← Session 25 — ii′-5 R-bound: design + level-2 EM disc bound
thinking (2832 chars)
Hmm — the Field/Basic.lean doesn't have the a/c ≤ a/b lemma by that grep (maybe different file). The strong toolbox: `gcongr` is the modern way — Zeta23 uses it. In the sphere bound, replace the calc step:
```lean
calc ‖scaledZeta t w‖ / ‖collProd ht w‖
≤ (9 / 4) * |t| ^ 2 / gapD t ^ collMultSum ht := by
gcongr
· exact norm_nonneg _ -- 0 ≤ ‖F‖? gcongr side conditions
· exact hF
· exact hPpos
· exact hP
```
gcongr for a/d ≤ b/c with pattern _/_ ≤ _/_: it reduces to a ≤ b and c ≤ d plus side conditions 0 ≤ a? 0 < c... gcongr discharges what it can and leaves the rest. Let me structure: `gcongr` then supply remaining goals. Uncertain about exact side goals — safer to do it in two explicit steps:
Step 1: ‖F‖/‖P‖ ≤ ‖F‖/δ^M: numerator fixed ≥ 0, 0 < δ^M ≤ ‖P‖: the lemma `div_le_div_of_nonneg_left`... not found. Current name? Let me search for `div_le_div_of_le_left` or use div_le_div_iff: ‖F‖/‖P‖ ≤ ‖F‖/δ^M ⟺ (div_le_div_iff (hb : 0 < b) (hc : 0 < c)?) `div_le_div_iff (hb : 0 < b) (hd : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b` ✓ that exists I'm fairly sure. So:
‖F‖/‖P‖ ≤ X/δ^M ⟺ ‖F‖·δ^M ≤ X·‖P‖: from ‖F‖ ≤ X and δ^M ≤ ‖P‖ and nonneg: nlinarith [hF, hP, hPpos, norm_nonneg, ...] — products of ≤: nlinarith can do: (X − ‖F‖)·(‖P‖) + ‖F‖·(‖P‖ − δ^M) ≥ 0 etc. nlinarith [hF, hP, hPpos, hXnonneg, hnormP] with hXnonneg : 0 ≤ (9/4)|t|². Let me just do:
```lean
have hXnn : (0:ℝ) ≤ (9 / 4) * |t| ^ 2 := by positivity
have hnormP : 0 < ‖collProd ht w‖ := hPpos.trans_le hP
rw [div_le_div_iff hnormP hPpos] -- hmm: div_le_div_iff (0<b) (0<d) : a/b ≤ c/d ↔ a*d ≤ c*b
-- goal: ‖F‖ * δ^M ≤ X * ‖P‖
nlinarith [hF, hP, hPpos, hXnn, norm_nonneg (scaledZeta t w)]
```
div_le_div_iff signature: `div_le_div_iff (hb : 0 < b) (hd : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b` ✓ (exists in Field/Basic usually). With a := ‖F‖, b := ‖P‖, c := X, d := δ^M: goal becomes ‖F‖ * δ^M ≤ X * ‖P‖. nlinarith: X·‖P‖ − ‖F‖·δ^M = X·(‖P‖ − δ^M) + (X − ‖F‖)·δ^M ≥ 0: X ≥ 0, ‖P‖−δ^M ≥ 0, X−‖F‖ ≥ 0, δ^M > 0: products needed: nlinarith [mul_nonneg hXnn (sub_nonneg.mpr hP), mul_nonneg (sub_nonneg.mpr hF) hPpos.le] — provide the product hints explicitly. ✓ robust.
5. **Metric.mem_sphere_iff_norm unknown** — the lemma is just `mem_sphere_iff_norm` (root namespace, seen in AbsMax.lean usage `mem_sphere_iff_norm.1`). Fix: `rw [mem_sphere_iff_norm, sub_zero] at hz`. In DiskBound they used `rw[mem_sphere_iff_norm, sub_zero] at hw` ✓ root namespace.
Also check: did the earlier `simp only [hfin, ↓reduceDIte]` + `rw [dif_neg hw]` + congrArg + prod_congr compile in Cf_eq_div_collProd? No error reported for it — ✓ compiled. And Cf_eq_zeroFactor_div's fix (remove trailing exact since congr 1 closes) — the error at 707 was "No goals" — let me view the current text around 700-712 to fix precisely.
Fixing: `gcongr`-free two-step div bound, root-namespace `mem_sphere_iff_norm`, forward `prod_pow_eq_pow_sum`, and the zeroFactor congr.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean",
"offset": 694,
"limit": 18
}
result 2026-08-22 21:45:56Z → 2026-08-26 03:24:44Z
{
"type": "text",
"file": {
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"content": " linarith\n have hsf : (4.56 * ((eA + eB) + (s + 1) * (-a * eA - b * eB))) / s =\n 4.56 * ((eA + eB) / s + (s + 1) * (-a * eA - b * eB) / s) := by\n field_simp [hs0.ne']\n rw [hsf]\n have hsf2 : (s + 1) * (-a * eA - b * eB) / s = -((s + 1) * (a * eA + b * eB) / s) := by\n field_simp [hs0.ne']\n ring\n rw [hsf2]\n linarith\n -- step 2: factor out b\u00b2 and b/s termwise\n have hstep2 :\n 4.56 * ((s + 1) * (a ^ 2 * eA + b ^ 2 * eB) +\n (s + 1) * (a * eA + b * eB) / s) \u2264\n 4.56 * (s + 1) * (b ^ 2 + b / s) * (eA + eB) := by\n have hsq : a ^ 2 \u2264 b ^ 2 := by nlinarith [hab, hapos]\n have h1 : a ^ 2 * eA + b ^ 2 * eB \u2264 b ^ 2 * (eA + eB) := by\n nlinarith [mul_le_mul_of_nonneg_right hsq heA.le]\n have h2 : (a * eA + b * eB) / s \u2264 b * (eA + eB) / s := by\n have h := add_le_add (mul_le_mul_of_nonneg_right hab heA.le) (le_refl (b * eB))\n have h' : a * eA + b * eB \u2264 b * (eA + eB) := by nlinarith [h]\n gcongr\n have m1 : (s + 1) * (a ^ 2 * eA + b ^ 2 * eB) \u2264 (s + 1) * (b ^ 2 * (eA + eB)) :=\n mul_le_mul_of_nonneg_left h1 (by linarith)\n have m2 : (s + 1) * (a * eA + b * eB) / s \u2264 (s + 1) * (b * (eA + eB)) / s := by\n have h3 := mul_le_mul_of_nonneg_left h2 (show (0 : \u211d) \u2264 s + 1 by linarith)\n rwa [\u2190 mul_div_assoc, \u2190 mul_div_assoc] at h3\n have e3 : 4.56 * (s + 1) * (b ^ 2 + b / s) * (eA + eB) =\n 4.56 * ((s + 1) * (b ^ 2 * (eA + eB)) + (s + 1) * (b * (eA + eB)) / s) := by\n field_simp [hs0.ne']\n rw [e3]\n exact mul_le_mul_of_nonneg_left (add_le_add m1 m2) (by norm_num)\n -- step 3: the three-factor split\n have hstep3 :\n 4.56 * (s + 1) * (b ^ 2 + b / s) * (eA + eB) =\n 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) := by\n have heB' : eB = eA * Real.exp (-s / 2) := by\n rw [heBdef, heAdef, \u2190 Real.exp_add]\n congr 1\n rw [hadef, hbdef]\n ring\n rw [heB']\n field_simp [hs0.ne']\n -- step 4: the three monotone factors at s = \u221a20\n have hu1 : (s + 1) / s \u2264 (1.2237 : \u211d) := by\n have h1 : (1 : \u211d) / s \u2264 1 / 4.4721 := one_div_le_one_div_of_le (by norm_num) hs4\n have h2 : (1 : \u211d) / 4.4721 \u2264 0.2237 := by\n rw [div_le_iff\u2080 (by norm_num : (0 : \u211d) < 4.4721)]\n norm_num\n have h3 : (s + 1) / s = 1 + 1 / s := by field_simp [hs0.ne']\n rw [h3]\n linarith\n have hu2fac : (b ^ 2 * s + b) * eA \u2264 (0.2124 : \u211d) := by\n have hAid : iimA = a := by unfold iimA; rw [\u2190 hLdef, \u2190 hadef]\n have hBid : iimB = b := by unfold iimB; rw [\u2190 hLdef, \u2190 hbdef]\n have hanti := iim_u2_antitone\n rw [hAid, hBid] at hanti\n have hu2le : (b ^ 2 * s + b) * Real.exp (-a * s) \u2264\n (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) :=\n hanti (Set.mem_Ici.mpr (le_refl _)) (Set.mem_Ici.mpr hs) hs\n have hu2val : (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) \u2264\n (0.2124 : \u211d) := by\n have hb2up : b ^ 2 \u2264 (1.92182 : \u211d) := by nlinarith [hb, hbpos]\n have hexp : Real.exp (-a * Real.sqrt 20) \u2264 (0.0212766 : \u211d) := by\n have ht : (3.9635 : \u211d) \u2264 a * Real.sqrt 20 := by\n calc (3.9635 : \u211d) \u2264 0.8862943606 * 4.4721 := by norm_num\n _ \u2264 a * Real.sqrt 20 :=\n mul_le_mul ha hsqrtlo (by norm_num) hapos.le\n have hsum : (47 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) := by\n have h1 := Real.sum_le_exp_of_nonneg (x := (3.9635 : \u211d)) (by norm_num) 7\n have h2 : Real.exp (3.9635 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) :=\n Real.exp_le_exp.mpr ht\n have h3 : (47 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7,\n (3.9635 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n norm_num [Finset.sum_range_succ]\n linarith\n rw [show -a * Real.sqrt 20 = -(a * Real.sqrt 20) by ring, Real.exp_neg]\n calc (Real.exp (a * Real.sqrt 20))\u207b\u00b9 \u2264 1 / 47 := by\n rw [\u2190 one_div]\n exact one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 47) hsum\n _ \u2264 0.0212766 := by\n rw [div_le_iff\u2080 (by norm_num : (0 : \u211d) < 47)]; norm_num\n have hpoly : b ^ 2 * Real.sqrt 20 + b \u2264 (9.9811 : \u211d) := by\n have h1 : b ^ 2 * Real.sqrt 20 \u2264 1.92182 * 4.4722 :=\n mul_le_mul hb2up hsqrthi (Real.sqrt_nonneg 20) (by norm_num)\n linarith\n exact le_trans (mul_le_mul hpoly hexp (Real.exp_pos _).le (by norm_num))\n (by norm_num)\n calc (b ^ 2 * s + b) * eA = (b ^ 2 * s + b) * Real.exp (-a * s) := rfl\n _ \u2264 (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) := hu2le\n _ \u2264 0.2124 := hu2val\n have hu3 : 1 + Real.exp (-s / 2) \u2264 (1.1099 : \u211d) := by\n have hexp : Real.exp (-s / 2) \u2264 (0.10982 : \u211d) := by\n have ht : (2.23605 : \u211d) \u2264 s / 2 := by linarith\n have hsum : (9.1064 : \u211d) \u2264 Real.exp (s / 2) := by\n have h1 := Real.sum_le_exp_of_nonneg (x := (2.23605 : \u211d)) (by norm_num) 6\n have h2 : Real.exp (2.23605 : \u211d) \u2264 Real.exp (s / 2) := Real.exp_le_exp.mpr ht\n have h3 : (9.1064 : \u211d) \u2264 \u2211 i \u2208 Finset.range 6,\n (2.23605 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n norm_num [Finset.sum_range_succ]\n linarith\n rw [show -s / 2 = -(s / 2) by ring, Real.exp_neg]\n calc (Real.exp (s / 2))\u207b\u00b9 \u2264 1 / 9.1064 := by\n rw [\u2190 one_div]\n exact one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 9.1064) hsum\n _ \u2264 0.10982 := by\n rw [div_le_iff\u2080 (by norm_num : (0 : \u211d) < 9.1064)]; norm_num\n linarith\n -- final assembly\n have hpos1 : (0 : \u211d) \u2264 (b ^ 2 * s + b) * eA :=\n mul_nonneg (by nlinarith [hbpos, hs0, sq_nonneg b]) heA.le\n have hpos2 : (0 : \u211d) \u2264 1 + Real.exp (-s / 2) := by positivity\n have hpos3 : (0 : \u211d) \u2264 (s + 1) / s := by positivity\n have hchain : 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) \u2264\n 4.56 * 1.2237 * 0.2124 * 1.1099 := by\n have h1 : ((s + 1) / s) * ((b ^ 2 * s + b) * eA) \u2264 1.2237 * 0.2124 :=\n mul_le_mul hu1 hu2fac hpos1 (by norm_num)\n have h2 : ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) \u2264\n 1.2237 * 0.2124 * 1.1099 :=\n mul_le_mul h1 hu3 hpos2 (by positivity)\n have h3 := mul_le_mul_of_nonneg_left h2 (by norm_num : (0 : \u211d) \u2264 4.56)\n have e : (4.56 : \u211d) * (((s + 1) / s) * ((b ^ 2 * s + b) * eA) *\n (1 + Real.exp (-s / 2))) =\n 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) := by ring\n have e' : (4.56 : \u211d) * (1.2237 * 0.2124 * 1.1099) = 4.56 * 1.2237 * 0.2124 * 1.1099 := by ring\n rwa [e, e'] at h3\n calc 4.56 * (2 * (-a * eA - b * eB) + (s + 1) * (a ^ 2 * eA + b ^ 2 * eB)) -\n (4.56 * ((eA + eB) + (s + 1) * (-a * eA - b * eB))) / s\n \u2264 4.56 * (s + 1) * (b ^ 2 + b / s) * (eA + eB) := le_trans hstep1 hstep2\n _ = 4.56 * ((s + 1) / s) * ((b ^ 2 * s + b) * eA) * (1 + Real.exp (-s / 2)) := hstep3\n _ \u2264 4.56 * 1.2237 * 0.2124 * 1.1099 := hchain\n _ < 2 := by norm_num\n\n/-! ### Assembly: regularity bundles -/\n\n/-- [PROVED session 011] The blocks are C\u00b2 on (\u22121, \u221e) (all\nL1/L2-singularities lie at T \u2264 \u22121). -/\ntheorem contDiffOn_L1 : ContDiffOn \u211d 2 (fun t => Real.log (t + 2)) (Set.Ioi (-1)) :=\n (contDiffOn_id.add contDiffOn_const).log (fun x hx => by\n have h : -1 < x := hx\n exact ne_of_gt (by linarith : (0 : \u211d) < x + 2))\n\ntheorem contDiffOn_L2 :\n ContDiffOn \u211d 2 (fun t => Real.log (Real.log (t + 2))) (Set.Ioi (-1)) :=\n contDiffOn_L1.log (fun x hx => by\n have h : -1 < x := hx\n exact ne_of_gt (Real.log_pos (by linarith : (1 : \u211d) < x + 2)))\n\ntheorem contDiffOn_L1sq :\n ContDiffOn \u211d 2 (fun t => Real.log (t + 2) ^ 2) (Set.Ioi (-1)) :=\n contDiffOn_L1.pow 2\n\ntheorem contDiffOn_L1L2 : ContDiffOn \u211d 2\n (fun t => Real.log (t + 2) * Real.log (Real.log (t + 2))) (Set.Ioi (-1)) :=\n contDiffOn_L1.mul contDiffOn_L2\n\ntheorem contDiffOn_L2sq :\n ContDiffOn \u211d 2 (fun t => Real.log (Real.log (t + 2)) ^ 2) (Set.Ioi (-1)) :=\n contDiffOn_L2.pow 2\n\ntheorem contDiffOn_TL1 :\n ContDiffOn \u211d 2 (fun t => t * Real.log (t + 2)) (Set.Ioi (-1)) :=\n contDiffOn_id.mul contDiffOn_L1\n\ntheorem contDiffOn_TL2 :\n ContDiffOn \u211d 2 (fun t => t * Real.log (Real.log (t + 2))) (Set.Ioi (-1)) :=\n contDiffOn_id.mul contDiffOn_L2\n\n/-- [PROVED session 011] T(m) = \u221am + 1 is C\u00b2 on (20, \u221e) and maps it\ninto (\u22121, \u221e). -/\ntheorem contDiffOn_sqrt_shift :\n ContDiffOn \u211d 2 (fun m => Real.sqrt m + 1) (Set.Ioi 20) :=\n (contDiffOn_id.sqrt (fun x hx => by\n have h : 20 < x := hx\n exact ne_of_gt (by linarith : (0 : \u211d) < x))).add contDiffOn_const",
"numLines": 175,
"startLine": 1547,
"totalLines": 2793
}
}
raw content_json (database row)
[{"type":"thinking","thinking":"Hmm — the Field/Basic.lean doesn't have the a/c ≤ a/b lemma by that grep (maybe different file). The strong toolbox: `gcongr` is the modern way — Zeta23 uses it. In the sphere bound, replace the calc step:\n\n```lean\n calc ‖scaledZeta t w‖ / ‖collProd ht w‖\n ≤ (9 / 4) * |t| ^ 2 / gapD t ^ collMultSum ht := by\n gcongr\n · exact norm_nonneg _ -- 0 ≤ ‖F‖? gcongr side conditions\n · exact hF\n · exact hPpos\n · exact hP\n```\ngcongr for a/d ≤ b/c with pattern _/_ ≤ _/_: it reduces to a ≤ b and c ≤ d plus side conditions 0 ≤ a? 0 < c... gcongr discharges what it can and leaves the rest. Let me structure: `gcongr` then supply remaining goals. Uncertain about exact side goals — safer to do it in two explicit steps:\n\nStep 1: ‖F‖/‖P‖ ≤ ‖F‖/δ^M: numerator fixed ≥ 0, 0 < δ^M ≤ ‖P‖: the lemma `div_le_div_of_nonneg_left`... not found. Current name? Let me search for `div_le_div_of_le_left` or use div_le_div_iff: ‖F‖/‖P‖ ≤ ‖F‖/δ^M ⟺ (div_le_div_iff (hb : 0 < b) (hc : 0 < c)?) `div_le_div_iff (hb : 0 < b) (hd : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b` ✓ that exists I'm fairly sure. So:\n‖F‖/‖P‖ ≤ X/δ^M ⟺ ‖F‖·δ^M ≤ X·‖P‖: from ‖F‖ ≤ X and δ^M ≤ ‖P‖ and nonneg: nlinarith [hF, hP, hPpos, norm_nonneg, ...] — products of ≤: nlinarith can do: (X − ‖F‖)·(‖P‖) + ‖F‖·(‖P‖ − δ^M) ≥ 0 etc. nlinarith [hF, hP, hPpos, hXnonneg, hnormP] with hXnonneg : 0 ≤ (9/4)|t|². Let me just do:\n\n```lean\n have hXnn : (0:ℝ) ≤ (9 / 4) * |t| ^ 2 := by positivity\n have hnormP : 0 < ‖collProd ht w‖ := hPpos.trans_le hP\n rw [div_le_div_iff hnormP hPpos] -- hmm: div_le_div_iff (0<b) (0<d) : a/b ≤ c/d ↔ a*d ≤ c*b\n -- goal: ‖F‖ * δ^M ≤ X * ‖P‖\n nlinarith [hF, hP, hPpos, hXnn, norm_nonneg (scaledZeta t w)]\n```\ndiv_le_div_iff signature: `div_le_div_iff (hb : 0 < b) (hd : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b` ✓ (exists in Field/Basic usually). With a := ‖F‖, b := ‖P‖, c := X, d := δ^M: goal becomes ‖F‖ * δ^M ≤ X * ‖P‖. nlinarith: X·‖P‖ − ‖F‖·δ^M = X·(‖P‖ − δ^M) + (X − ‖F‖)·δ^M ≥ 0: X ≥ 0, ‖P‖−δ^M ≥ 0, X−‖F‖ ≥ 0, δ^M > 0: products needed: nlinarith [mul_nonneg hXnn (sub_nonneg.mpr hP), mul_nonneg (sub_nonneg.mpr hF) hPpos.le] — provide the product hints explicitly. ✓ robust.\n\n5. **Metric.mem_sphere_iff_norm unknown** — the lemma is just `mem_sphere_iff_norm` (root namespace, seen in AbsMax.lean usage `mem_sphere_iff_norm.1`). Fix: `rw [mem_sphere_iff_norm, sub_zero] at hz`. In DiskBound they used `rw[mem_sphere_iff_norm, sub_zero] at hw` ✓ root namespace.\n\nAlso check: did the earlier `simp only [hfin, ↓reduceDIte]` + `rw [dif_neg hw]` + congrArg + prod_congr compile in Cf_eq_div_collProd? No error reported for it — ✓ compiled. And Cf_eq_zeroFactor_div's fix (remove trailing exact since congr 1 closes) — the error at 707 was \"No goals\" — let me view the current text around 700-712 to fix precisely.","signature":""},{"type":"text","text":"Fixing: `gcongr`-free two-step div bound, root-namespace `mem_sphere_iff_norm`, forward `prod_pow_eq_pow_sum`, and the zeroFactor congr."},{"type":"tool_use","id":"Read_70","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","offset":694,"limit":18},"extra_content":null}]