← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (851 chars)
The `end NguyenRH1` is MISSING — the compile succeeded earlier because... wait, the last successful compile was BEFORE I added the iteratedDerivWithin_dpsiTermC section? No — the last clean compile included sections up to 7. Then my edits added the zpow lemma + iteratedDerivWithin_dpsiTermC — and the `end NguyenRH1` was consumed in one of my edits (the big Section-7 edit replaced text ending at "end NguyenRH1" with content NOT ending in it... looking at my section-7 edit — the new_string ended with `rw [zero_add]\n\nend NguyenRH1` — hmm no. Whatever — the file currently lacks `end NguyenRH1`. The last compile (which produced errors at 682-704) parsed fine because Lean allows the namespace to remain open at EOF? No — it errors on unclosed namespaces usually ("unclosed" errors are warnings in Lean 4?). Anyway — append the new sections + end.
The `end NguyenRH1` got consumed in an edit. Appending the new sections and the closer.
tool: Bash ok
input
{
"command": "cd \"D:/Python/Quantum\" && cat >> research/lean/NguyenRH1/NguyenRH1/LiDigamma.lean << 'EOF'\n\n/-- The k-th derivative family (k \u2265 1) is summable locally uniformly on the domain. -/\nlemma dpsiTermC_deriv_summableLocallyUniformlyOn (k : \u2115) (hk : 1 \u2264 k) :\n SummableLocallyUniformlyOn\n (fun m => (iteratedDerivWithin k (fun s : \u2102 => dpsiTermC s m) dpsiDomain)) dpsiDomain := by\n apply SummableLocallyUniformlyOn_of_locally_bounded dpsiDomain_isOpen\n intro K hK hKc\n refine \u27e8fun m => (k ! : \u211d) / ((m : \u211d) + 1) ^ (k + 1),\n summable_div_add_one_pow _ (k + 1) (by omega), fun m s hs => ?_\u27e9\n have hsd : s \u2208 dpsiDomain := hK hs\n rw [iteratedDerivWithin_dpsiTermC k hk m hsd]\n have hre : 0 < s.re := dpsiDomain_re_pos hsd\n have hnormS : (m : \u211d) + 1 \u2264 \u2016s + \u2191m + 1\u2016 := by\n have h6 : (s + \u2191m + 1).re = s.re + m + 1 := by\n have h3 : (\u2191m : \u2102).re = (m : \u211d) := by simp\n simp [Complex.add_re, Complex.one_re, h3]\n have h5 : s.re + m + 1 \u2264 \u2016s + \u2191m + 1\u2016 := by\n rw [\u2190 h6]\n exact Complex.re_le_norm _\n linarith [h5, hre]\n rw [norm_mul, norm_mul]\n have hsign : \u2016-((-1 : \u2102)) ^ k\u2016 = 1 := by\n have h7 : \u2016((-1 : \u2102))\u2016 = 1 := by rw [norm_neg, norm_one]\n rw [norm_neg, norm_pow, h7, one_pow]\n rw [hsign, one_mul, RCLike.norm_natCast, norm_zpow]\n rw [show (-(k : \u2124) - 1 : \u2124) = -(((k + 1 : \u2115) : \u2124)) by push_cast; ring, zpow_neg, zpow_natCast]\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [show (((m : \u211d) + 1) ^ (k + 1))\u207b\u00b9 = 1 / (((m : \u211d) + 1) ^ (k + 1)) from by rw [one_div]]\n rw [show (\u2016s + \u2191m + 1\u2016 ^ (k + 1))\u207b\u00b9 = 1 / (\u2016s + \u2191m + 1\u2016 ^ (k + 1)) from by rw [one_div]]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left\u2080 (by positivity : (0 : \u211d) \u2264 (m : \u211d) + 1) hnormS (k + 1)\n\n/-- The iterated derivative of the series function at 1/2 (termwise). -/\nlemma iteratedDerivWithin_tsum_dpsiTermC (k : \u2115) (hk : 1 \u2264 k) :\n iteratedDerivWithin k (fun s : \u2102 => \u2211' m : \u2115, dpsiTermC s m) dpsiDomain ((1 : \u2102) / 2) =\n \u2211' m : \u2115, -((-1 : \u2102)) ^ k * (k ! : \u2102) * ((1 / 2 : \u2102) + \u2191m + 1) ^ (-(k : \u2124) - 1) := by\n rw [iteratedDerivWithin_tsum k dpsiDomain_isOpen dpsiDomain_half\n (fun s hs => dpsiTermC_summable hs)\n (fun j hj1 _ => dpsiTermC_deriv_summableLocallyUniformlyOn j hj1)\n ?_]\n \u00b7 apply tsum_congr\n intro m\n exact iteratedDerivWithin_dpsiTermC k hk m dpsiDomain_half\n \u00b7 intro n j r hjm hr\n rcases Nat.eq_zero_or_pos j with hj0 | hj0\n \u00b7 subst hj0\n rw [iteratedDerivWithin_zero]\n have hsm : r + n + 1 \u2260 0 := dpsiTermC_pole_free hr n\n have heq : (fun s : \u2102 => dpsiTermC s n) = fun s => 1 / ((n : \u2102) + 1) - 1 / (s + \u2191n + 1) := by\n funext s; rw [dpsiTermC]\n rw [heq]\n apply DifferentiableAt.sub (differentiableAt_const _)\n have h3 : DifferentiableAt \u2102 (fun s : \u2102 => s + \u2191n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (differentiableAt_const (1 : \u2102)).div h3 hsm\n \u00b7 rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]\n have hsm : r + n + 1 \u2260 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt \u2102 (fun s : \u2102 => s + \u2191n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : \u2124) - 1) (r + \u2191n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)\n\nend NguyenRH1\nEOF\ncd research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiDigamma.lean 2>&1 | grep -v \"^$\" | head -30",
"description": "Append section 8 and compile",
"timeout": 600000
}
result 2026-08-22 08:48:57Z → 2026-08-26 09:23:49Z
NguyenRH1/LiDogLeg.lean:1087:12: error: No goals to be solved
NguyenRH1/LiDogLeg.lean:1119:68: error: unsolved goals
NguyenRH1/LiDogLeg.lean:1119:68: error: unsolved goals
n : ℕ
n : ℕ
hn : 1 ≤ n
hn : 1 ≤ n
m : ℕ
hm : 3 ≤ m
hXre : (↑(2 * √↑n)).re = 2 * √↑n
h2re : re 2 = 2
hm0 : ¬m = 0
hΛ : ‖↑(ArithmeticFunction.vonMangoldt m)‖ = ArithmeticFunction.vonMangoldt m
NguyenRH1/LiDogLeg.lean:1165:8: error(lean.unknownIdentifier): Unknown constant `ArithmeticFunction.vonMangoldt_prime`
NguyenRH1/LiDogLeg.lean:1167:35: error: Application type mismatch: The argument
NguyenRH1/LiDogLeg.lean:1167:35: error: Application type mismatch: The argument
LT.lt.le Real.log_two_gt_d9
LT.lt.le Real.log_two_gt_d9
has type
0.6931471803 ≤ Real.log 2
but is expected to have type
0 ≤ Real.log 2
in the application
NguyenRH1/LiDogLeg.lean:1168:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
‖↑(ArithmeticFunction.vonMangoldt 2)‖
in the target expression
|ArithmeticFunction.vonMangoldt 2| / 2 ^ (2 * √↑n) = Real.log 2 * 2 ^ (-(2 * √↑n))
n : ℕ
hn : 1 ≤ n
NguyenRH1/LiDogLeg.lean:1232:62: error: Application type mismatch: The argument
?m.144
has type
0 ≤ 2.2254
but is expected to have type
0 ≤ Real.exp 0.8
in the application
NguyenRH1/LiDogLeg.lean:1236:46: error: Application type mismatch: The argument
hprod
has type
9 / 4 ≤ Real.exp 0.811
but is expected to have type
Real.exp ?m.187 ≤ ?m.186
in the application
NguyenRH1/LiDogLeg.lean:1263:21: error: unsolved goals
x : ℝ
hx : 2.236 ≤ x
hlog49 : Real.log (4 / 9) ≤ -0.811
hlogneg : Real.log (4 / 9) < 0
h1 : (4 / 9) ^ x = Real.exp (Real.log (4 / 9) * x)
h2a : Real.log (4 / 9) * x ≤ Real.log (4 / 9) * 2.236
h2b : Real.log (4 / 9) * 2.236 ≤ -0.811 * 2.236
⊢ False
NguyenRH1/LiDogLeg.lean:1275:7: error: invalid 'calc' step, left-hand side is
Real.exp (x * Real.log (4 / 9)) : ℝ
but is expected to be
Real.exp (Real.log (4 / 9) * x) : ℝ
NguyenRH1/LiDogLeg.lean:1299:4: error: Type mismatch
Real.rpow_def_of_pos
(Mathlib.Meta.NormNum.isNNRat_lt_true
(Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))
(Mathlib.Meta.NormNum.isNNRat_div
raw content_json (database row)
[{"type":"thinking","thinking":"The `end NguyenRH1` is MISSING — the compile succeeded earlier because... wait, the last successful compile was BEFORE I added the iteratedDerivWithin_dpsiTermC section? No — the last clean compile included sections up to 7. Then my edits added the zpow lemma + iteratedDerivWithin_dpsiTermC — and the `end NguyenRH1` was consumed in one of my edits (the big Section-7 edit replaced text ending at \"end NguyenRH1\" with content NOT ending in it... looking at my section-7 edit — the new_string ended with `rw [zero_add]\\n\\nend NguyenRH1` — hmm no. Whatever — the file currently lacks `end NguyenRH1`. The last compile (which produced errors at 682-704) parsed fine because Lean allows the namespace to remain open at EOF? No — it errors on unclosed namespaces usually (\"unclosed\" errors are warnings in Lean 4?). Anyway — append the new sections + end.","signature":""},{"type":"text","text":"The `end NguyenRH1` got consumed in an edit. Appending the new sections and the closer."},{"type":"tool_use","id":"Bash_430","name":"Bash","input":{"command":"cd \"D:/Python/Quantum\" && cat >> research/lean/NguyenRH1/NguyenRH1/LiDigamma.lean << 'EOF'\n\n/-- The k-th derivative family (k ≥ 1) is summable locally uniformly on the domain. -/\nlemma dpsiTermC_deriv_summableLocallyUniformlyOn (k : ℕ) (hk : 1 ≤ k) :\n SummableLocallyUniformlyOn\n (fun m => (iteratedDerivWithin k (fun s : ℂ => dpsiTermC s m) dpsiDomain)) dpsiDomain := by\n apply SummableLocallyUniformlyOn_of_locally_bounded dpsiDomain_isOpen\n intro K hK hKc\n refine ⟨fun m => (k ! : ℝ) / ((m : ℝ) + 1) ^ (k + 1),\n summable_div_add_one_pow _ (k + 1) (by omega), fun m s hs => ?_⟩\n have hsd : s ∈ dpsiDomain := hK hs\n rw [iteratedDerivWithin_dpsiTermC k hk m hsd]\n have hre : 0 < s.re := dpsiDomain_re_pos hsd\n have hnormS : (m : ℝ) + 1 ≤ ‖s + ↑m + 1‖ := by\n have h6 : (s + ↑m + 1).re = s.re + m + 1 := by\n have h3 : (↑m : ℂ).re = (m : ℝ) := by simp\n simp [Complex.add_re, Complex.one_re, h3]\n have h5 : s.re + m + 1 ≤ ‖s + ↑m + 1‖ := by\n rw [← h6]\n exact Complex.re_le_norm _\n linarith [h5, hre]\n rw [norm_mul, norm_mul]\n have hsign : ‖-((-1 : ℂ)) ^ k‖ = 1 := by\n have h7 : ‖((-1 : ℂ))‖ = 1 := by rw [norm_neg, norm_one]\n rw [norm_neg, norm_pow, h7, one_pow]\n rw [hsign, one_mul, RCLike.norm_natCast, norm_zpow]\n rw [show (-(k : ℤ) - 1 : ℤ) = -(((k + 1 : ℕ) : ℤ)) by push_cast; ring, zpow_neg, zpow_natCast]\n rw [div_eq_mul_inv]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [show (((m : ℝ) + 1) ^ (k + 1))⁻¹ = 1 / (((m : ℝ) + 1) ^ (k + 1)) from by rw [one_div]]\n rw [show (‖s + ↑m + 1‖ ^ (k + 1))⁻¹ = 1 / (‖s + ↑m + 1‖ ^ (k + 1)) from by rw [one_div]]\n apply one_div_le_one_div_of_le (by positivity)\n exact pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1) hnormS (k + 1)\n\n/-- The iterated derivative of the series function at 1/2 (termwise). -/\nlemma iteratedDerivWithin_tsum_dpsiTermC (k : ℕ) (hk : 1 ≤ k) :\n iteratedDerivWithin k (fun s : ℂ => ∑' m : ℕ, dpsiTermC s m) dpsiDomain ((1 : ℂ) / 2) =\n ∑' m : ℕ, -((-1 : ℂ)) ^ k * (k ! : ℂ) * ((1 / 2 : ℂ) + ↑m + 1) ^ (-(k : ℤ) - 1) := by\n rw [iteratedDerivWithin_tsum k dpsiDomain_isOpen dpsiDomain_half\n (fun s hs => dpsiTermC_summable hs)\n (fun j hj1 _ => dpsiTermC_deriv_summableLocallyUniformlyOn j hj1)\n ?_]\n · apply tsum_congr\n intro m\n exact iteratedDerivWithin_dpsiTermC k hk m dpsiDomain_half\n · intro n j r hjm hr\n rcases Nat.eq_zero_or_pos j with hj0 | hj0\n · subst hj0\n rw [iteratedDerivWithin_zero]\n have hsm : r + n + 1 ≠ 0 := dpsiTermC_pole_free hr n\n have heq : (fun s : ℂ => dpsiTermC s n) = fun s => 1 / ((n : ℂ) + 1) - 1 / (s + ↑n + 1) := by\n funext s; rw [dpsiTermC]\n rw [heq]\n apply DifferentiableAt.sub (differentiableAt_const _)\n have h3 : DifferentiableAt ℂ (fun s : ℂ => s + ↑n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (differentiableAt_const (1 : ℂ)).div h3 hsm\n · rw [iteratedDerivWithin_dpsiTermC j (by omega) n hr]\n have hsm : r + n + 1 ≠ 0 := dpsiTermC_pole_free hr n\n apply DifferentiableAt.mul (differentiableAt_const _)\n have h4 : DifferentiableAt ℂ (fun s : ℂ => s + ↑n + 1) r :=\n (differentiableAt_id.add (differentiableAt_const _)).add (differentiableAt_const _)\n exact (((hasDerivAt_zpow (-(j : ℤ) - 1) (r + ↑n + 1) (Or.inl hsm)).differentiableAt).comp r\n h4.hasDerivAt)\n\nend NguyenRH1\nEOF\ncd research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiDigamma.lean 2>&1 | grep -v \"^$\" | head -30","description":"Append section 8 and compile","timeout":600000},"extra_content":null}]