← Session 29 — ii′-5e-v: circle bound (far-tail + series bounds)
thinking (2198 chars)
The heredoc with single quotes inside the python string — the issue: the python code contains `'''` triple-quotes and the content has no problem, but the bash heredoc `<< 'EOF'` should be literal... The error "unexpected EOF looking for matching ''" at line 161 — the content contains an apostrophe that closed something? With `<< 'EOF'` (quoted delimiter) the content is literal, no expansion — apostrophes are fine. Hmm, line 161... the content includes `(by omega : 1 ≤ j)` etc. Wait — the outer Bash command itself is wrapped in double quotes?? No, I passed it as the command string. The issue may be the python `'''` inside interacting with... Actually the error is from bash parsing — the heredoc terminator must be exactly `EOF` at line start. My content contains lines starting with `EOF`? No.
Hmm — actually the problem: my command had `<< 'EOF'` and the content includes the two SINGLE-QUOTE characters in `<< 'EOF'`... that's standard. Let me look — "unexpected EOF while looking for matching `''" — there's an unbalanced single quote in the COMMAND portion. The python string uses `'''` delimiters and the Lean content includes `‖`, fine — but the content also includes `'` characters? Lean code: `Nat.lt_pow_succ_log_self`... `pow_succ'` — THERE: `pow_succ'` contains a single quote! Inside the python triple-quoted string that's fine, but bash heredoc with quoted delimiter is literal so `'` is fine there too...
Wait — the REAL issue: the heredoc is inside `python - << 'EOF'` — that's fine. But my Bash tool call wraps everything... The tool said "Shell command failed /usr/bin/bash: -c: line 161: unexpected EOF". Line 161 of the script. The script content with `pow_succ'` — bash heredoc with quoted delimiter should NOT care. UNLESS the heredoc delimiter got consumed early: my content has a line `open(p, 'w', encoding='utf-8').write(src)` etc. Hmm.
Let me just avoid the heredoc-in-bash for large content: use the Write tool to create a temp python file, or better — append the new section via the Edit tool directly (safer). The Edit tool with old_string `end NguyenRH1\n\nend` and new_string = the section + ends. The section is large but Edit handles it. Let me do that.
The bash heredoc choked on the content. Using the Edit tool directly instead.
tool: Edit
input
{
"replace_all": false,
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"old_string": "end NguyenRH1\n\nend",
"new_string": "/-! ## 8b. The inner-high and upper shells -/\n\n/-- **The inner-high shell** (R/2 < \u2016\u03c1\u2016 \u2264 R\u22122): \u2264 60\u00b7A\u2080\u00b7(log R)\u00b2.\nDyadic in d := R\u2212\u2016\u03c1\u2016 \u2208 [2, R/2). -/\nlemma shell_inhi {s : \u2102} {R : \u211d} (hs : \u2016s\u2016 = R) (hsz : \u00ac IsNontrivialZero s)\n (hR : (16002 : \u211d) \u2264 R) :\n \u2211 \u03c1 \u2208 (circleZeros R).filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n \u00ac \u2016(\u03c1 : \u2102)\u2016 \u2264 R / 2 \u2227 \u2016(\u03c1 : \u2102)\u2016 \u2264 R - 2), \u2016seriesTerm s \u03c1\u2016\n \u2264 60 * A0count * (Real.log R) ^ 2 := by\n classical\n have hRpos : (0 : \u211d) < R := by linarith [hR]\n have hlogR : (9 : \u211d) \u2264 Real.log R := nine_le_log_of_16002_le hR\n have hlogRnn : (0 : \u211d) \u2264 Real.log R := by linarith [hlogR]\n set J := Nat.log 2 \u230aR\u230b\u208a with hJ\n have hJle : (J : \u211d) \u2264 2 * Real.log R := Nat.log_floor_two_le hR\n have h2JR : (2 : \u211d) ^ J \u2264 R := two_pow_log_floor_le hR\n set FS := (circleZeros R).filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n \u00ac \u2016(\u03c1 : \u2102)\u2016 \u2264 R / 2 \u2227 \u2016(\u03c1 : \u2102)\u2016 \u2264 R - 2) with hFS\n have hdj : \u2200 \u03c1 \u2208 FS,\n 1 \u2264 Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2227 Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2264 J := by\n intro \u03c1 h\u03c1\n rw [hFS, Finset.mem_filter] at h\u03c1\n obtain \u27e8-, h\u03c1R2, h\u03c1Rm2\u27e9 := h\u03c1\n have hd2 : (2 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1Rm2]\n have hfl2 : (2 : \u2115) \u2264 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a := by\n rw [Nat.le_floor_iff (by linarith [hd2] : (0 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016)]\n exact_mod_cast hd2\n have h1 : (1 : \u2115) \u2264 Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a := by\n have h2 := Nat.log_mono_right (b := 2) hfl2\n have h3 : Nat.log 2 2 = 1 := by decide\n rw [h3] at h2\n exact h2\n have h2 : Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2264 J := by\n apply Nat.log_mono_right (b := 2)\n apply Nat.floor_mono\n have hnn : (0 : \u211d) \u2264 \u2016(\u03c1 : \u2102)\u2016 := norm_nonneg _\n linarith [hnn]\n exact \u27e8h1, h2\u27e9\n have hterm : \u2200 \u03c1 \u2208 FS, \u2016seriesTerm s \u03c1\u2016\n \u2264 2 * (zeroMult (\u03c1 : \u2102) : \u211d) / (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a) := by\n intro \u03c1 h\u03c1\n rw [hFS, Finset.mem_filter] at h\u03c1\n obtain \u27e8-, h\u03c1R2, h\u03c1Rm2\u27e9 := h\u03c1\n have h\u03c1R2' : (R / 2 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := lt_of_not_ge h\u03c1R2\n have hd2 : (2 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1Rm2]\n have hpowle : (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by\n have hfl : \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2260 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n have h2 : ((2 ^ Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u2115) : \u211d) \u2264 (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) := by\n exact_mod_cast h1\n have h3 : ((2 ^ Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u2115) : \u211d)\n = (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a) := by\n push_cast\n ring\n rw [h3] at h2\n have h4 : (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n have hsr : R - \u2016(\u03c1 : \u2102)\u2016 \u2264 \u2016s - (\u03c1 : \u2102)\u2016 := by\n have h := abs_norm_sub_norm_le s (\u03c1 : \u2102)\n rw [hs, abs_of_nonneg (by linarith [hd2] : (0 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016)] at h\n exact h\n have h\u03c1pos : (0 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1R2', hR]\n have hsrpos : (0 : \u211d) < \u2016s - (\u03c1 : \u2102)\u2016 := by linarith [hsr, hd2]\n have hid := seriesTerm_norm_eq_div hs hsz \u03c1\n rw [hid]\n have hm : (0 : \u211d) \u2264 (zeroMult (\u03c1 : \u2102) : \u211d) := Nat.cast_nonneg _\n have h1 : R / (\u2016(\u03c1 : \u2102)\u2016 * \u2016s - (\u03c1 : \u2102)\u2016)\n \u2264 R / (\u2016(\u03c1 : \u2102)\u2016 * (R - \u2016(\u03c1 : \u2102)\u2016)) := by\n apply div_le_div_of_nonneg_left hRpos.le (mul_pos h\u03c1pos (by linarith [hd2]))\n exact mul_le_mul_of_nonneg_left hsr h\u03c1pos.le\n have h2 : R / (\u2016(\u03c1 : \u2102)\u2016 * (R - \u2016(\u03c1 : \u2102)\u2016)) \u2264 2 / (R - \u2016(\u03c1 : \u2102)\u2016) := by\n rw [div_le_div_iff\u2080 (mul_pos h\u03c1pos (by linarith [hd2] :\n (0 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 * (R - \u2016(\u03c1 : \u2102)\u2016)))\n (by linarith [hd2] : (0 : \u211d) < R - \u2016(\u03c1 : \u2102)\u2016)]\n have h5 : R \u2264 2 * \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1R2']\n have h6 : (0 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by linarith [hd2]\n nlinarith [h5, h6, h\u03c1pos]\n have h3 : (2 : \u211d) / (R - \u2016(\u03c1 : \u2102)\u2016) \u2264 2 / (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a) := by\n apply div_le_div_of_nonneg_left (by norm_num : (0 : \u211d) \u2264 2) (by positivity)\n exact hpowle\n calc (zeroMult (\u03c1 : \u2102) : \u211d) * (R / (\u2016(\u03c1 : \u2102)\u2016 * \u2016s - (\u03c1 : \u2102)\u2016))\n \u2264 (zeroMult (\u03c1 : \u2102) : \u211d) * (R / (\u2016(\u03c1 : \u2102)\u2016 * (R - \u2016(\u03c1 : \u2102)\u2016))) :=\n mul_le_mul_of_nonneg_left h1 hm\n _ \u2264 (zeroMult (\u03c1 : \u2102) : \u211d) * (2 / (R - \u2016(\u03c1 : \u2102)\u2016)) :=\n mul_le_mul_of_nonneg_left h2 hm\n _ \u2264 (zeroMult (\u03c1 : \u2102) : \u211d) * (2 / (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a)) :=\n mul_le_mul_of_nonneg_left h3 hm\n _ = 2 * (zeroMult (\u03c1 : \u2102) : \u211d) / (2 : \u211d) ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a) := by ring\n have hcount : \u2200 j : \u2115,\n \u2211 \u03c1 \u2208 FS.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a = j), (zeroMult (\u03c1 : \u2102) : \u211d)\n \u2264 3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R) := by\n intro j\n have hRHSnn : (0 : \u211d) \u2264 3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R) :=\n mul_nonneg (mul_nonneg (mul_nonneg (by norm_num : (0 : \u211d) \u2264 3) A0count_pos.le)\n (by positivity)) (mul_nonneg (by norm_num : (0 : \u211d) \u2264 2) hlogRnn)\n by_cases hj : j \u2208 Finset.Ico 1 (J + 1)\n \u00b7 rw [Finset.mem_Ico] at hj\n by_cases hbig : R / 2 \u2264 (2 : \u211d) ^ j\n \u00b7 -- the fiber is empty: d \u2265 2^j \u2265 R/2 contradicts d < R/2\n have hempty : FS.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a = j) = \u2205 := by\n apply Finset.eq_empty_iff_forall_notMem.mpr\n intro \u03c1 h\u03c1\n rw [Finset.mem_filter] at h\u03c1\n obtain \u27e8h\u03c1F, h\u03c1j\u27e9 := h\u03c1\n rw [hFS, Finset.mem_filter] at h\u03c1F\n obtain \u27e8-, h\u03c1R2, h\u03c1Rm2\u27e9 := h\u03c1F\n have h\u03c1R2' : (R / 2 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := lt_of_not_ge h\u03c1R2\n have hd2 : (2 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1Rm2]\n have hpowle : (2 : \u211d) ^ j \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by\n have hfl : \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2260 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n rw [h\u03c1j] at h1\n have h2 : ((2 ^ j : \u2115) : \u211d) \u2264 (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) := by exact_mod_cast h1\n have h3 : ((2 ^ j : \u2115) : \u211d) = (2 : \u211d) ^ j := by push_cast; ring\n rw [h3] at h2\n have h4 : (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n linarith [hpowle, hbig, h\u03c1R2']\n rw [hempty, Finset.sum_empty]\n exact hRHSnn\n \u00b7 push Not at hbig\n have hj2 : (2 : \u211d) \u2264 (2 : \u211d) ^ j := by\n have h1 : (2 : \u211d) ^ 1 \u2264 (2 : \u211d) ^ j :=\n pow_le_pow_right\u2080 (by norm_num) (by omega : 1 \u2264 j)\n linarith [h1]\n have hM1 : (1 : \u211d) \u2264 max (R - (2 : \u211d) ^ (j + 1) - 1) (R / 2 - 1) := by\n have h1 : (1 : \u211d) \u2264 R / 2 - 1 := by linarith [hR]\n exact le_trans h1 (le_max_right _ _)\n have hMb : max (R - (2 : \u211d) ^ (j + 1) - 1) (R / 2 - 1) \u2264 R - (2 : \u211d) ^ j := by\n apply max_le_iff.mpr\n constructor\n \u00b7 have h1 : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n have h2 : (0 : \u211d) < (2 : \u211d) ^ j := by positivity\n linarith [h1, h2]\n \u00b7 linarith [hbig]\n have hsub : \u2200 \u03c1 \u2208 FS.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a = j),\n R - (2 : \u211d) ^ (j + 1) - 1 \u2264 |(\u03c1 : \u2102).im| \u2227 |(\u03c1 : \u2102).im| \u2264 R - (2 : \u211d) ^ j \u2227\n R / 2 - 1 \u2264 |(\u03c1 : \u2102).im| := by\n intro \u03c1 h\u03c1\n rw [Finset.mem_filter] at h\u03c1\n obtain \u27e8h\u03c1F, h\u03c1j\u27e9 := h\u03c1\n rw [hFS, Finset.mem_filter] at h\u03c1F\n obtain \u27e8-, h\u03c1R2, h\u03c1Rm2\u27e9 := h\u03c1F\n have h\u03c1R2' : (R / 2 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := lt_of_not_ge h\u03c1R2\n have hd2 : (2 : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by linarith [h\u03c1Rm2]\n have hpowle : (2 : \u211d) ^ j \u2264 R - \u2016(\u03c1 : \u2102)\u2016 := by\n have hfl : \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a \u2260 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n rw [h\u03c1j] at h1\n have h2 : ((2 ^ j : \u2115) : \u211d) \u2264 (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) := by exact_mod_cast h1\n have h3 : ((2 ^ j : \u2115) : \u211d) = (2 : \u211d) ^ j := by push_cast; ring\n rw [h3] at h2\n have h4 : (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) \u2264 R - \u2016(\u03c1 : \u2102)\u2016 :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n have hpowup : R - \u2016(\u03c1 : \u2102)\u2016 < (2 : \u211d) ^ (j + 1) := by\n have h1 : \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a < 2 ^ (Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a + 1) :=\n Nat.lt_pow_succ_log_self (by norm_num) _\n rw [h\u03c1j] at h1\n have h2 : \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a + 1 \u2264 2 ^ (j + 1) := h1\n have h3 : R - \u2016(\u03c1 : \u2102)\u2016 < (\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a : \u211d) + 1 := Nat.lt_floor_add_one _\n have h4 : ((\u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a + 1 : \u2115) : \u211d) \u2264 ((2 ^ (j + 1) : \u2115) : \u211d) := by\n exact_mod_cast h2\n have h5 : ((2 ^ (j + 1) : \u2115) : \u211d) = (2 : \u211d) ^ (j + 1) := by push_cast; ring\n rw [h5] at h4\n push_cast at h4\n linarith [h3, h4]\n have him1 := abs_im_ge_norm_sub_one \u03c1\n have him2 : |(\u03c1 : \u2102).im| \u2264 \u2016(\u03c1 : \u2102)\u2016 := abs_im_le_norm _\n exact \u27e8by linarith [him1, hpowup], by linarith [him2, hpowle],\n by linarith [him1, h\u03c1R2']\u27e9\n have hbrick := sum_zeroMult_le_of_im_band\n (FS.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a = j)) hM1 hMb hsub\n have hab : R - (2 : \u211d) ^ j - (R - (2 : \u211d) ^ (j + 1) - 1) + 2 = (2 : \u211d) ^ j + 3 := by\n have h1 : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n linarith [h1]\n rw [hab] at hbrick\n -- the log bound: max|a||b| \u2264 3R\n have hmaxle : max |R - (2 : \u211d) ^ (j + 1) - 1| |R - (2 : \u211d) ^ j| \u2264 3 * R := by\n have h1 : |R - (2 : \u211d) ^ j| = R - (2 : \u211d) ^ j := by\n apply abs_of_nonneg\n have h2 : (2 : \u211d) ^ j \u2264 (2 : \u211d) ^ J :=\n pow_le_pow_right\u2080 (by norm_num) (by omega : j \u2264 J)\n linarith [h2, h2JR, hbig]\n have h2 : |R - (2 : \u211d) ^ (j + 1) - 1| \u2264 3 * R := by\n have h3 : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n have h4 : (2 : \u211d) ^ j \u2264 (2 : \u211d) ^ J :=\n pow_le_pow_right\u2080 (by norm_num) (by omega : j \u2264 J)\n have h5 : R - (2 : \u211d) ^ (j + 1) - 1 \u2264 R := by\n have h6 : (0 : \u211d) < (2 : \u211d) ^ (j + 1) := by positivity\n linarith [h6]\n have h6 : (2 : \u211d) ^ (j + 1) - 1 \u2264 2 * R + 1 := by\n nlinarith [h3, h4, h2JR, hR]\n rw [abs_le]\n constructor <;> linarith [h5, h6, hR]\n have h3 : |R - (2 : \u211d) ^ j| \u2264 3 * R := by\n rw [h1]\n have h4 : (0 : \u211d) < (2 : \u211d) ^ j := by positivity\n linarith [h4, hR]\n exact max_le h2 h3\n have hlogle : Real.log (3 * max |R - (2 : \u211d) ^ (j + 1) - 1| |R - (2 : \u211d) ^ j| + 5)\n \u2264 2 * Real.log R :=\n log_three_le_two_log hR\n (le_trans (abs_nonneg _) (le_max_left _ _) |>.trans hmaxle) hmaxle\n exact hbrick.trans (mul_le_mul_of_nonneg_left hlogle\n (mul_nonneg (mul_nonneg (by norm_num : (0 : \u211d) \u2264 3) A0count_pos.le)\n (by positivity)))\n \u00b7 have hempty : FS.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n Nat.log 2 \u230aR - \u2016(\u03c1 : \u2102)\u2016\u230b\u208a = j) = \u2205 := by\n apply Finset.eq_empty_iff_forall_notMem.mpr\n intro \u03c1 h\u03c1\n rw [Finset.mem_filter] at h\u03c1\n obtain \u27e8h\u03c1F, h\u03c1j\u27e9 := h\u03c1\n have hdj\u03c1 := hdj \u03c1 h\u03c1F\n rw [h\u03c1j] at hdj\u03c1\n rw [Finset.mem_Ico] at hj\n omega\n rw [hempty, Finset.sum_empty]\n exact hRHSnn\n -- the closed-form per-shell bound\n have hper : \u2200 j \u2208 Finset.Ico 1 (J + 1),\n (2 / (2 : \u211d) ^ j) * (3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R))\n \u2264 30 * A0count * Real.log R := by\n intro j hj\n rw [Finset.mem_Ico] at hj\n have h2j0 : (0 : \u211d) < (2 : \u211d) ^ j := by positivity\n have hj2 : (2 : \u211d) \u2264 (2 : \u211d) ^ j := by\n have h1 : (2 : \u211d) ^ 1 \u2264 (2 : \u211d) ^ j :=\n pow_le_pow_right\u2080 (by norm_num) (by omega : 1 \u2264 j)\n linarith [h1]\n have hA0 := A0count_pos.le\n have h1 : (2 / (2 : \u211d) ^ j) * (3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R))\n = 12 * A0count * Real.log R * (1 + 3 / (2 : \u211d) ^ j) := by\n field_simp\n ring\n rw [h1]\n have h2 : 3 / (2 : \u211d) ^ j \u2264 3 / 2 := by\n apply div_le_div_of_nonneg_left (by norm_num : (0 : \u211d) \u2264 3) (by norm_num) hj2\n have h3 : (0 : \u211d) \u2264 1 + 3 / (2 : \u211d) ^ j := by\n have h4 : (0 : \u211d) \u2264 3 / (2 : \u211d) ^ j := div_nonneg (by norm_num) h2j0.le\n linarith [h4]\n have h4 : (1 : \u211d) + 3 / (2 : \u211d) ^ j \u2264 5 / 2 := by linarith [h2]\n have h5 : (12 : \u211d) * A0count * Real.log R * (1 + 3 / (2 : \u211d) ^ j)\n \u2264 (12 * A0count * Real.log R) * (5 / 2) :=\n mul_le_mul_of_nonneg_left h4\n (mul_nonneg (mul_nonneg (by norm_num : (0 : \u211d) \u2264 12) hA0) hlogRnn)\n have h6 : (12 : \u211d) * A0count * Real.log R * (5 / 2) = 30 * A0count * Real.log R := by\n ring\n rw [h6] at h5\n exact h5\n -- assembly via dyadic_family_le\n have hmain := dyadic_family_le FS (fun \u03c1 => R - \u2016(\u03c1 : \u2102)\u2016) J\n (fun \u03c1 => \u2016seriesTerm s \u03c1\u2016) 2\n (fun j => 3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R))\n (by norm_num) hdj hterm hcount\n calc \u2211 \u03c1 \u2208 FS, \u2016seriesTerm s \u03c1\u2016\n \u2264 \u2211 j \u2208 Finset.Ico 1 (J + 1), (2 / (2 : \u211d) ^ j) *\n (3 * A0count * ((2 : \u211d) ^ j + 3) * (2 * Real.log R)) := hmain\n _ \u2264 \u2211 j \u2208 Finset.Ico 1 (J + 1), 30 * A0count * Real.log R := Finset.sum_le_sum hper\n _ = (J : \u211d) * (30 * A0count * Real.log R) := by\n rw [Finset.sum_const, Nat.card_Ico, Nat.add_sub_cancel, nsmul_eq_mul]\n _ \u2264 60 * A0count * (Real.log R) ^ 2 := by\n have hA0 := A0count_pos.le\n have h1 : (J : \u211d) * (30 * A0count * Real.log R)\n \u2264 (2 * Real.log R) * (30 * A0count * Real.log R) :=\n mul_le_mul_of_nonneg_right hJle\n (mul_nonneg (mul_nonneg (by norm_num : (0 : \u211d) \u2264 30) hA0) hlogRnn)\n have h2 : (2 * Real.log R) * (30 * A0count * Real.log R)\n = 60 * A0count * (Real.log R) ^ 2 := by ring\n rw [h2] at h1\n exact h1\n\nend NguyenRH1\n\nend"
}
raw content_json (database row)
[{"type":"thinking","thinking":"The heredoc with single quotes inside the python string — the issue: the python code contains `'''` triple-quotes and the content has no problem, but the bash heredoc `<< 'EOF'` should be literal... The error \"unexpected EOF looking for matching ''\" at line 161 — the content contains an apostrophe that closed something? With `<< 'EOF'` (quoted delimiter) the content is literal, no expansion — apostrophes are fine. Hmm, line 161... the content includes `(by omega : 1 ≤ j)` etc. Wait — the outer Bash command itself is wrapped in double quotes?? No, I passed it as the command string. The issue may be the python `'''` inside interacting with... Actually the error is from bash parsing — the heredoc terminator must be exactly `EOF` at line start. My content contains lines starting with `EOF`? No. \n\nHmm — actually the problem: my command had `<< 'EOF'` and the content includes the two SINGLE-QUOTE characters in `<< 'EOF'`... that's standard. Let me look — \"unexpected EOF while looking for matching `''\" — there's an unbalanced single quote in the COMMAND portion. The python string uses `'''` delimiters and the Lean content includes `‖`, fine — but the content also includes `'` characters? Lean code: `Nat.lt_pow_succ_log_self`... `pow_succ'` — THERE: `pow_succ'` contains a single quote! Inside the python triple-quoted string that's fine, but bash heredoc with quoted delimiter is literal so `'` is fine there too...\n\nWait — the REAL issue: the heredoc is inside `python - << 'EOF'` — that's fine. But my Bash tool call wraps everything... The tool said \"Shell command failed /usr/bin/bash: -c: line 161: unexpected EOF\". Line 161 of the script. The script content with `pow_succ'` — bash heredoc with quoted delimiter should NOT care. UNLESS the heredoc delimiter got consumed early: my content has a line `open(p, 'w', encoding='utf-8').write(src)` etc. Hmm.\n\nLet me just avoid the heredoc-in-bash for large content: use the Write tool to create a temp python file, or better — append the new section via the Edit tool directly (safer). The Edit tool with old_string `end NguyenRH1\\n\\nend` and new_string = the section + ends. The section is large but Edit handles it. Let me do that.","signature":""},{"type":"text","text":"The bash heredoc choked on the content. Using the Edit tool directly instead."},{"type":"tool_use","id":"Edit_91","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"end NguyenRH1\n\nend","new_string":"/-! ## 8b. The inner-high and upper shells -/\n\n/-- **The inner-high shell** (R/2 < ‖ρ‖ ≤ R−2): ≤ 60·A₀·(log R)².\nDyadic in d := R−‖ρ‖ ∈ [2, R/2). -/\nlemma shell_inhi {s : ℂ} {R : ℝ} (hs : ‖s‖ = R) (hsz : ¬ IsNontrivialZero s)\n (hR : (16002 : ℝ) ≤ R) :\n ∑ ρ ∈ (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2), ‖seriesTerm s ρ‖\n ≤ 60 * A0count * (Real.log R) ^ 2 := by\n classical\n have hRpos : (0 : ℝ) < R := by linarith [hR]\n have hlogR : (9 : ℝ) ≤ Real.log R := nine_le_log_of_16002_le hR\n have hlogRnn : (0 : ℝ) ≤ Real.log R := by linarith [hlogR]\n set J := Nat.log 2 ⌊R⌋₊ with hJ\n have hJle : (J : ℝ) ≤ 2 * Real.log R := Nat.log_floor_two_le hR\n have h2JR : (2 : ℝ) ^ J ≤ R := two_pow_log_floor_le hR\n set FS := (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2) with hFS\n have hdj : ∀ ρ ∈ FS,\n 1 ≤ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ∧ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ≤ J := by\n intro ρ hρ\n rw [hFS, Finset.mem_filter] at hρ\n obtain ⟨-, hρR2, hρRm2⟩ := hρ\n have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n have hfl2 : (2 : ℕ) ≤ ⌊R - ‖(ρ : ℂ)‖⌋₊ := by\n rw [Nat.le_floor_iff (by linarith [hd2] : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖)]\n exact_mod_cast hd2\n have h1 : (1 : ℕ) ≤ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ := by\n have h2 := Nat.log_mono_right (b := 2) hfl2\n have h3 : Nat.log 2 2 = 1 := by decide\n rw [h3] at h2\n exact h2\n have h2 : Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ≤ J := by\n apply Nat.log_mono_right (b := 2)\n apply Nat.floor_mono\n have hnn : (0 : ℝ) ≤ ‖(ρ : ℂ)‖ := norm_nonneg _\n linarith [hnn]\n exact ⟨h1, h2⟩\n have hterm : ∀ ρ ∈ FS, ‖seriesTerm s ρ‖\n ≤ 2 * (zeroMult (ρ : ℂ) : ℝ) / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n intro ρ hρ\n rw [hFS, Finset.mem_filter] at hρ\n obtain ⟨-, hρR2, hρRm2⟩ := hρ\n have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n have hpowle : (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) ≤ R - ‖(ρ : ℂ)‖ := by\n have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n have h2 : ((2 ^ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by\n exact_mod_cast h1\n have h3 : ((2 ^ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ : ℕ) : ℝ)\n = (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n push_cast\n ring\n rw [h3] at h2\n have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n have hsr : R - ‖(ρ : ℂ)‖ ≤ ‖s - (ρ : ℂ)‖ := by\n have h := abs_norm_sub_norm_le s (ρ : ℂ)\n rw [hs, abs_of_nonneg (by linarith [hd2] : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖)] at h\n exact h\n have hρpos : (0 : ℝ) < ‖(ρ : ℂ)‖ := by linarith [hρR2', hR]\n have hsrpos : (0 : ℝ) < ‖s - (ρ : ℂ)‖ := by linarith [hsr, hd2]\n have hid := seriesTerm_norm_eq_div hs hsz ρ\n rw [hid]\n have hm : (0 : ℝ) ≤ (zeroMult (ρ : ℂ) : ℝ) := Nat.cast_nonneg _\n have h1 : R / (‖(ρ : ℂ)‖ * ‖s - (ρ : ℂ)‖)\n ≤ R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)) := by\n apply div_le_div_of_nonneg_left hRpos.le (mul_pos hρpos (by linarith [hd2]))\n exact mul_le_mul_of_nonneg_left hsr hρpos.le\n have h2 : R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)) ≤ 2 / (R - ‖(ρ : ℂ)‖) := by\n rw [div_le_div_iff₀ (mul_pos hρpos (by linarith [hd2] :\n (0 : ℝ) < ‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)))\n (by linarith [hd2] : (0 : ℝ) < R - ‖(ρ : ℂ)‖)]\n have h5 : R ≤ 2 * ‖(ρ : ℂ)‖ := by linarith [hρR2']\n have h6 : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hd2]\n nlinarith [h5, h6, hρpos]\n have h3 : (2 : ℝ) / (R - ‖(ρ : ℂ)‖) ≤ 2 / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 2) (by positivity)\n exact hpowle\n calc (zeroMult (ρ : ℂ) : ℝ) * (R / (‖(ρ : ℂ)‖ * ‖s - (ρ : ℂ)‖))\n ≤ (zeroMult (ρ : ℂ) : ℝ) * (R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖))) :=\n mul_le_mul_of_nonneg_left h1 hm\n _ ≤ (zeroMult (ρ : ℂ) : ℝ) * (2 / (R - ‖(ρ : ℂ)‖)) :=\n mul_le_mul_of_nonneg_left h2 hm\n _ ≤ (zeroMult (ρ : ℂ) : ℝ) * (2 / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊)) :=\n mul_le_mul_of_nonneg_left h3 hm\n _ = 2 * (zeroMult (ρ : ℂ) : ℝ) / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by ring\n have hcount : ∀ j : ℕ,\n ∑ ρ ∈ FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j), (zeroMult (ρ : ℂ) : ℝ)\n ≤ 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R) := by\n intro j\n have hRHSnn : (0 : ℝ) ≤ 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R) :=\n mul_nonneg (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 3) A0count_pos.le)\n (by positivity)) (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hlogRnn)\n by_cases hj : j ∈ Finset.Ico 1 (J + 1)\n · rw [Finset.mem_Ico] at hj\n by_cases hbig : R / 2 ≤ (2 : ℝ) ^ j\n · -- the fiber is empty: d ≥ 2^j ≥ R/2 contradicts d < R/2\n have hempty : FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j) = ∅ := by\n apply Finset.eq_empty_iff_forall_notMem.mpr\n intro ρ hρ\n rw [Finset.mem_filter] at hρ\n obtain ⟨hρF, hρj⟩ := hρ\n rw [hFS, Finset.mem_filter] at hρF\n obtain ⟨-, hρR2, hρRm2⟩ := hρF\n have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n have hpowle : (2 : ℝ) ^ j ≤ R - ‖(ρ : ℂ)‖ := by\n have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n rw [hρj] at h1\n have h2 : ((2 ^ j : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by exact_mod_cast h1\n have h3 : ((2 ^ j : ℕ) : ℝ) = (2 : ℝ) ^ j := by push_cast; ring\n rw [h3] at h2\n have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n linarith [hpowle, hbig, hρR2']\n rw [hempty, Finset.sum_empty]\n exact hRHSnn\n · push Not at hbig\n have hj2 : (2 : ℝ) ≤ (2 : ℝ) ^ j := by\n have h1 : (2 : ℝ) ^ 1 ≤ (2 : ℝ) ^ j :=\n pow_le_pow_right₀ (by norm_num) (by omega : 1 ≤ j)\n linarith [h1]\n have hM1 : (1 : ℝ) ≤ max (R - (2 : ℝ) ^ (j + 1) - 1) (R / 2 - 1) := by\n have h1 : (1 : ℝ) ≤ R / 2 - 1 := by linarith [hR]\n exact le_trans h1 (le_max_right _ _)\n have hMb : max (R - (2 : ℝ) ^ (j + 1) - 1) (R / 2 - 1) ≤ R - (2 : ℝ) ^ j := by\n apply max_le_iff.mpr\n constructor\n · have h1 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n have h2 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n linarith [h1, h2]\n · linarith [hbig]\n have hsub : ∀ ρ ∈ FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j),\n R - (2 : ℝ) ^ (j + 1) - 1 ≤ |(ρ : ℂ).im| ∧ |(ρ : ℂ).im| ≤ R - (2 : ℝ) ^ j ∧\n R / 2 - 1 ≤ |(ρ : ℂ).im| := by\n intro ρ hρ\n rw [Finset.mem_filter] at hρ\n obtain ⟨hρF, hρj⟩ := hρ\n rw [hFS, Finset.mem_filter] at hρF\n obtain ⟨-, hρR2, hρRm2⟩ := hρF\n have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n have hpowle : (2 : ℝ) ^ j ≤ R - ‖(ρ : ℂ)‖ := by\n have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n linarith [hd2]\n have h1 := Nat.pow_log_le_self 2 hfl\n rw [hρj] at h1\n have h2 : ((2 ^ j : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by exact_mod_cast h1\n have h3 : ((2 ^ j : ℕ) : ℝ) = (2 : ℝ) ^ j := by push_cast; ring\n rw [h3] at h2\n have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n Nat.floor_le (by linarith [hd2])\n linarith [h2, h4]\n have hpowup : R - ‖(ρ : ℂ)‖ < (2 : ℝ) ^ (j + 1) := by\n have h1 : ⌊R - ‖(ρ : ℂ)‖⌋₊ < 2 ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ + 1) :=\n Nat.lt_pow_succ_log_self (by norm_num) _\n rw [hρj] at h1\n have h2 : ⌊R - ‖(ρ : ℂ)‖⌋₊ + 1 ≤ 2 ^ (j + 1) := h1\n have h3 : R - ‖(ρ : ℂ)‖ < (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) + 1 := Nat.lt_floor_add_one _\n have h4 : ((⌊R - ‖(ρ : ℂ)‖⌋₊ + 1 : ℕ) : ℝ) ≤ ((2 ^ (j + 1) : ℕ) : ℝ) := by\n exact_mod_cast h2\n have h5 : ((2 ^ (j + 1) : ℕ) : ℝ) = (2 : ℝ) ^ (j + 1) := by push_cast; ring\n rw [h5] at h4\n push_cast at h4\n linarith [h3, h4]\n have him1 := abs_im_ge_norm_sub_one ρ\n have him2 : |(ρ : ℂ).im| ≤ ‖(ρ : ℂ)‖ := abs_im_le_norm _\n exact ⟨by linarith [him1, hpowup], by linarith [him2, hpowle],\n by linarith [him1, hρR2']⟩\n have hbrick := sum_zeroMult_le_of_im_band\n (FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j)) hM1 hMb hsub\n have hab : R - (2 : ℝ) ^ j - (R - (2 : ℝ) ^ (j + 1) - 1) + 2 = (2 : ℝ) ^ j + 3 := by\n have h1 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n linarith [h1]\n rw [hab] at hbrick\n -- the log bound: max|a||b| ≤ 3R\n have hmaxle : max |R - (2 : ℝ) ^ (j + 1) - 1| |R - (2 : ℝ) ^ j| ≤ 3 * R := by\n have h1 : |R - (2 : ℝ) ^ j| = R - (2 : ℝ) ^ j := by\n apply abs_of_nonneg\n have h2 : (2 : ℝ) ^ j ≤ (2 : ℝ) ^ J :=\n pow_le_pow_right₀ (by norm_num) (by omega : j ≤ J)\n linarith [h2, h2JR, hbig]\n have h2 : |R - (2 : ℝ) ^ (j + 1) - 1| ≤ 3 * R := by\n have h3 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n have h4 : (2 : ℝ) ^ j ≤ (2 : ℝ) ^ J :=\n pow_le_pow_right₀ (by norm_num) (by omega : j ≤ J)\n have h5 : R - (2 : ℝ) ^ (j + 1) - 1 ≤ R := by\n have h6 : (0 : ℝ) < (2 : ℝ) ^ (j + 1) := by positivity\n linarith [h6]\n have h6 : (2 : ℝ) ^ (j + 1) - 1 ≤ 2 * R + 1 := by\n nlinarith [h3, h4, h2JR, hR]\n rw [abs_le]\n constructor <;> linarith [h5, h6, hR]\n have h3 : |R - (2 : ℝ) ^ j| ≤ 3 * R := by\n rw [h1]\n have h4 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n linarith [h4, hR]\n exact max_le h2 h3\n have hlogle : Real.log (3 * max |R - (2 : ℝ) ^ (j + 1) - 1| |R - (2 : ℝ) ^ j| + 5)\n ≤ 2 * Real.log R :=\n log_three_le_two_log hR\n (le_trans (abs_nonneg _) (le_max_left _ _) |>.trans hmaxle) hmaxle\n exact hbrick.trans (mul_le_mul_of_nonneg_left hlogle\n (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 3) A0count_pos.le)\n (by positivity)))\n · have hempty : FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j) = ∅ := by\n apply Finset.eq_empty_iff_forall_notMem.mpr\n intro ρ hρ\n rw [Finset.mem_filter] at hρ\n obtain ⟨hρF, hρj⟩ := hρ\n have hdjρ := hdj ρ hρF\n rw [hρj] at hdjρ\n rw [Finset.mem_Ico] at hj\n omega\n rw [hempty, Finset.sum_empty]\n exact hRHSnn\n -- the closed-form per-shell bound\n have hper : ∀ j ∈ Finset.Ico 1 (J + 1),\n (2 / (2 : ℝ) ^ j) * (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n ≤ 30 * A0count * Real.log R := by\n intro j hj\n rw [Finset.mem_Ico] at hj\n have h2j0 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n have hj2 : (2 : ℝ) ≤ (2 : ℝ) ^ j := by\n have h1 : (2 : ℝ) ^ 1 ≤ (2 : ℝ) ^ j :=\n pow_le_pow_right₀ (by norm_num) (by omega : 1 ≤ j)\n linarith [h1]\n have hA0 := A0count_pos.le\n have h1 : (2 / (2 : ℝ) ^ j) * (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n = 12 * A0count * Real.log R * (1 + 3 / (2 : ℝ) ^ j) := by\n field_simp\n ring\n rw [h1]\n have h2 : 3 / (2 : ℝ) ^ j ≤ 3 / 2 := by\n apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 3) (by norm_num) hj2\n have h3 : (0 : ℝ) ≤ 1 + 3 / (2 : ℝ) ^ j := by\n have h4 : (0 : ℝ) ≤ 3 / (2 : ℝ) ^ j := div_nonneg (by norm_num) h2j0.le\n linarith [h4]\n have h4 : (1 : ℝ) + 3 / (2 : ℝ) ^ j ≤ 5 / 2 := by linarith [h2]\n have h5 : (12 : ℝ) * A0count * Real.log R * (1 + 3 / (2 : ℝ) ^ j)\n ≤ (12 * A0count * Real.log R) * (5 / 2) :=\n mul_le_mul_of_nonneg_left h4\n (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 12) hA0) hlogRnn)\n have h6 : (12 : ℝ) * A0count * Real.log R * (5 / 2) = 30 * A0count * Real.log R := by\n ring\n rw [h6] at h5\n exact h5\n -- assembly via dyadic_family_le\n have hmain := dyadic_family_le FS (fun ρ => R - ‖(ρ : ℂ)‖) J\n (fun ρ => ‖seriesTerm s ρ‖) 2\n (fun j => 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n (by norm_num) hdj hterm hcount\n calc ∑ ρ ∈ FS, ‖seriesTerm s ρ‖\n ≤ ∑ j ∈ Finset.Ico 1 (J + 1), (2 / (2 : ℝ) ^ j) *\n (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R)) := hmain\n _ ≤ ∑ j ∈ Finset.Ico 1 (J + 1), 30 * A0count * Real.log R := Finset.sum_le_sum hper\n _ = (J : ℝ) * (30 * A0count * Real.log R) := by\n rw [Finset.sum_const, Nat.card_Ico, Nat.add_sub_cancel, nsmul_eq_mul]\n _ ≤ 60 * A0count * (Real.log R) ^ 2 := by\n have hA0 := A0count_pos.le\n have h1 : (J : ℝ) * (30 * A0count * Real.log R)\n ≤ (2 * Real.log R) * (30 * A0count * Real.log R) :=\n mul_le_mul_of_nonneg_right hJle\n (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 30) hA0) hlogRnn)\n have h2 : (2 * Real.log R) * (30 * A0count * Real.log R)\n = 60 * A0count * (Real.log R) ^ 2 := by ring\n rw [h2] at h1\n exact h1\n\nend NguyenRH1\n\nend"},"extra_content":null}]