[{"type":"thinking","thinking":"Now I have the exact structures. The plan for the patches:\n\n**Bn_FAlt_80000_le** (≤ 329,000 → ≤ 332,000):\n- Add `hDL : 2 * DogLegRight 80000 ≤ (3184.2 : ℝ)` with the certified computation:\n  - e^{5/2}+1 ≤ 13.186 (via e²·e^{0.5}: exp_one_pow_le 2 ≤ 7.3890562²... hmm — need e² ≤ 7.3891 (exp_one_pow_le 2 gives e² ≤ 2.7182818286² = 7.3890564 → ≤ 7.3891? 7.3890564 ≤ 7.3891 ✓) and e^{0.5} ≤ 1.6488 (partial sums? — for an UPPER bound on e^{0.5}: the (1−x/16)⁻¹⁶ trick: (1−0.5/16)⁻¹⁶ = (0.96875)⁻¹⁶ = 1.66209?? — compute: (1/0.96875)^16 = (1.032258)^16 = e^{16·0.031749} = e^{0.507984} = 1.662096. So e^{0.5} ≤ 1.6621. Then e^{5/2} ≤ 7.3891·1.6621 = 12.2814?? — hmm — 7.3891·1.6621 = 12.2814. So e^{5/2}+1 ≤ 13.2814?? — vs my earlier 13.186 — the e^{0.5} trick is fat (1.6621 vs true 1.64872). Better: e^{0.5} = e/e^{0.5}... no. e^{0.5} ≤ 1.6488 via partial sums?? — partial sums give LOWER bounds. For e^{0.5} upper: e^{0.5} = e^{1}/e^{0.5}: circular. Or e^{0.5} ≤ (sum+tail): the (1−x/16)⁻¹⁶ at 0.5: 1.6621 (0.8% fat). Hmm — or e^{5/2} = e²·e^{0.5} with e^{0.5} ≤ 1.6621: e^{5/2} ≤ 12.2814. Or: e^{5/2} ≤ e³/e^{0.5}·... — e³ ≤ 20.0856 (exp_one_pow_le 3: e³ ≤ 2.7182818286³ = 20.085537 → ≤ 20.0856); e^{0.5} ≥ 1.64872 (partial sums: 1+0.5+0.125+0.02083+0.002604+0.00026+0.0000217 = 1.648716 ≥ 1.6487): e^{5/2} = e³/e^{0.5} ≤ 20.0856/1.6487 = 12.1826?? — 20.0856/1.6487 = 12.1827 — then e^{5/2}+1 ≤ 13.1827 ✓ TIGHT!! Use the e³/e^{0.5} route: e^{5/2} = e³/e^{0.5} ≤ 20.0856/1.6487 = 12.1827 → e^{5/2}+1 ≤ 13.1827 → charge 13.183.\n  - Then ((√80000+1)/2)·(e^{5/2}+1)/2 ≤ (284/2)·13.183/2 = 142·6.5915 = 936.0?? — 142·6.5915 = 936.0 — hmm: 6.5915·142 = 935.99. So ≤ 936.\n  - IIMaj/4 ≤ 2590.08/4 = 647.52 ✓ (via hII / 4).\n  - So DogLegRight(8e4) ≤ 936 + 8.38 + 647.52 = 1,591.9 → ≤ 1,592. 2·DogLegRight ≤ 3,184.\n  - hDL : 2·DogLegRight 80000 ≤ 3,184.\n- Change hterm4: the old `2 * (Real.exp 4 + 1) * (7 * Hhat (Real.sqrt 80000 + 1) + 3.3) ≤ 2 * (54.6015 + 1) * (7 * 251.46 + 3.3)` — the new Bn has `2 * (Real.exp 4 + 1) * (7 * Hhat (Real.sqrt 80000 + 1))` — the proof's `apply mul_le_mul _ _ (by linarith [hHnn]) (by norm_num); · apply mul_le_mul_of_nonneg_left _ ...; linarith [he4]; · linarith [hH]` — the second bullet `linarith [hH]` bounds `7·Ĥ+3.3 ≤ 7·251.46+3.3` via hH : Ĥ ≤ 251.46 ✓ — for the new form (no 3.3): `linarith [hH]` still bounds 7·Ĥ ≤ 7·251.46 ✓ — so hterm4 needs its STATEMENT changed to the new form `2 * (Real.exp 4 + 1) * (7 * Hhat (Real.sqrt 80000 + 1)) ≤ 2 * (54.6015 + 1) * (7 * 251.46)` — the proof structure is identical (drop the 3.3).\n- Final linarith: [hFAlt, hterm2, hterm4, hDL, hII] with the target 332,000:\n  Total: 4·273.82 + 4·284·112.9637 + 53.408 + (2·55.6015·7·251.46) + 3,184 + 2590.08 + 1 = 1095.28 + 128,326.77 + 53.41 + 195,754.3 + 3,184 + 2590.08 + 1 = 331,004.8 → ≤ 332,000 ✓ (slack 995).\n\n**Bn_FAlt_80000_ge** (≥ 274,000 → ≥ 273,000):\n- Change hterm4 to the new form WITHOUT 3.3: `(2 * (54.5978 + 1) * (7 * 191.85) : ℝ) ≤ 2 * (Real.exp 4 + 1) * (7 * Hhat (Real.sqrt 80000 + 1))` — the proof `linarith [hHlo]` bounds 7·191.85 ≤ 7·Ĥ ✓ (drop the 3.3 from both sides).\n- Add `hDLnn : (0 : ℝ) ≤ 2 * DogLegRight 80000 := by positivity`? — hmm — 2·DogLegRight ≥ 0 via positivity on the DogLegRight form (unfold DogLegRight; positivity — the terms are positive: √n+1 > 0, e^{5/2}+1 > 0, 8.38 > 0, IIMaj/4 ≥ 0 (IIMaj_nonneg) ✓ positivity might need help with IIMaj_nonneg — provide via `by unfold DogLegRight; have h := IIMaj_nonneg 80000; positivity`?? — positivity knows IIMaj ≥ 0? — IIMaj is a def — positivity can't unfold it — but IIMaj_nonneg exists: `have hIIMnn := IIMaj_nonneg 80000` then `unfold DogLegRight; positivity` — hmm positivity with the hIIMnn in context... — or `by unfold DogLegRight; have hIIMnn := IIMaj_nonneg 80000; positivity`. Might work; fallback: compute the nonneg explicitly via mul_nonneg/add_nonneg with IIMaj_nonneg.\n- Final linarith: [hFAlt0, hterm2, hterm4-new, hII, hDLnn] with the target 273,000:\n  New total-lower: 0 + 4·283·110.84 + (2·55.5978·7·191.85) + 0 + 0 + 40·1.3352 + 1 + 0 — the old proof's total was ≥ 274,000 — the new total = old − (the 3.3-part of the old hterm4-lo = 2·55.5978·3.3·... — wait — the old hterm4-lo : `2 * (54.5978 + 1) * (7 * 191.85 + 3.3)` = 2·55.5978·1346.25 = 111.1956·1346.25 = 149,723.5. The new hterm4-lo : 2·55.5978·7·191.85 = 111.1956·1342.95 = 149,356.7. Difference: −366.8. So new total-lower = 274,000 − 366.8 = 273,633.2 → ≥ 273,000 ✓ (slack 633).\n\n**Bn_FAlt_100000_le** (≤ 350,000 → ≤ 352,000):\n- Add hDL : 2·DogLegRight 100000 ≤ 3,564 (from (318/2)·13.183/2 + 8.38 + 2900.16/4 = 159·6.5915+8.38+725.04 = 1,048.4+8.38+725.04 = 1,781.8 → 1,782 → 2·DogLegRight = 3,564).\n- Change hterm4 similarly (drop 3.3 from both sides).\n- Final linarith target 352,000: 1,245.8 + 4·318·114.0656 + 53.41 + (2·55.6015·7·255.05) + 3,564 + 2900.16 + 1 = 1245.8 + 145,091.4 + 53.41 + 198,510.7 + 3,564 + 2900.16 + 1 = 351,366.5 → ≤ 352,000 ✓ (slack 634).\n\nThen marginTilde_FAlt_80000_pos and marginTilde_FAlt_80000_lt_100000 — those proofs are just `rw [marginTilde_eq_sub]; have hS := sInftyLower_80000_lo; have hB := Bn_FAlt_80000_le; linarith` and the lt version with up/ge bounds — they only reference the Bn bounds (not internals) ✓ so they work once the bounds are re-established with updated targets:\n- marginTilde_FAlt_80000_pos : 0 < margin(8e4) — with Bn ≤ 332,000 and sInftyLower ≥ 336,000: margin ≥ 4,001 > 0 ✓ (the theorem statement is 0 < marginTilde FAlt 80000 — proof: rw [marginTilde_eq_sub]; linarith [sInftyLower_80000_lo, Bn_FAlt_80000_le] ✓ works with the new bound).\n- marginTilde_FAlt_80000_lt_100000 : margin(8e4) < margin(10⁵) — margin(8e4) ≤ 339,000 − 273,000 + 1 = 66,001; margin(10⁵) ≥ 432,500 − 352,000 + 1 = 80,501 ✓ 66,001 < 80,501 ✓ works.\n\nAlso — the doc comments on the Bn evaluation theorems (the \"certified ≈ ...\" notes) need updating for honesty, and the margin theorems' comments.\n\nLet me now write the patches. First the e^{5/2}+1 ≤ 13.183 certification (shared by all three? — 8e4 and 10⁵ use the same DogLegRight form with different √n bounds — the e^{5/2}+1 bound is shared ✓ — make it a standalone lemma at the top of §4).\n\nFor e^{5/2}+1 ≤ 13.183: e^{5/2} = e³/e^{0.5}: e³ ≤ 20.0856 (exp_one_pow_le 3 : e^1³ ≤ 2.7182818286³ — compute 2.7182818286³ = 20.085537 → ≤ 20.0856 ✓); e^{0.5} ≥ 1.6487 (partial sums Σ₀⁷(0.5^i/i!) = 1+0.5+0.125+0.020833+0.002604+0.000260+0.0000217+0.00000155 = 1.648720 ≥ 1.6487 ✓): e^{5/2} ≤ 20.0856/1.6487 = 12.1827?? — compute: 20.0856/1.6487 = 12.1827?? — 1.6487·12.1827 = 20.08566 ≈ 20.0856 ✓ — hmm: 20.0856/1.6487 = 12.18275. So e^{5/2} ≤ 12.1828 → e^{5/2}+1 ≤ 13.1828 → charge 13.183 ✓ (slack 0.0005 vs true 12.1825 — tight but the certified form 20.0856/1.6487 = 12.18275 ≤ 13.183 ✓ norm_num-checkable).\n\nHmm wait — e³/e^{0.5}: e^{5/2} = e^{3 − 1/2} = e³/e^{0.5} ✓. div_le_div₀ (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hdb : d ≤ b) : a/b ≤ c/d — with a := exp 3, b := exp 0.5, c := 20.0856, d := 1.6487: exp 3/exp 0.5 ≤ 20.0856/1.6487 ✓ then 20.0856/1.6487 ≤ 13.1828 ⟺ 20.0856 ≤ 13.1828·1.6487 = 21.7304?? — hmm — that's the WRONG direction!! I want 20.0856/1.6487 ≤ 13.1828?? — no: 20.0856/1.6487 = 12.18275 ≤ 13.1828 ✓ trivially (12.18 ≤ 13.18) ✓ — oh wait, I conflated: 20.0856/1.6487 ≈ 12.18, and I want e^{5/2} ≤ 12.1828?? — no — e^{5/2} ≤ 20.0856/1.6487 = 12.18275 — and then e^{5/2}+1 ≤ 13.18275 → charge 13.183 ✓.\n\nHmm — actually let me redo: e^{5/2} TRUE value = 12.182494. The certified route gives ≤ 20.0856/1.6487 = 12.18275 — slack 0.00026 over true ✓ tight ✓. Charge e^{5/2}+1 ≤ 13.183 ✓.\n\nWait — but is the e³/e^{0.5} division even the right algebra? e^{5/2} = e^{2.5} = e^{3−0.5} = e³·e^{−0.5} = e³/e^{0.5} ✓. And the bound: e³ ≤ 20.0856 (upper ✓), e^{0.5} ≥ 1.6487 (lower ✓): e³/e^{0.5} ≤ 20.0856/1.6487 ✓ (numerator upper, denominator lower ✓ div_le_div₀ ✓).\n\nThen the DogLegRight bound:\n```\ntheorem dogLegRight_le (n : ℝ) (hII : IIMaj n ≤ (4.56 * (Real.sqrt n + 1) * 2)) (hT : Real.sqrt n + 1 ≤ c) :\n    2 * DogLegRight n ≤ (c / 2) * (13.183 / 2) * 2 + 16.76 + 4.56 * (Real.sqrt n + 1) * 2 / 2 * ... \n```\nhmm — let me just inline the bounds in each proof (simpler than a shared lemma). The pattern:\n```\n  have he52 : Real.exp (5 / 2) + 1 ≤ (13.183 : ℝ) := by\n    have h3 : Real.exp 3 ≤ (20.0856 : ℝ) := by\n      rw [show Real.exp 3 = Real.exp 1 ^ 3 by rw [← Real.exp_nat_mul]; congr 1; norm_num]\n      exact le_trans (exp_one_pow_le 3) (by norm_num)\n    have h05 : (1.6487 : ℝ) ≤ Real.exp 0.5 :=\n      le_trans (by norm_num [Finset.sum_range_succ])\n        (Real.sum_le_exp_of_nonneg (by norm_num : (0 : ℝ) ≤ 0.5) 7)\n    have h1 : Real.exp (5 / 2) = Real.exp 3 / Real.exp 0.5 := by\n      rw [show (5 / 2 : ℝ) = 3 - 0.5 by norm_num, Real.exp_sub]\n    rw [h1]\n    calc Real.exp 3 / Real.exp 0.5 ≤ 20.0856 / 1.6487 :=\n        div_le_div₀ (by norm_num) h3 (by norm_num) h05\n      _ ≤ 13.183 - 1 := by norm_num\n```\nHmm — `Real.exp 3 = Real.exp 1 ^ 3` — via exp_nat_mul (like exp_eleven/exp_five/exp_four in the file ✓ — there's a pattern: `rw [← Real.exp_nat_mul 1 3]; congr 1; norm_num` ✓ same as the existing exp_five/exp_four lemmas ✓).\n\nThen div_le_div₀ (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hdb : d ≤ b) : a/b ≤ c/d — with a := exp 3, b := exp 0.5, c := 20.0856, d := 1.6487: hc : 0 ≤ 20.0856 ✓ norm_num; hac : exp 3 ≤ 20.0856 ✓ h3; hd : 0 < 1.6487 ✓ norm_num; hdb : 1.6487 ≤ exp 0.5 ✓ h05 ✓. Then `20.0856/1.6487 ≤ 13.183 − 1 = 12.183` ⟺ 20.0856 ≤ 12.183·1.6487 = 20.0861 ⟺ norm_num ✓ (12.183·1.6487 = 20.08612 ≥ 20.0856 ✓ slack 0.0005).\n\nHmm wait — I want e^{5/2}+1 ≤ 13.183: e^{5/2} ≤ 12.183 ⟹ +1 ≤ 13.183 ✓ — the last calc step should be `20.0856/1.6487 ≤ 12.183 := by norm_num` then `linarith` for the +1. Let me restructure:\n```\n  have he52 : Real.exp (5 / 2) ≤ (12.183 : ℝ) := by\n    ... (h3, h05)\n    rw [h1]\n    calc Real.exp 3 / Real.exp 0.5 ≤ 20.0856 / 1.6487 :=\n        div_le_div₀ (by norm_num) h3 (by norm_num) h05\n      _ ≤ 12.183 := by norm_num\n  -- then: (284/2)·(e^{5/2}+1)/2 ≤ ...\n```\nAnd DogLegRight ≤ ...:\n```\n  have hDL : 2 * DogLegRight 80000 ≤ (3184 : ℝ) := by\n    unfold DogLegRight\n    have hT2 : Real.sqrt 80000 + 1 ≤ 284 := hT\n    have hII4 : IIMaj 80000 / 4 ≤ (2590.08 : ℝ) / 4 := by linarith [hII]\n    have h1 : ((Real.sqrt 80000 + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 ≤\n        (284 / 2) * (13.183 : ℝ) / 2 := by\n      apply mul_le_mul (div_le_div_of_nonneg_right hT2 (by norm_num)) _\n        (by positivity) (by positivity)\n      · apply div_le_div_of_nonneg_right _ (by norm_num)\n        linarith [he52]\n    linarith [h1, hII4]\n```\nHmm — messy. Let me simplify: DogLegRight(8e4) = ((√80000+1)/2)·(e^{5/2}+1)/2 + 8.38 + IIMaj(8e4)/4. \n- ((√80000+1)/2)·(e^{5/2}+1)/2 ≤ (284/2)·(13.183/2): via mul_le_mul ((√80000+1)/2 ≤ 284/2) ((e^{5/2}+1)/2 ≤ 13.183/2) — each via div_le_div_of_nonneg_right (by norm_num : 0 ≤ 2) with hT2 and (e^{5/2}+1 ≤ 13.183 by linarith [he52]).\n- 8.38 exact.\n- IIMaj/4 ≤ 2590.08/4 = 647.52 by linarith [hII].\nTotal ≤ 936 + 8.38 + 647.52 = 1,591.9 → 2·DogLegRight ≤ 3,183.8 → 3,184.\n\nThe unfold DogLegRight exposes the form; then the bound. Since `simp only [Bn]` already ran, the goal's DogLegRight appears as `2 * DogLegRight 80000` in the unfolded Bn — the final linarith uses hDL.\n\nLet me write the three patches now. For the e^{5/2}+1 bound, make it a standalone theorem before §4 (shared). Let me also check the exp_nat_mul pattern from the existing exp_eleven/exp_five/exp_four (lines 66-79):\n```\ntheorem exp_eleven : Real.exp 11 = Real.exp 1 ^ 11 := by\n  rw [← Real.exp_nat_mul 1 11]\n  congr 1\n  norm_num\n```\n✓ same pattern for exp_three.\n\nOK — let me write the patches. Start with the shared e^{5/2} bound + the three proof patches.","signature":""},{"type":"text","text":"Now patching the three Bn evaluation proofs with certified DogLegRight bounds + the shared e^{5/2} certification."},{"type":"tool_use","id":"Read_624","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","offset":60,"limit":30},"extra_content":null}]