrong thing. hz.1 : z ∈ Metric.ball 1 2 — that's `dist z 1 < 2`, not what hδ wants. I need `dist z 1 < δ` — I don't have it! The ball(1,2) might be bigger than δ. Restructure: use the punctured ball around 1 of radius min δ 2: s := Metric.ball 1 (min δ 2). Simpler: extract the δ first, then use ball 1 (δ/2) as the neighborhood s:
```
have hT := riemannZeta_residue_one
rw [Metric.tendsto_nhdsWithin_nhds] at hT
obtain ⟨δ, hδ0, hδ⟩ := hT 1 one_pos
-- work on s := Metric.ball 1 (δ/2)
have hdiff : DifferentiableOn ℂ (fun w => (w - 1) * riemannZeta w) (Metric.ball 1 (δ/2) \ {1}) := ...
have hbd : BddAbove ((fun z => ‖(z - 1) * riemannZeta z‖) '' (Metric.ball 1 (δ/2) \ {1})) := by
refine ⟨2, ?_⟩
rintro _ ⟨z, hz, rfl⟩
have hz1 : z ≠ 1 := hz.2
have hdist : dist z 1 < δ := by
have := hz.1
rw [Metric.mem_ball] at this
linarith [this, hδ0]
have h1 := hδ (Set.mem_compl_singleton_iff.mpr hz1) hdist
rw [dist_eq_norm] at h1
have h2 : ‖(z - 1) * riemannZeta z‖ ≤ ‖(z - 1) * riemannZeta z - 1‖ + ‖(1 : ℂ)‖ :=
norm_sub_le ((z - 1) * riemannZeta z) 1
rw [norm_one] at h2
linarith [h1]
have h := differentiableOn_update_limUnder_of_bddAbove
(Metric.ball_mem_nhds 1 (by positivity : (0 : ℝ) < δ / 2)) hdiff hbd
```
Note: hδ's signature from the display: `hδ : ∀ ⦃x : ℂ⦄, x ∈ {1}ᶜ → dist x 1 < δ → dist ((x - 1) * riemannZeta x) 1 < 1` — needs x ∈ {1}ᶜ (the nhdsWithin set) ✓ Set.mem_compl_singleton_iff.mpr hz1.
And hdiff on ball 1 (δ/2) — same proof.
2. Line 157: `NormedAddCommGroup ℕ` + `HSub ℂ ℂ ℕ` — in A_iter, which line? 157 is the `rw [show (fun w : ℂ => ...) = ...]` — the show's funext proof: `funext w; simp [Pi.sub_apply]` — hmm, no. Let me count lines: A_iter starts ~line 150. Line 157: likely `rw [iteratedDeriv_pow]` in h1, or the show line. The `HSub ℂ ℂ ℕ` — a subtraction ℂ − ℂ with expected result ℕ — bizarre. OH WAIT — I bet it's in h1: `iteratedDeriv i (fun w : ℂ => w ^ n) 1 = n.descFactorial i` — the RHS: `n.descFactorial i` where the expected type is ℂ — cast ✓. Hmm no. Let me look at lines 155–165 directly.
3. Line 163: the `rw [show ... , iteratedDeriv_sub hf hg]` — pattern `fun w => w ^ n - (w - 1) ^ n` not found — the `show`-based rw — the LHS of my show is `(fun w : ℂ => w ^ n - (w - 1) ^ n)` — the goal has `fun w => w ^ n - (w - 1) ^ n` — should match... unless elaboration differs. Let me look at the actual current state of the A_iter block in the file and rewrite it wholesale instead of patching. Also hmul's rwa has the same issue at 204.
Let me read the current A_iter block and rewrite it + the hmul block carefully.
Let me look at the current state of the broken blocks and rewrite them directly.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
"offset": 148,
"limit": 75
}
result 2026-08-21 21:11:56Z → 2026-08-22 17:27:08Z
{
"type": "text",
"file": {
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean",
"content": "\n/-- The unit-interval IBP: over each integer interval [k, k+1] (k \u2265 1),\n\u222b B\u0303\u2081\u00b7x^{\u2212s\u22121} = (s+1)\u00b7\u222b B\u0303\u2082\u00b7x^{\u2212s\u22122} (boundary terms vanish: B\u0303\u2082 = 0 at\nthe integers). -/\ntheorem intervalIntegral_b1_cpow (k : \u2115) (hk : 1 \u2264 k) (s : \u2102) (hs : s \u2260 -1) :\n \u222b x in (k : \u211d)..(k + 1 : \u211d), (B1 x : \u2102) * (x : \u2102) ^ (-s - 1)\n = (s + 1) * \u222b x in (k : \u211d)..(k + 1 : \u211d), (b2Fun x : \u2102) * (x : \u2102) ^ (-s - 2) := by\n have hk0 : (0:\u211d) < (k : \u211d) := by exact_mod_cast hk\n have hu : ContinuousOn (fun x : \u211d => (b2Fun x : \u2102)) (Set.Icc (k : \u211d) (k + 1 : \u211d)) :=\n (continuous_ofReal.comp continuous_b2Fun).continuousOn\n have hv : ContinuousOn (fun x : \u211d => (x : \u2102) ^ (-s - 1)) (Set.Icc (k : \u211d) (k + 1 : \u211d)) := by\n intro x hx\n have hx0 : x \u2260 0 := by\n have : (k : \u211d) \u2264 x := hx.1\n linarith",
"numLines": 15,
"startLine": 98,
"totalLines": 146
}
}
raw content_json (database row)
[{"type":"thinking","thinking":"Fixing remaining issues:\n\n1. Line 61 (hbd): the hδ application — `hδ (by rwa [dist_eq_norm, sub_zero] exact hz.1)` — the structure: hδ needs `z ∈ {1}ᶜ` and `dist z 1 < δ` — my by-block produced the wrong thing. hz.1 : z ∈ Metric.ball 1 2 — that's `dist z 1 < 2`, not what hδ wants. I need `dist z 1 < δ` — I don't have it! The ball(1,2) might be bigger than δ. Restructure: use the punctured ball around 1 of radius min δ 2: s := Metric.ball 1 (min δ 2). Simpler: extract the δ first, then use ball 1 (δ/2) as the neighborhood s:\n```\n have hT := riemannZeta_residue_one\n rw [Metric.tendsto_nhdsWithin_nhds] at hT\n obtain ⟨δ, hδ0, hδ⟩ := hT 1 one_pos\n -- work on s := Metric.ball 1 (δ/2)\n have hdiff : DifferentiableOn ℂ (fun w => (w - 1) * riemannZeta w) (Metric.ball 1 (δ/2) \\ {1}) := ...\n have hbd : BddAbove ((fun z => ‖(z - 1) * riemannZeta z‖) '' (Metric.ball 1 (δ/2) \\ {1})) := by\n refine ⟨2, ?_⟩\n rintro _ ⟨z, hz, rfl⟩\n have hz1 : z ≠ 1 := hz.2\n have hdist : dist z 1 < δ := by\n have := hz.1\n rw [Metric.mem_ball] at this\n linarith [this, hδ0]\n have h1 := hδ (Set.mem_compl_singleton_iff.mpr hz1) hdist\n rw [dist_eq_norm] at h1\n have h2 : ‖(z - 1) * riemannZeta z‖ ≤ ‖(z - 1) * riemannZeta z - 1‖ + ‖(1 : ℂ)‖ :=\n norm_sub_le ((z - 1) * riemannZeta z) 1\n rw [norm_one] at h2\n linarith [h1]\n have h := differentiableOn_update_limUnder_of_bddAbove\n (Metric.ball_mem_nhds 1 (by positivity : (0 : ℝ) < δ / 2)) hdiff hbd\n```\nNote: hδ's signature from the display: `hδ : ∀ ⦃x : ℂ⦄, x ∈ {1}ᶜ → dist x 1 < δ → dist ((x - 1) * riemannZeta x) 1 < 1` — needs x ∈ {1}ᶜ (the nhdsWithin set) ✓ Set.mem_compl_singleton_iff.mpr hz1.\n\nAnd hdiff on ball 1 (δ/2) — same proof.\n\n2. Line 157: `NormedAddCommGroup ℕ` + `HSub ℂ ℂ ℕ` — in A_iter, which line? 157 is the `rw [show (fun w : ℂ => ...) = ...]` — the show's funext proof: `funext w; simp [Pi.sub_apply]` — hmm, no. Let me count lines: A_iter starts ~line 150. Line 157: likely `rw [iteratedDeriv_pow]` in h1, or the show line. The `HSub ℂ ℂ ℕ` — a subtraction ℂ − ℂ with expected result ℕ — bizarre. OH WAIT — I bet it's in h1: `iteratedDeriv i (fun w : ℂ => w ^ n) 1 = n.descFactorial i` — the RHS: `n.descFactorial i` where the expected type is ℂ — cast ✓. Hmm no. Let me look at lines 155–165 directly.\n\n3. Line 163: the `rw [show ... , iteratedDeriv_sub hf hg]` — pattern `fun w => w ^ n - (w - 1) ^ n` not found — the `show`-based rw — the LHS of my show is `(fun w : ℂ => w ^ n - (w - 1) ^ n)` — the goal has `fun w => w ^ n - (w - 1) ^ n` — should match... unless elaboration differs. Let me look at the actual current state of the A_iter block in the file and rewrite it wholesale instead of patching. Also hmul's rwa has the same issue at 204.\n\nLet me read the current A_iter block and rewrite it + the hmul block carefully.","signature":""},{"type":"text","text":"Let me look at the current state of the broken blocks and rewrite them directly."},{"type":"tool_use","id":"Read_203","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean","offset":148,"limit":75},"extra_content":null}]