← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (2128 chars)
The hsplit0 residual: after the rw chain, the goal is
`∑ k ∈ range (m−j₀−1), ‖(w + ↑(j₀ + (k+1)))⁻¹‖ + ‖(w + ↑j₀)‖⁻¹ = ‖(w + ↑j₀)‖⁻¹ + ∑ i ∈ range (m−j₀−1), ‖(w + ↑(j₀+(i+1)))⁻¹‖`
The LHS has the sum FIRST then the term (sum_range_succ' produces `f 0 + ∑ ...`?? no — sum_range_succ' : ∑ i ∈ range (n+1), f i = f 0 + ∑ i ∈ range n, f (i+1) — so the first term is FIRST: `f 0 + ∑...`. But the residual shows `∑ ... + ‖(w + ↑j₀)‖⁻¹` on the LHS — sum FIRST. So sum_range_succ' in this Mathlib might be `∑ i ∈ range (n+1), f i = (∑ i ∈ range n, f (i+1)) + f 0`?? — the displayed LHS: `∑ k, ... + ‖(w + ↑j₀)‖⁻¹` — sum then term. So sum_range_succ' here : ∑_{range (n+1)} f = (∑_{range n} f (i+1)) + f 0? Hmm — that contradicts my memory but the display is clear: after rw, the LHS is `∑ (shifted) + f 0`. So sum_range_succ' : ∑ i ∈ range (n+1), f i = (∑ i ∈ range n, f (i+1)) + f 0?? Hmm — wait maybe it's sum_range_succ' : ∑ i ∈ range (n+1), f i = ∑ i ∈ range n, f (i+1) + f 0 — with the sum on the left of the +... whatever — the residual LHS has the sum first. So my hsplit0's stated form (term + sum) mismatches (sum + term). Fix: state hsplit0 with the sum first:
```
have hsplit0 : ∑ j ∈ Finset.Ico j₀ m, ‖(w + (j : ℂ))⁻¹‖
= (∑ i ∈ Finset.range (m - j₀ - 1), ‖(w + ((j₀ + (i + 1) : ℕ)) : ℂ)⁻¹‖)
+ ‖(w + (j₀ : ℂ))⁻¹‖ := by
```
and the downstream add_le_add hfirst_r hrest' gives 1/c + (1+log) form — the components swapped: (sum) + term ≤ (1+log) + 1/c = need add_comm at the end — the final combine: my hright's conclusion is 1/c + (1 + log m): from (sum ≤ 1+log) + (term ≤ 1/c): (sum) + term ≤ (1+log m) + 1/c = 1/c + (1+log m) by ring/comm. Let me write:
```
rw [hsplit0]
-- goal: (∑ ...) + ‖(w+↑j₀)‖⁻¹ ≤ 1/c + (1 + log m)
calc (∑ i ∈ Finset.range (m - j₀ - 1), ‖(w + ((j₀ + (i + 1) : ℕ)) : ℂ)⁻¹‖) + ‖(w + (j₀ : ℂ))⁻¹‖
≤ (1 + Real.log m) + 1 / c := add_le_add hrest' hfirst_r
_ = 1 / c + (1 + Real.log m) := by ring
```
with hrest' defined before (the sum bound). Reorder: hrest' first (it's independent), then hsplit0, then the calc. Let me rewrite hright's block accordingly.
`sum_range_succ'` here appends the term on the right. Adjusting hright's structure.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean",
"offset": 617,
"limit": 28
}
result 2026-08-23 11:06:01Z → 2026-08-23 11:06:01Z
{
"type": "text",
"file": {
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean",
"content": " rw [Finset.sum_range_succ']\n rw [add_zero]\n rw [hsplit0]\n have hrest' : \u2211 i \u2208 Finset.range (m - j\u2080 - 1), \u2016(w + ((j\u2080 + (i + 1) : \u2115)) : \u2102)\u207b\u00b9\u2016\n \u2264 1 + Real.log m := by\n calc \u2211 i \u2208 Finset.range (m - j\u2080 - 1), \u2016(w + ((j\u2080 + (i + 1) : \u2115)) : \u2102)\u207b\u00b9\u2016\n \u2264 \u2211 i \u2208 Finset.range (m - j\u2080 - 1), ((i : \u211d) + 1)\u207b\u00b9 := by\n apply Finset.sum_le_sum\n intro i _\n have h1 := recip_norm_le_right hdist hj\u2080 (j := j\u2080 + (i + 1)) (by omega)\n have h2 : (j\u2080 + (i + 1)) - j\u2080 = i + 1 := by omega\n rw [h2] at h1\n have h3 : (((i + 1 : \u2115)) : \u211d)\u207b\u00b9 = ((i : \u211d) + 1)\u207b\u00b9 := by rw [Nat.cast_add_one]\n rwa [h3] at h1\n _ \u2264 1 + Real.log (\u2191(m - j\u2080 - 1) : \u211d) := hHarm _\n _ \u2264 1 + Real.log m := hlogmono0 (show m - j\u2080 - 1 \u2264 m by omega) hm1\n exact add_le_add hfirst_r hrest'\n -- LEFT side\n have hleft : \u2211 j \u2208 Finset.range j\u2080, \u2016(w + (j : \u2102))\u207b\u00b9\u2016 \u2264 1 / c + (1 + Real.log m) := by\n by_cases hj\u20800 : j\u2080 = 0\n \u00b7 rw [hj\u20800]\n simp only [Finset.range_zero, Finset.sum_empty]\n have h1 : (0 : \u211d) \u2264 1 / c := by positivity\n have h2 : (0 : \u211d) \u2264 Real.log m := Real.log_nonneg (by exact_mod_cast hm1)\n linarith [h1, h2]\n \u00b7 have hj\u20801 : 1 \u2264 j\u2080 := Nat.one_le_iff_ne_zero.mpr hj\u20800\n have hdecomp : \u2211 j \u2208 Finset.range j\u2080, \u2016(w + (j : \u2102))\u207b\u00b9\u2016\n = \u2211 j \u2208 Finset.range (j\u2080 - 1), \u2016(w + (j : \u2102))\u207b\u00b9\u2016",
"numLines": 28,
"startLine": 617,
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raw content_json (database row)
[{"type":"thinking","thinking":"The hsplit0 residual: after the rw chain, the goal is\n`∑ k ∈ range (m−j₀−1), ‖(w + ↑(j₀ + (k+1)))⁻¹‖ + ‖(w + ↑j₀)‖⁻¹ = ‖(w + ↑j₀)‖⁻¹ + ∑ i ∈ range (m−j₀−1), ‖(w + ↑(j₀+(i+1)))⁻¹‖`\n\nThe LHS has the sum FIRST then the term (sum_range_succ' produces `f 0 + ∑ ...`?? no — sum_range_succ' : ∑ i ∈ range (n+1), f i = f 0 + ∑ i ∈ range n, f (i+1) — so the first term is FIRST: `f 0 + ∑...`. But the residual shows `∑ ... + ‖(w + ↑j₀)‖⁻¹` on the LHS — sum FIRST. So sum_range_succ' in this Mathlib might be `∑ i ∈ range (n+1), f i = (∑ i ∈ range n, f (i+1)) + f 0`?? — the displayed LHS: `∑ k, ... + ‖(w + ↑j₀)‖⁻¹` — sum then term. So sum_range_succ' here : ∑_{range (n+1)} f = (∑_{range n} f (i+1)) + f 0? Hmm — that contradicts my memory but the display is clear: after rw, the LHS is `∑ (shifted) + f 0`. So sum_range_succ' : ∑ i ∈ range (n+1), f i = (∑ i ∈ range n, f (i+1)) + f 0?? Hmm — wait maybe it's sum_range_succ' : ∑ i ∈ range (n+1), f i = ∑ i ∈ range n, f (i+1) + f 0 — with the sum on the left of the +... whatever — the residual LHS has the sum first. So my hsplit0's stated form (term + sum) mismatches (sum + term). Fix: state hsplit0 with the sum first:\n```\n have hsplit0 : ∑ j ∈ Finset.Ico j₀ m, ‖(w + (j : ℂ))⁻¹‖\n = (∑ i ∈ Finset.range (m - j₀ - 1), ‖(w + ((j₀ + (i + 1) : ℕ)) : ℂ)⁻¹‖)\n + ‖(w + (j₀ : ℂ))⁻¹‖ := by\n```\nand the downstream add_le_add hfirst_r hrest' gives 1/c + (1+log) form — the components swapped: (sum) + term ≤ (1+log) + 1/c = need add_comm at the end — the final combine: my hright's conclusion is 1/c + (1 + log m): from (sum ≤ 1+log) + (term ≤ 1/c): (sum) + term ≤ (1+log m) + 1/c = 1/c + (1+log m) by ring/comm. Let me write:\n```\n rw [hsplit0]\n -- goal: (∑ ...) + ‖(w+↑j₀)‖⁻¹ ≤ 1/c + (1 + log m)\n calc (∑ i ∈ Finset.range (m - j₀ - 1), ‖(w + ((j₀ + (i + 1) : ℕ)) : ℂ)⁻¹‖) + ‖(w + (j₀ : ℂ))⁻¹‖\n ≤ (1 + Real.log m) + 1 / c := add_le_add hrest' hfirst_r\n _ = 1 / c + (1 + Real.log m) := by ring\n```\nwith hrest' defined before (the sum bound). Reorder: hrest' first (it's independent), then hsplit0, then the calc. Let me rewrite hright's block accordingly.","signature":""},{"type":"text","text":"`sum_range_succ'` here appends the term on the right. Adjusting hright's structure."},{"type":"tool_use","id":"Read_420","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean","offset":617,"limit":28},"extra_content":null}]