← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (889 chars)
`intervalIntegral_conj : ∫ x in a..b, conj (f x) ∂μ = conj (∫ x in a..b, f x ∂μ)` exists (Basic.lean:902). Use that instead of `integral_conj`:
```
unfold HIntegral
rw [← intervalIntegral_conj]
apply intervalIntegral.integral_congr
...
```
`← intervalIntegral_conj` rewrites `conj (∫ x in a..b, f x ∂μ)` → `∫ x in a..b, conj (f x) ∂μ` ✓ — the pattern is on the interval form ✓ will match the goal's RHS `starRingEnd ℂ (∫ x in 3..2√↑n, Jlag n (↑x + ↑T·I))` — the lemma's `conj` = starRingEnd ℂ ✓ and the interval form matches ✓. Then integral_congr with the pointwise identity (Jlag n (↑σ + ↑(-T)·I) = conj (Jlag n (↑σ + ↑T·I))) ✓.
Wait — after `rw [← intervalIntegral_conj]`, the goal's RHS becomes `∫ x in 3..2√↑n, conj (Jlag n (↑x + ↑T·I))` — and LHS is `∫ x in 3..2√↑n, Jlag n (↑x + ↑(-T)·I)` — then integral_congr with the pointwise identity ✓.
Apply all four fixes.
`intervalIntegral_conj` is the right one. Applying all fixes.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " have hVl2 : \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n apply mul_le_mul hlen (le_refl _) (by positivity) hlen0",
"new_string": " have hVl2 : \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n apply mul_le_mul hlen (le_refl _) (by positivity) (by positivity)"
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " have hK : (0 : \u211d) < (Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by\n apply mul_pos (by positivity)\n apply mul_pos (by positivity)\n positivity",
"new_string": " have hK : (0 : \u211d) < (Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by positivity"
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " _ \u2264 ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) hlen0\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 \u2264\n (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2 (by positivity) (by positivity)",
"new_string": " _ \u2264 ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) (by positivity)\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 \u2264\n (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2\n (mul_nonneg (Real.rpow_nonneg (by norm_num) _) dirichletC2_nonneg)\n (by positivity)"
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " unfold HIntegral\n rw [\u2190 integral_conj]\n apply intervalIntegral.integral_congr",
"new_string": " unfold HIntegral\n rw [\u2190 intervalIntegral_conj]\n apply intervalIntegral.integral_congr"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`intervalIntegral_conj : ∫ x in a..b, conj (f x) ∂μ = conj (∫ x in a..b, f x ∂μ)` exists (Basic.lean:902). Use that instead of `integral_conj`:\n\n```\n unfold HIntegral\n rw [← intervalIntegral_conj]\n apply intervalIntegral.integral_congr\n ...\n```\n`← intervalIntegral_conj` rewrites `conj (∫ x in a..b, f x ∂μ)` → `∫ x in a..b, conj (f x) ∂μ` ✓ — the pattern is on the interval form ✓ will match the goal's RHS `starRingEnd ℂ (∫ x in 3..2√↑n, Jlag n (↑x + ↑T·I))` — the lemma's `conj` = starRingEnd ℂ ✓ and the interval form matches ✓. Then integral_congr with the pointwise identity (Jlag n (↑σ + ↑(-T)·I) = conj (Jlag n (↑σ + ↑T·I))) ✓.\n\nWait — after `rw [← intervalIntegral_conj]`, the goal's RHS becomes `∫ x in 3..2√↑n, conj (Jlag n (↑x + ↑T·I))` — and LHS is `∫ x in 3..2√↑n, Jlag n (↑x + ↑(-T)·I)` — then integral_congr with the pointwise identity ✓.\n\nApply all four fixes.","signature":""},{"type":"text","text":"`intervalIntegral_conj` is the right one. Applying all fixes."},{"type":"tool_use","id":"Edit_381","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" have hVl2 : ‖VIntegral (Jlag n) 3 T ((3 / 2) * T)‖ ≤\n ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n apply mul_le_mul hlen (le_refl _) (by positivity) hlen0","new_string":" have hVl2 : ‖VIntegral (Jlag n) 3 T ((3 / 2) * T)‖ ≤\n ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n apply mul_le_mul hlen (le_refl _) (by positivity) (by positivity)"},"extra_content":null},{"type":"tool_use","id":"Edit_382","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" have hK : (0 : ℝ) < (Real.sqrt (n : ℝ) + 1) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n ((2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n apply mul_pos (by positivity)\n apply mul_pos (by positivity)\n positivity","new_string":" have hK : (0 : ℝ) < (Real.sqrt (n : ℝ) + 1) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n ((2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by positivity"},"extra_content":null},{"type":"tool_use","id":"Edit_383","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" _ ≤ ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n ((2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * (1 : ℝ)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) ≤\n ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) hlen0\n have h2 : (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * dirichletC2 ≤\n (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * (1 : ℝ) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2 (by positivity) (by positivity)","new_string":" _ ≤ ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n ((2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * (1 : ℝ)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) ≤\n ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) (by positivity)\n have h2 : (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * dirichletC2 ≤\n (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) * (1 : ℝ) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2\n (mul_nonneg (Real.rpow_nonneg (by norm_num) _) dirichletC2_nonneg)\n (by positivity)"},"extra_content":null},{"type":"tool_use","id":"Edit_384","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" unfold HIntegral\n rw [← integral_conj]\n apply intervalIntegral.integral_congr","new_string":" unfold HIntegral\n rw [← intervalIntegral_conj]\n apply intervalIntegral.integral_congr"},"extra_content":null}]