[{"type":"thinking","thinking":"Errors:\n1. `Set.uIcc_of_le hx` gives `Set.uIcc 0 x = Set.Icc 0 x`, and `ht.1` after rw gives `min 0 x ≤ t` — the rw didn't fire because ht.1 is already the projected membership... Actually `ht : t ∈ Set.uIcc 0 x`; `ht.1` — membership in uIcc isn't a structure with .1. Let me fix: `rw [Set.uIcc_of_le hx] at ht` then `ht.1`/`ht.2`.\n\n2. The HasDerivAt.comp chain has issues — the `convert ... using 2 <;> ring` approach produced garbage. Let me simplify: avoid convert entirely; construct the HasDerivAt directly with the right function form.\n\nFor log_one_add_ge's hF: the function is `fun y => Real.log (1 + y)`. `(Real.hasDerivAt_log hne).comp t ((hasDerivAt_id' t).const_add 1)`: HasDerivAt (fun y => 1 + y) 1 t (const_add of id' gives derivative 1? `HasDerivAt.const_add (c : 𝕜) : HasDerivAt f f' x → HasDerivAt (fun x => c + f x) f' x` — so (hasDerivAt_id' t).const_add 1 : HasDerivAt (fun x => 1 + x) 1 t ✓. Then .comp t : HasDerivAt (Real.log ∘ (fun y => 1+y)) ((1+t)⁻¹ • 1) t — the outer function applied... HasDerivAt.comp : HasDerivAt f f' (g x) → HasDerivAt g g' x → HasDerivAt (f ∘ g) (g' • f') x. Here f = Real.log, f' = (1+t)⁻¹, g = fun y => 1+y, g' = 1. So comp gives HasDerivAt (Real.log ∘ fun y => 1+y) (1 • (1+t)⁻¹)... the smul form: (1 : ℝ) • (1+t)⁻¹. The target is HasDerivAt (fun y => Real.log (1+y)) (1/(1+t)) t. `Real.log ∘ fun y => 1+y` vs `fun y => Real.log (1+y)` — defeq but might need show. And the derivative value 1 • (1+t)⁻¹ = (1+t)⁻¹ — but I want 1/(1+t): (1+t)⁻¹ = 1/(1+t) via one_div. Let me write it more carefully with `show` + simpa.\n\nSimplest robust pattern (from this file's hasDerivAt_neg_log_add_one_div): build via .add/.div/.sub and rewrite the function/derivative forms with `rw [show ...]` then exact.\n\nFor hF: \n```\nhave h1 : HasDerivAt (fun y : ℝ => 1 + y) (1 : ℝ) t := by\n  simpa using (hasDerivAt_id' t).const_add 1\nhave h2 := (Real.hasDerivAt_log (by linarith : (1:ℝ)+t ≠ 0)).comp t h1\n-- h2 : HasDerivAt (Real.log ∘ fun y => 1+y) (1 • (1+t)⁻¹) t\nsimpa [Function.comp_def, one_div] using h2\n```\nHmm — simpa needs to turn `1 • (1+t)⁻¹` into `1/(1+t)`: one_smul, one_div. `simpa [Function.comp_def]` might leave (1+t)⁻¹ vs 1/(1+t): add one_div to the simp set or use show to state the goal derivative as (1+t)⁻¹:\nActually state hF's conclusion with derivative (1+t)⁻¹? The goal says 1/(1+t). `show HasDerivAt (fun y => Real.log (1+y)) ((1+t)⁻¹) t` — one_div rewrite first. Let me just: `rw [one_div]; simpa [Function.comp_def] using h2`.\n\nFor hG (the polynomial antiderivative): the messy convert chain broke. Write it cleanly:\n```\nhave h1 : HasDerivAt (fun y : ℝ => y - y^2/2 + y^3/3 - y^4/4) (1 - t + t^2 - t^3) t := by\n  have ha : HasDerivAt (fun y : ℝ => y) 1 t := hasDerivAt_id' t\n  have hb : HasDerivAt (fun y : ℝ => y^2/2) (2*t/2) t := (hasDerivAt_pow 2 t).div_const 2\n  have hc : HasDerivAt (fun y : ℝ => y^3/3) (3*t^2/3) t := (hasDerivAt_pow 3 t).div_const 3\n  have hd : HasDerivAt (fun y : ℝ => y^4/4) (4*t^3/4) t := (hasDerivAt_pow 4 t).div_const 4\n  have h2 := ((ha.sub hb).add hc).sub hd\n  -- the function form: (fun y => ((y - y^2/2) + y^3/3) - y^4/4) vs assoc\n  rw [show (1 - 2*t/2 + 3*t^2/3 - 4*t^3/4) = 1 - t + t^2 - t^3 from by ring] at h2\n  exact h2\n```\nThe associativity: `ha.sub hb` : HasDerivAt (fun y => y - y^2/2) ...; `.add hc` : (fun y => (y - y^2/2) + y^3/3); `.sub hd` : (fun y => ((y - y^2/2) + y^3/3) - y^4/4). The goal fun is `y - y^2/2 + y^3/3 - y^4/4` which parses as ((y - y^2/2) + y^3/3) - y^4/4 ✓ same. The derivative: ((1 - 2t/2) + 3t²/3) - 4t³/4 — the rw [show] at h2 handles.\n\nWait — the earlier error was because I wrote the function-equality rw with Pi.sub_apply forms that didn't match. The direct construction avoids that.\n\nSame for hG in neg_log_one_sub_le.\n\n3. hInt1's integrability: `continuous_const.div (continuous_const.add continuous_id) (fun t => by positivity)` — positivity can't show 1+t ≠ 0 for all t (t can be −1). The function fun t => 1/(1+t) is continuous only off −1; on the interval [0,x] it's fine but `Continuous.intervalIntegrable` needs global continuity. Use `ContinuousOn.intervalIntegrable` with the domain Icc 0 x instead! Pattern from log_diff_le's hconti: `ContinuousOn.intervalIntegrable` + `Continuous.continuousOn`? No — that also needs global continuity. For 1/(1+t) on Icc 0 x: `ContinuousOn.div continuousOn_const (continuousOn_const.add continuousOn_id) (fun t ht => ne_of_gt ...)`. Let me write:\n```\napply ContinuousOn.intervalIntegrable\napply ContinuousOn.div continuousOn_const (continuousOn_const.add continuousOn_id)\nintro t ht\nrw [Set.mem_Icc] at ht\nlinarith [ht.1]\n```\nSimilarly for 1/(1−t) on Icc 0 y.\n\nActually — even better: use `intervalIntegral.integral_mono_on`'s integrability hypotheses via `ContinuousOn.intervalIntegrable` (which takes ContinuousOn on the Icc). The deriv-based integral evaluation (integral_eq_sub_of_hasDerivAt) also wants IntervalIntegrable of the derivative — ContinuousOn works.\n\nAlso `intervalIntegral.integral_eq_sub_of_hasDerivAt` signature: (h : ∀ x ∈ uIcc a b, HasDerivAt F f' x) (hf : IntervalIntegrable f' volume a b) : ∫ = F b − F a ✓ as used.\n\n4. The `Set.uIcc_of_le` issue: fix with rw at ht.\n\nLet me rewrite the two big lemmas (log_one_add_ge, neg_log_one_sub_le) more carefully. Also there was a suspicious `(fun t => by ...)` block in hInt1 of neg_log_one_sub_le — replace with ContinuousOn.\n\nLet me rewrite both lemmas wholesale.","signature":""},{"type":"text","text":"Several structural issues. Let me rewrite the two integral-comparison lemmas with cleaner HasDerivAt construction and ContinuousOn integrability."},{"type":"tool_use","id":"Edit_41","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean","old_string":"/-- log(1+x) ≥ x − x²/2 + x³/3 − x⁴/4 for x ≥ 0 (1/(1+t) ≥ 1−t+t²−t³ on\n[0, x], since 1 − (1−t+t²−t³)(1+t) = t⁴ ≥ 0). -/\nlemma log_one_add_ge {x : ℝ} (hx : 0 ≤ x) :\n    x - x ^ 2 / 2 + x ^ 3 / 3 - x ^ 4 / 4 ≤ Real.log (1 + x) := by\n  have hF : ∀ t ∈ Set.uIcc 0 x,\n      HasDerivAt (fun y => Real.log (1 + y)) (1 / (1 + t)) t := by\n    intro t ht\n    have ht0 : (0 : ℝ) ≤ t := by\n      have h := ht.1\n      rwa [Set.uIcc_of_le hx, Set.mem_Icc] at h\n    exact (Real.hasDerivAt_log (by linarith : (1 : ℝ) + t ≠ 0)).comp t\n      ((hasDerivAt_id' t).const_add 1)\n  have hG : ∀ t ∈ Set.uIcc 0 x,\n      HasDerivAt (fun y => y - y ^ 2 / 2 + y ^ 3 / 3 - y ^ 4 / 4)\n        (1 - t + t ^ 2 - t ^ 3) t := by\n    intro t ht\n    have h1 := (hasDerivAt_id' t).sub (((hasDerivAt_pow 2 t).div_const 2))\n    have h2 := h1.add (((hasDerivAt_pow 3 t).div_const 3))\n    have h3 := h2.sub (((hasDerivAt_pow 4 t).div_const 4))\n    rw [show (fun y => y - y ^ 2 / 2 + y ^ 3 / 3 - y ^ 4 / 4)\n        = (fun y => y - y ^ 2 / 2 + y ^ 3 / 3) - fun y => y ^ 4 / 4 from by\n          ext y; simp only [Pi.sub_apply]] at *\n    rw [show (1 : ℝ) - t + t ^ 2 - t ^ 3\n        = (1 - 2 * t / 2 + 3 * t ^ 2 / 3) - 4 * t ^ 3 / 4 from by ring]\n    convert h3 using 2 <;> ring\n  have hInt1 : ∫ t in (0 : ℝ)..x, 1 / (1 + t) = Real.log (1 + x) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF]\n    · simp [Real.log_one]\n    · apply Continuous.intervalIntegrable\n      exact continuous_const.div (continuous_const.add continuous_id)\n        (fun t => by positivity)\n  have hInt2 : ∫ t in (0 : ℝ)..x, (1 - t + t ^ 2 - t ^ 3)\n      = x - x ^ 2 / 2 + x ^ 3 / 3 - x ^ 4 / 4 := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hG]\n    · ring_nf\n    · exact Continuous.intervalIntegrable (by fun_prop)\n  have hmono : ∫ t in (0 : ℝ)..x, (1 - t + t ^ 2 - t ^ 3)\n      ≤ ∫ t in (0 : ℝ)..x, 1 / (1 + t) := by\n    apply intervalIntegral.integral_mono_on hx.le\n    · exact Continuous.intervalIntegrable (by fun_prop)\n    · apply Continuous.intervalIntegrable\n      exact continuous_const.div (continuous_const.add continuous_id)\n        (fun t => by positivity)\n    · intro t ht\n      rw [Set.mem_Icc] at ht\n      have h1 : (0 : ℝ) < 1 + t := by linarith [ht.1]\n      rw [div_le_iff₀ h1]\n      have h2 : (1 - t + t ^ 2 - t ^ 3) * (1 + t) = 1 - t ^ 4 := by ring\n      rw [h2]\n      have h3 : t ^ 4 ≥ 0 := by positivity\n      linarith [ht.1]\n  rw [hInt1, hInt2] at hmono\n  exact hmono\n\n/-- −log(1−y) ≤ y + y²/2 + y³/(3(1−y)) for 0 ≤ y < 1\n(1/(1−t) ≤ 1 + t + t²/(1−y) on [0, y]). -/\nlemma neg_log_one_sub_le {y : ℝ} (hy0 : 0 ≤ y) (hy1 : y < 1) :\n    -Real.log (1 - y) ≤ y + y ^ 2 / 2 + y ^ 3 / (3 * (1 - y)) := by\n  have hF : ∀ t ∈ Set.uIcc 0 y,\n      HasDerivAt (fun x => -Real.log (1 - x)) (1 / (1 - t)) t := by\n    intro t ht\n    have ht1 : t < 1 := by\n      have h := ht.2\n      rwa [Set.uIcc_of_le hy0, Set.mem_Icc] at h\n    have h1 := (Real.hasDerivAt_log (by linarith : (1 : ℝ) - t ≠ 0)).comp t\n      ((hasDerivAt_id' t).const_sub 1)\n    have h2 := h1.neg\n    rw [show (fun x => -Real.log (1 - x))\n        = -(Real.log ∘ fun x => 1 - x) from by ext x; simp [Function.comp_def]] at *\n    rw [show (1 : ℝ) / (1 - t) = -((1 - t)⁻¹ * -1) from by field_simp]\n    convert h2 using 2 <;> ring\n  have hG : ∀ t ∈ Set.uIcc 0 y,\n      HasDerivAt (fun x => x + x ^ 2 / 2 + x ^ 3 / (3 * (1 - y)))\n        (1 + t + t ^ 2 / (1 - y)) t := by\n    intro t ht\n    have h1 := (hasDerivAt_id' t).add (((hasDerivAt_pow 2 t).div_const 2))\n    have h2 := h1.add (((hasDerivAt_pow 3 t).div_const (3 * (1 - y))))\n    rw [show (fun x => x + x ^ 2 / 2 + x ^ 3 / (3 * (1 - y)))\n        = (fun x => x + x ^ 2 / 2) + fun x => x ^ 3 / (3 * (1 - y)) from by\n          ext x; simp only [Pi.add_apply]] at *\n    rw [show (1 : ℝ) + t + t ^ 2 / (1 - y)\n        = (1 + 2 * t / 2) + 3 * t ^ 2 / (3 * (1 - y)) from by ring]\n    convert h2 using 2 <;> ring\n  have hInt1 : ∫ t in (0 : ℝ)..y, 1 / (1 - t) = -Real.log (1 - y) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF]\n    · simp [Real.log_one]\n    · apply Continuous.intervalIntegrable\n      exact continuous_const.div (continuous_const.sub continuous_id)\n        (fun t => by\n          have h1 : t ≠ 1 ∨ (1 : ℝ) - t ≠ 0 := by\n            by_contra h\n            push_neg at h\n            exact absurd h.1 (sub_ne_zero.mp h.2)\n          obtain h1 | h1 := h1\n          · intro h2\n            rw [sub_eq_zero] at h2\n            exact h1 h2\n          · exact h1)\n  have hInt2 : ∫ t in (0 : ℝ)..y, (1 + t + t ^ 2 / (1 - y))\n      = y + y ^ 2 / 2 + y ^ 3 / (3 * (1 - y)) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hG]\n    · ring_nf\n    · exact Continuous.intervalIntegrable (by fun_prop)\n  have hmono : ∫ t in (0 : ℝ)..y, 1 / (1 - t)\n      ≤ ∫ t in (0 : ℝ)..y, (1 + t + t ^ 2 / (1 - y)) := by\n    apply intervalIntegral.integral_mono_on hy0\n    · apply Continuous.intervalIntegrable\n      exact continuous_const.div (continuous_const.sub continuous_id)\n        (fun t => sub_ne_zero.mpr (by linarith [hy1] : t ≠ 1))\n    · exact Continuous.intervalIntegrable (by fun_prop)\n    · intro t ht\n      rw [Set.mem_Icc] at ht\n      have h1 : (0 : ℝ) < 1 - t := by linarith [ht.2, hy1]\n      have h2 : (0 : ℝ) < 1 - y := by linarith [hy1]\n      rw [le_div_iff₀ h1]\n      have h3 : (1 + t + t ^ 2 / (1 - y)) * (1 - t)\n          = 1 - t ^ 2 + t ^ 2 * (1 - t) / (1 - y) := by field_simp; ring\n      rw [h3]\n      have h4 : t ^ 2 * (1 - t) / (1 - y) ≥ t ^ 2 := by\n        have h5 : (1 - y) ≤ (1 - t) := by linarith [ht.2]\n        have h6 : (0 : ℝ) ≤ t ^ 2 := by positivity\n        rw [div_le_iff₀ h2]\n        calc t ^ 2 * (1 - y) ≤ t ^ 2 * (1 - t) :=\n              mul_le_mul_of_nonneg_left h5 h6\n          _ = t ^ 2 * (1 - t) * 1 := (mul_one _).symm\n      linarith\n  rw [hInt1, hInt2] at hmono\n  exact hmono","new_string":"/-- log(1+x) ≥ x − x²/2 + x³/3 − x⁴/4 for x ≥ 0 (1/(1+t) ≥ 1−t+t²−t³ on\n[0, x], since 1 − (1−t+t²−t³)(1+t) = t⁴ ≥ 0). -/\nlemma log_one_add_ge {x : ℝ} (hx : 0 ≤ x) :\n    x - x ^ 2 / 2 + x ^ 3 / 3 - x ^ 4 / 4 ≤ Real.log (1 + x) := by\n  have hcont1 : ContinuousOn (fun t : ℝ => 1 / (1 + t)) (Set.Icc 0 x) := by\n    apply ContinuousOn.div continuousOn_const (continuousOn_const.add continuousOn_id)\n    intro t ht\n    rw [Set.mem_Icc] at ht\n    linarith [ht.1]\n  have hF : ∀ t ∈ Set.uIcc 0 x,\n      HasDerivAt (fun y => Real.log (1 + y)) (1 / (1 + t)) t := by\n    intro t ht\n    rw [Set.uIcc_of_le hx, Set.mem_Icc] at ht\n    have h1 : HasDerivAt (fun y : ℝ => 1 + y) (1 : ℝ) t := by\n      simpa using (hasDerivAt_id' t).const_add 1\n    have h2 := (Real.hasDerivAt_log (by linarith : (1 : ℝ) + t ≠ 0)).comp t h1\n    rw [one_div]\n    simpa [Function.comp_def] using h2\n  have hG : ∀ t ∈ Set.uIcc 0 x,\n      HasDerivAt (fun y => y - y ^ 2 / 2 + y ^ 3 / 3 - y ^ 4 / 4)\n        (1 - t + t ^ 2 - t ^ 3) t := by\n    intro t _\n    have ha : HasDerivAt (fun y : ℝ => y) (1 : ℝ) t := hasDerivAt_id' t\n    have hb : HasDerivAt (fun y : ℝ => y ^ 2 / 2) (2 * t / 2) t :=\n      (hasDerivAt_pow 2 t).div_const 2\n    have hc : HasDerivAt (fun y : ℝ => y ^ 3 / 3) (3 * t ^ 2 / 3) t :=\n      (hasDerivAt_pow 3 t).div_const 3\n    have hd : HasDerivAt (fun y : ℝ => y ^ 4 / 4) (4 * t ^ 3 / 4) t :=\n      (hasDerivAt_pow 4 t).div_const 4\n    have h2 := ((ha.sub hb).add hc).sub hd\n    rw [show (1 : ℝ) - 2 * t / 2 + 3 * t ^ 2 / 3 - 4 * t ^ 3 / 4\n        = 1 - t + t ^ 2 - t ^ 3 from by ring] at h2\n    exact h2\n  have hInt1 : ∫ t in (0 : ℝ)..x, 1 / (1 + t) = Real.log (1 + x) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF\n      (ContinuousOn.intervalIntegrable hcont1)]\n    simp [Real.log_one]\n  have hInt2 : ∫ t in (0 : ℝ)..x, (1 - t + t ^ 2 - t ^ 3)\n      = x - x ^ 2 / 2 + x ^ 3 / 3 - x ^ 4 / 4 := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hG\n      (ContinuousOn.intervalIntegrable (by fun_prop))]\n    ring_nf\n  have hmono : ∫ t in (0 : ℝ)..x, (1 - t + t ^ 2 - t ^ 3)\n      ≤ ∫ t in (0 : ℝ)..x, 1 / (1 + t) := by\n    apply intervalIntegral.integral_mono_on hx.le\n      (ContinuousOn.intervalIntegrable (by fun_prop))\n      (ContinuousOn.intervalIntegrable hcont1)\n    intro t ht\n    rw [Set.mem_Icc] at ht\n    have h1 : (0 : ℝ) < 1 + t := by linarith [ht.1]\n    rw [div_le_iff₀ h1]\n    have h2 : (1 - t + t ^ 2 - t ^ 3) * (1 + t) = 1 - t ^ 4 := by ring\n    rw [h2]\n    have h3 : t ^ 4 ≥ 0 := by positivity\n    linarith\n  rw [hInt1, hInt2] at hmono\n  exact hmono\n\n/-- −log(1−y) ≤ y + y²/2 + y³/(3(1−y)) for 0 ≤ y < 1\n(1/(1−t) ≤ 1 + t + t²/(1−y) on [0, y]). -/\nlemma neg_log_one_sub_le {y : ℝ} (hy0 : 0 ≤ y) (hy1 : y < 1) :\n    -Real.log (1 - y) ≤ y + y ^ 2 / 2 + y ^ 3 / (3 * (1 - y)) := by\n  have hcont1 : ContinuousOn (fun t : ℝ => 1 / (1 - t)) (Set.Icc 0 y) := by\n    apply ContinuousOn.div continuousOn_const (continuousOn_const.sub continuousOn_id)\n    intro t ht\n    rw [Set.mem_Icc] at ht\n    linarith [ht.2, hy1]\n  have hF : ∀ t ∈ Set.uIcc 0 y,\n      HasDerivAt (fun x => -Real.log (1 - x)) (1 / (1 - t)) t := by\n    intro t ht\n    rw [Set.uIcc_of_le hy0, Set.mem_Icc] at ht\n    have h1 : HasDerivAt (fun x : ℝ => 1 - x) (-1 : ℝ) t := by\n      simpa using (hasDerivAt_id' t).const_sub 1\n    have h2 := ((Real.hasDerivAt_log (by linarith : (1 : ℝ) - t ≠ 0)).comp t h1).neg\n    rw [one_div]\n    simpa [Function.comp_def, neg_div] using h2\n  have hG : ∀ t ∈ Set.uIcc 0 y,\n      HasDerivAt (fun x => x + x ^ 2 / 2 + x ^ 3 / (3 * (1 - y)))\n        (1 + t + t ^ 2 / (1 - y)) t := by\n    intro t _\n    have ha : HasDerivAt (fun x : ℝ => x) (1 : ℝ) t := hasDerivAt_id' t\n    have hb : HasDerivAt (fun x : ℝ => x ^ 2 / 2) (2 * t / 2) t :=\n      (hasDerivAt_pow 2 t).div_const 2\n    have hc : HasDerivAt (fun x : ℝ => x ^ 3 / (3 * (1 - y))) (3 * t ^ 2 / (3 * (1 - y))) t :=\n      (hasDerivAt_pow 3 t).div_const (3 * (1 - y))\n    have h2 := (ha.add hb).add hc\n    rw [show (1 : ℝ) + 2 * t / 2 + 3 * t ^ 2 / (3 * (1 - y))\n        = 1 + t + t ^ 2 / (1 - y) from by ring] at h2\n    exact h2\n  have hInt1 : ∫ t in (0 : ℝ)..y, 1 / (1 - t) = -Real.log (1 - y) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF\n      (ContinuousOn.intervalIntegrable hcont1)]\n    simp [Real.log_one]\n  have hInt2 : ∫ t in (0 : ℝ)..y, (1 + t + t ^ 2 / (1 - y))\n      = y + y ^ 2 / 2 + y ^ 3 / (3 * (1 - y)) := by\n    rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hG\n      (ContinuousOn.intervalIntegrable (by fun_prop))]\n    ring_nf\n  have hmono : ∫ t in (0 : ℝ)..y, 1 / (1 - t)\n      ≤ ∫ t in (0 : ℝ)..y, (1 + t + t ^ 2 / (1 - y)) := by\n    apply intervalIntegral.integral_mono_on hy0\n      (ContinuousOn.intervalIntegrable hcont1)\n      (ContinuousOn.intervalIntegrable (by fun_prop))\n    intro t ht\n    rw [Set.mem_Icc] at ht\n    have h1 : (0 : ℝ) < 1 - t := by linarith [ht.2, hy1]\n    have h2 : (0 : ℝ) < 1 - y := by linarith [hy1]\n    rw [le_div_iff₀ h1]\n    have h3 : (1 + t + t ^ 2 / (1 - y)) * (1 - t)\n        = 1 - t ^ 2 + t ^ 2 * (1 - t) / (1 - y) := by field_simp; ring\n    rw [h3]\n    have h4 : (t : ℝ) ^ 2 ≤ t ^ 2 * (1 - t) / (1 - y) := by\n      have h5 : (1 - y) ≤ (1 - t) := by linarith [ht.2]\n      have h6 : (0 : ℝ) ≤ t ^ 2 := by positivity\n      rw [div_le_iff₀ h2]\n      calc t ^ 2 * (1 - y) ≤ t ^ 2 * (1 - t) := mul_le_mul_of_nonneg_left h5 h6\n        _ = t ^ 2 * (1 - t) * 1 := (mul_one _).symm\n    linarith\n  rw [hInt1, hInt2] at hmono\n  exact hmono"},"extra_content":null}]