[{"type":"thinking","thinking":"`div_le_div₀ (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hdb : d ≤ b) : a / b ≤ c / d` — the four-argument form. For h48: `Real.exp (4/3) / (2 * (Real.log 2 - 4/9)) ≤ 3.8073 / (2 * 0.2487)`: div_le_div₀ (hc : 0 ≤ 3.8073) (hac : exp(4/3) ≤ 3.8073) (hd : 0 < 2·0.2487) (hdb : 2·0.2487 ≤ 2·(log2−4/9)) ✓.\n\nAnd the nonneg: `intervalIntegral.integral_nonneg (hab : a ≤ b) (hf : ∀ u, u ∈ Icc a b → 0 ≤ f u)` ✓.\n\nNow the fixes:\n1. hlog2 : (0.6931471803 : ℝ) ≤ Real.log 2 := Real.log_two_gt_d9.le\n2. div_le_div → div_le_div₀ with the reordered args: div_le_div₀ (by norm_num : (0:ℝ) ≤ 3.8073) exp_four_thirds_le (by norm_num : (0:ℝ) < 2 * 0.2487) (by linarith : 2 * 0.2487 ≤ 2 * (Real.log 2 - 4 / 9)).\n3. The ∫-nonneg for mul_le_mul: provide via intervalIntegral.integral_nonneg.\n4. The ring failure at ~676 (the e1's `push_cast; ring` in h3b's h3l2 — the warning says push_cast did nothing and ring failed? The first error block shows a `ring` failure with ring_nf suggestion — need the exact location. It's probably h3b's e1: `rw [show ((1:ℝ)/8) = (2^3:ℝ)⁻¹ by norm_num, Real.log_inv, Real.log_pow]; push_cast; ring` — the goal: `(-3 * Real.log 2) = Real.log ((2^3)⁻¹)` → after log_inv: `−(Real.log (2^3))` → after log_pow: `−(3·Real.log 2)` — wait: Real.log_pow : Real.log (x^n) = ↑n * Real.log x — so Real.log ((2^3)⁻¹) = −Real.log (2^3) = −(3·Real.log 2) — the goal: `−3·Real.log 2 = −(3·Real.log 2)`?? — push_cast + ring should close... the push_cast warning at 474 was earlier (hw1's). The ring failure location — let me get it precisely after fixing the other three. Actually — maybe it's the he3c's `congr 1; ring` at 654 or h3l2's e1's push_cast; ring at 651/675 — the `(2^3:ℝ)⁻¹` — hmm the show `((1:ℝ)/8) = (2^3:ℝ)⁻¹` by norm_num — is (2^3)⁻¹ = 1/8 by norm_num ✓. After the rewrites: log((2^3)⁻¹) → −log(2^3) → −(3·log 2) — wait Real.log_inv : Real.log x⁻¹ = −Real.log x ✓ then Real.log_pow : Real.log (2^3) = 3·Real.log 2 ✓ — so the goal: `−3·Real.log 2 = −(3·Real.log 2)`?? hmm — wait, e1's goal is `(−3 * Real.log 2) = Real.log (1/8)` — after rw [show (1/8) = (2^3)⁻¹]: `−3·log2 = log((2^3)⁻¹)` → log_inv: `−3·log2 = −log(2^3)` → log_pow: `−3·log2 = −(↑3·log2)` — the ↑3 is a NAT CAST (log_pow has n : ℕ: `↑n * log x`) — so RHS is `−((3:ℕ):ℝ)·log2` vs LHS `−3·log2` (3 as a real literal) — push_cast normalizes ((3:ℕ):ℝ) → 3 ✓ then ring: −3·log2 = −3·log2 ✓ — should close. The push_cast \"does nothing\" warning suggests the cast wasn't there — and the ring failure is elsewhere. Let me just recompile after the three fixes and see what remains.","signature":""},{"type":"text","text":"`div_le_div₀` and `intervalIntegral.integral_nonneg` are the names. Applying all fixes."},{"type":"tool_use","id":"Edit_298","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"  -- step 4: the certified assembly\n  have hlog2 : (0.6931471803 : ℝ) ≤ Real.log 2 := Real.log_two_gt_d9\n  have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) ≤ (7.6545 : ℝ) := by\n    have hcge := cdog_gt\n    have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) ≤ 3.8073 / (2 * 0.2487) :=\n      div_le_div exp_four_thirds_le (by linarith) (by norm_num) (by positivity)","new_string":"  -- step 4: the certified assembly\n  have hlog2 : (0.6931471803 : ℝ) ≤ Real.log 2 := Real.log_two_gt_d9.le\n  have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) ≤ (7.6545 : ℝ) := by\n    have hcge := cdog_gt\n    have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) ≤ 3.8073 / (2 * 0.2487) :=\n      div_le_div₀ (by norm_num : (0 : ℝ) ≤ 3.8073) exp_four_thirds_le\n        (by norm_num : (0 : ℝ) < 2 * 0.2487) (by linarith [hcge])"},"extra_content":null},{"type":"tool_use","id":"Edit_299","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"    calc dirichletC2 * ((∫ σ in (3 : ℝ)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ)) +\n          (∫ σ in (3 : ℝ)..XR, 4 * (2 : ℝ) ^ (-σ)))\n        ≤ 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n          apply mul_le_mul hC2 (add_le_add h3a h3b) (by positivity) (by norm_num)","new_string":"    have hIntAB : (0 : ℝ) ≤ (∫ σ in (3 : ℝ)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ)) +\n        (∫ σ in (3 : ℝ)..XR, 4 * (2 : ℝ) ^ (-σ)) := by\n      apply add_nonneg\n      · apply intervalIntegral.integral_nonneg hXR\n        intro σ hσ\n        positivity\n      · apply intervalIntegral.integral_nonneg hXR\n        intro σ hσ\n        positivity\n    calc dirichletC2 * ((∫ σ in (3 : ℝ)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ)) +\n          (∫ σ in (3 : ℝ)..XR, 4 * (2 : ℝ) ^ (-σ)))\n        ≤ 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n          apply mul_le_mul hC2 (add_le_add h3a h3b) hIntAB (by norm_num)"},"extra_content":null}]