← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (420 chars)
`one_le_pow₀` has a different signature (maybe (h : 1 ≤ a) : 1 ≤ a^n with n implicit). And `by positivity` for 1 ≤ 2^j — positivity proves 0 < or 0 ≤ — the second probe example didn't error?? Only the first line errored. So `by positivity` proves (1:ℝ) ≤ 2^j ✓ (positivity handles 1 ≤ via... it reported no error for line 4 ✓). Use `by positivity`.
Now let me rewrite the final calc step of fiber_mult_count completely.
`positivity` handles `1 ≤ 2^j`. Writing the final calc step properly.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"old_string": " _ \u2264 A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n have h1 := Ncount_le_of_interval_gen (a := (2 : \u211d) ^ j * X - 1)\n (b := (2 : \u211d) ^ (j + 1) * X) (by linarith)\n have h2 := Ncount_le_of_interval_gen (a := -(2 : \u211d) ^ (j + 1) * X)\n (b := -(2 : \u211d) ^ j * X) (by linarith)\n have hmax1 : max |(2 : \u211d) ^ j * X - 1| |(2 : \u211d) ^ (j + 1) * X|\n = (2 : \u211d) ^ (j + 1) * X := by\n rw [abs_of_nonneg hpos1, abs_of_nonneg hpos2.le]\n apply max_eq_right\n nlinarith [h2jX, h2j1X]\n sorry\n sorry",
"new_string": " _ \u2264 A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n have h2jX1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j * X := by\n have h1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j := by positivity\n nlinarith [h1, hX]\n have hlogpos : (0 : \u211d) \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_nonneg\n linarith [h2j1X]\n have h1 := Ncount_le_of_interval_gen (a := (2 : \u211d) ^ j * X - 1)\n (b := (2 : \u211d) ^ (j + 1) * X) (by linarith)\n have h2 := Ncount_le_of_interval_gen (a := -(2 : \u211d) ^ (j + 1) * X)\n (b := -(2 : \u211d) ^ j * X) (by linarith)\n -- the log arguments match L = log(3\u00b72^{j+1}X + 5)\n have hlog1 : Real.log (3 * max |(2 : \u211d) ^ j * X - 1| |(2 : \u211d) ^ (j + 1) * X| + 5)\n \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |(2 : \u211d) ^ j * X - 1| |(2 : \u211d) ^ (j + 1) * X|\n \u2264 (2 : \u211d) ^ (j + 1) * X := by\n apply max_le\n \u00b7 rw [abs_of_nonneg (by linarith [h2jX1])]\n nlinarith [h2jX, h2j1X]\n \u00b7 rw [abs_of_nonneg h2j1X.le]\n linarith [hmax]\n have hlog2 : Real.log (3 * max |-(2 : \u211d) ^ (j + 1) * X| |-(2 : \u211d) ^ j * X| + 5)\n \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |-(2 : \u211d) ^ (j + 1) * X| |-(2 : \u211d) ^ j * X|\n \u2264 (2 : \u211d) ^ (j + 1) * X := by\n apply max_le\n \u00b7 rw [abs_neg, abs_of_nonneg h2j1X.le]\n \u00b7 rw [abs_neg, abs_of_nonneg h2jX.le]\n nlinarith [h2jX, h2j1X]\n linarith [hmax]\n -- the lengths: b\u2212a+1 = 2^jX + 2 and 2^jX + 1\n have hlen1 : (2 : \u211d) ^ (j + 1) * X - ((2 : \u211d) ^ j * X - 1) + 1\n = (2 : \u211d) ^ j * X + 2 := by\n have hpow : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by ring\n rw [hpow]\n ring\n have hlen2 : -(2 : \u211d) ^ j * X - (-(2 : \u211d) ^ (j + 1) * X) + 1\n = (2 : \u211d) ^ j * X + 1 := by\n have hpow : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by ring\n rw [hpow]\n ring\n rw [hlen1] at h1\n rw [hlen2] at h2\n -- assemble: A\u2080(2^jX+2)L + A\u2080(2^jX+1)L = A\u2080(2^{j+1}X+3)L\n have hA0 := A0count_pos.le\n have h1' : (Zeta23.Ncount ((2 : \u211d) ^ j * X - 1) ((2 : \u211d) ^ (j + 1) * X) : \u211d)\n \u2264 A0count * ((2 : \u211d) ^ j * X + 2) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) :=\n h1.trans (mul_le_mul_of_nonneg_left hlog1 (by positivity))\n have h2' : (Zeta23.Ncount (-(2 : \u211d) ^ (j + 1) * X) (-((2 : \u211d) ^ j * X)) : \u211d)\n \u2264 A0count * ((2 : \u211d) ^ j * X + 1) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) :=\n h2.trans (mul_le_mul_of_nonneg_left hlog2 (by positivity))\n have hpow : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by ring\n linarith [h1', h2', hA0, hlogpos]"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`one_le_pow₀` has a different signature (maybe (h : 1 ≤ a) : 1 ≤ a^n with n implicit). And `by positivity` for 1 ≤ 2^j — positivity proves 0 < or 0 ≤ — the second probe example didn't error?? Only the first line errored. So `by positivity` proves (1:ℝ) ≤ 2^j ✓ (positivity handles 1 ≤ via... it reported no error for line 4 ✓). Use `by positivity`.\n\nNow let me rewrite the final calc step of fiber_mult_count completely.","signature":""},{"type":"text","text":"`positivity` handles `1 ≤ 2^j`. Writing the final calc step properly."},{"type":"tool_use","id":"Edit_383","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":" _ ≤ A0count * ((2 : ℝ) ^ (j + 1) * X + 3) * Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) := by\n have h1 := Ncount_le_of_interval_gen (a := (2 : ℝ) ^ j * X - 1)\n (b := (2 : ℝ) ^ (j + 1) * X) (by linarith)\n have h2 := Ncount_le_of_interval_gen (a := -(2 : ℝ) ^ (j + 1) * X)\n (b := -(2 : ℝ) ^ j * X) (by linarith)\n have hmax1 : max |(2 : ℝ) ^ j * X - 1| |(2 : ℝ) ^ (j + 1) * X|\n = (2 : ℝ) ^ (j + 1) * X := by\n rw [abs_of_nonneg hpos1, abs_of_nonneg hpos2.le]\n apply max_eq_right\n nlinarith [h2jX, h2j1X]\n sorry\n sorry","new_string":" _ ≤ A0count * ((2 : ℝ) ^ (j + 1) * X + 3) * Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) := by\n have h2jX1 : (1 : ℝ) ≤ (2 : ℝ) ^ j * X := by\n have h1 : (1 : ℝ) ≤ (2 : ℝ) ^ j := by positivity\n nlinarith [h1, hX]\n have hlogpos : (0 : ℝ) ≤ Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) := by\n apply Real.log_nonneg\n linarith [h2j1X]\n have h1 := Ncount_le_of_interval_gen (a := (2 : ℝ) ^ j * X - 1)\n (b := (2 : ℝ) ^ (j + 1) * X) (by linarith)\n have h2 := Ncount_le_of_interval_gen (a := -(2 : ℝ) ^ (j + 1) * X)\n (b := -(2 : ℝ) ^ j * X) (by linarith)\n -- the log arguments match L = log(3·2^{j+1}X + 5)\n have hlog1 : Real.log (3 * max |(2 : ℝ) ^ j * X - 1| |(2 : ℝ) ^ (j + 1) * X| + 5)\n ≤ Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |(2 : ℝ) ^ j * X - 1| |(2 : ℝ) ^ (j + 1) * X|\n ≤ (2 : ℝ) ^ (j + 1) * X := by\n apply max_le\n · rw [abs_of_nonneg (by linarith [h2jX1])]\n nlinarith [h2jX, h2j1X]\n · rw [abs_of_nonneg h2j1X.le]\n linarith [hmax]\n have hlog2 : Real.log (3 * max |-(2 : ℝ) ^ (j + 1) * X| |-(2 : ℝ) ^ j * X| + 5)\n ≤ Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |-(2 : ℝ) ^ (j + 1) * X| |-(2 : ℝ) ^ j * X|\n ≤ (2 : ℝ) ^ (j + 1) * X := by\n apply max_le\n · rw [abs_neg, abs_of_nonneg h2j1X.le]\n · rw [abs_neg, abs_of_nonneg h2jX.le]\n nlinarith [h2jX, h2j1X]\n linarith [hmax]\n -- the lengths: b−a+1 = 2^jX + 2 and 2^jX + 1\n have hlen1 : (2 : ℝ) ^ (j + 1) * X - ((2 : ℝ) ^ j * X - 1) + 1\n = (2 : ℝ) ^ j * X + 2 := by\n have hpow : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by ring\n rw [hpow]\n ring\n have hlen2 : -(2 : ℝ) ^ j * X - (-(2 : ℝ) ^ (j + 1) * X) + 1\n = (2 : ℝ) ^ j * X + 1 := by\n have hpow : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by ring\n rw [hpow]\n ring\n rw [hlen1] at h1\n rw [hlen2] at h2\n -- assemble: A₀(2^jX+2)L + A₀(2^jX+1)L = A₀(2^{j+1}X+3)L\n have hA0 := A0count_pos.le\n have h1' : (Zeta23.Ncount ((2 : ℝ) ^ j * X - 1) ((2 : ℝ) ^ (j + 1) * X) : ℝ)\n ≤ A0count * ((2 : ℝ) ^ j * X + 2) * Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) :=\n h1.trans (mul_le_mul_of_nonneg_left hlog1 (by positivity))\n have h2' : (Zeta23.Ncount (-(2 : ℝ) ^ (j + 1) * X) (-((2 : ℝ) ^ j * X)) : ℝ)\n ≤ A0count * ((2 : ℝ) ^ j * X + 1) * Real.log (3 * (2 : ℝ) ^ (j + 1) * X + 5) :=\n h2.trans (mul_le_mul_of_nonneg_left hlog2 (by positivity))\n have hpow : (2 : ℝ) ^ (j + 1) * X = 2 * ((2 : ℝ) ^ j * X) := by ring\n linarith [h1', h2', hA0, hlogpos]"},"extra_content":null}]