← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (808 chars)
The statement elaborates fine. So the loop is inside the proof. Now bisect the proof: I'll add the proof incrementally in the probe file. Most likely suspect: `apply SummableLocallyUniformlyOn_of_locally_bounded Metric.isOpen_ball` — maybe `Metric.isOpen_ball` unification: the lemma wants `IsOpen s` where s is the same ball; fine.
Second suspect: the `refine` with the dominator — the expected existential `∃ u, Summable u ∧ ∀ n k ∈ K, ‖f n k‖ ≤ u n` — the `∀ (n : α), ∀ k ∈ K, ...` — k : β = ℂ. My refine gives `fun ρ s hs => ?_` — `∀ k ∈ K` desugars to ∀ k, k ∈ K → ... so the lambda `fun ρ s hs` has hs : s ∈ K ✓.
Third suspect: the rw chain. Fourth: the final `ring` or `positivity`.
Let me bisect by building the proof step by step in the probe file. First chunk: up through the two derivative rws.
Statement is fine; the loop is in the proof. Bisecting incrementally.
tool: Bash
input
{
"command": "cd D:/Python/Quantum/research/lean/NguyenRH1 && python - << 'EOF'\nsrc = open('NguyenRH1/LiMoments.lean', encoding='utf-8').read()\nstart = src.index('lemma seriesTerm_deriv_summableLocallyUniformlyOn')\nstmt_end = src.index(':= by', start)\nend = src.index('end NguyenRH1', stmt_end)\nproof = ''' := by\n apply SummableLocallyUniformlyOn_of_locally_bounded Metric.isOpen_ball\n intro K hK _\n refine \u27e8fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n (zeroMult \u03c1 : \u211d) / \u2016(\u03c1 : \u2102)\u2016 ^ 2 * ((j ! : \u211d) * 2 ^ (j + 1) * (1 / \u03b4) ^ (j - 1)),\n summable_mult_div_norm_sq.mul_left _, fun \u03c1 s hs => ?_\u27e9\n have hsb : s \u2208 Metric.ball (0 : \u2102) (\u03b4 / 2) := hK hs\n rw [Metric.mem_ball, dist_zero_right] at hsb\n have h\u03c1ge : \u03b4 \u2264 \u2016(\u03c1 : \u2102)\u2016 := h\u03b4le \u03c1\n have h\u03c1pos : (0 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := lt_of_lt_of_le h\u03b4 h\u03c1ge\n have hnorm1 : \u2016(\u03c1 : \u2102)\u2016 / 2 \u2264 \u2016s - (\u03c1 : \u2102)\u2016 := by\n have h2 := (abs_le.mp (norm_sub_norm_le (\u03c1 : \u2102) s)).2\n rw [norm_sub_rev]\n linarith\n have hs\u03c1 : s - (\u03c1 : \u2102) \u2260 0 :=\n norm_pos_iff.mp (lt_of_lt_of_le (by linarith : (0 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 / 2) hnorm1)\n rw [(iteratedDerivWithin_of_isOpen Metric.isOpen_ball) hsb, iteratedDeriv_seriesTerm hj \u03c1 hs\u03c1]\n sorry\n\n'''\nprobe = src[:stmt_end] + proof + src[end:]\nopen('NguyenRH1/LiMoments_probe.lean', 'w', encoding='utf-8').write(probe)\nprint('written')\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiMoments_probe.lean 2>&1 | head -20",
"description": "Probe proof up to derivative rewrites",
"timeout": 420000
}
raw content_json (database row)
[{"type":"thinking","thinking":"The statement elaborates fine. So the loop is inside the proof. Now bisect the proof: I'll add the proof incrementally in the probe file. Most likely suspect: `apply SummableLocallyUniformlyOn_of_locally_bounded Metric.isOpen_ball` — maybe `Metric.isOpen_ball` unification: the lemma wants `IsOpen s` where s is the same ball; fine.\n\nSecond suspect: the `refine` with the dominator — the expected existential `∃ u, Summable u ∧ ∀ n k ∈ K, ‖f n k‖ ≤ u n` — the `∀ (n : α), ∀ k ∈ K, ...` — k : β = ℂ. My refine gives `fun ρ s hs => ?_` — `∀ k ∈ K` desugars to ∀ k, k ∈ K → ... so the lambda `fun ρ s hs` has hs : s ∈ K ✓.\n\nThird suspect: the rw chain. Fourth: the final `ring` or `positivity`.\n\nLet me bisect by building the proof step by step in the probe file. First chunk: up through the two derivative rws.","signature":""},{"type":"text","text":"Statement is fine; the loop is in the proof. Bisecting incrementally."},{"type":"tool_use","id":"Bash_44","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && python - << 'EOF'\nsrc = open('NguyenRH1/LiMoments.lean', encoding='utf-8').read()\nstart = src.index('lemma seriesTerm_deriv_summableLocallyUniformlyOn')\nstmt_end = src.index(':= by', start)\nend = src.index('end NguyenRH1', stmt_end)\nproof = ''' := by\n apply SummableLocallyUniformlyOn_of_locally_bounded Metric.isOpen_ball\n intro K hK _\n refine ⟨fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n (zeroMult ρ : ℝ) / ‖(ρ : ℂ)‖ ^ 2 * ((j ! : ℝ) * 2 ^ (j + 1) * (1 / δ) ^ (j - 1)),\n summable_mult_div_norm_sq.mul_left _, fun ρ s hs => ?_⟩\n have hsb : s ∈ Metric.ball (0 : ℂ) (δ / 2) := hK hs\n rw [Metric.mem_ball, dist_zero_right] at hsb\n have hρge : δ ≤ ‖(ρ : ℂ)‖ := hδle ρ\n have hρpos : (0 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_lt_of_le hδ hρge\n have hnorm1 : ‖(ρ : ℂ)‖ / 2 ≤ ‖s - (ρ : ℂ)‖ := by\n have h2 := (abs_le.mp (norm_sub_norm_le (ρ : ℂ) s)).2\n rw [norm_sub_rev]\n linarith\n have hsρ : s - (ρ : ℂ) ≠ 0 :=\n norm_pos_iff.mp (lt_of_lt_of_le (by linarith : (0 : ℝ) < ‖(ρ : ℂ)‖ / 2) hnorm1)\n rw [(iteratedDerivWithin_of_isOpen Metric.isOpen_ball) hsb, iteratedDeriv_seriesTerm hj ρ hsρ]\n sorry\n\n'''\nprobe = src[:stmt_end] + proof + src[end:]\nopen('NguyenRH1/LiMoments_probe.lean', 'w', encoding='utf-8').write(probe)\nprint('written')\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiMoments_probe.lean 2>&1 | head -20","description":"Probe proof up to derivative rewrites","timeout":420000},"extra_content":null}]