[{"type":"thinking","thinking":"Fixable issues:\n\n1. Line 829: hcont1's nonzero side-goal: `((fun x => 1) + id) t = 0 ⊢ False` — the ContinuousOn.div side condition: my `intro t ht; rw [Set.mem_Icc] at ht; linarith [ht.1]` — the goal is `(fun x => 1 + x) t ≠ 0` shown as `((fun x => 1) + id) t = 0 → False`. linarith needs the form 1 + t ≠ 0. The issue: `continuousOn_const.add continuousOn_id` produces the function `(fun x => 1) + id` and the goal is `((fun x => 1) + id) t ≠ 0` — simp only [Pi.add_apply, id] first, then linarith.\n\n2. div_const derivative: `(hasDerivAt_pow 2 t).div_const 2 : HasDerivAt (fun x => x^2/2) (↑2 * t^(2-1) / 2) t` — the derivative form is ↑2·t^(2−1)/2, not 2·t/2. Fix: don't ascribe the derivative in the have; instead state hb without the derivative and convert: \n```\nhave hb := (hasDerivAt_pow 2 t).div_const (2 : ℝ)\n-- hb : HasDerivAt (fun x => x^2/2) (↑2 * t^(2-1) / 2) t\n```\nThen the assembly h2 will have derivative in that raw form, and my final `rw [show raw-form = target-form from by ring]` — the raw form is (↑2*t^(2−1)/2 − ...) — the show needs to match the ACTUAL raw form. The banked gotcha from this very file (session 035, antitoneOn_log_div_sq): they used hsimp : ((2:ℕ):ℝ)·x^(2−1) = 2·x via Nat.cast + pow_one. So the pow derivative gives ↑n·x^(n−1). I'll follow: after building h2, the derivative is ((1 − ↑2*t^(2−1)/2) + ↑3*t^(3−1)/3) − ↑4*t^(4−1)/4. My show-pattern: `rw [show ((1:ℝ) - 2*t/2 + 3*t^2/3 - 4*t^3/4) = ...]` — WRONG raw form. Fix: normalize h2's derivative via simp only [Nat.cast_ofNat, ...] then a show-rw. Simpler: build each piece with the derivative already normalized:\n```\nhave hb : HasDerivAt (fun y : ℝ => y ^ 2 / 2) (t : ℝ) t := by\n  have h := (hasDerivAt_pow 2 t).div_const (2 : ℝ)\n  rw [show (↑(2:ℕ) * t ^ (2 - 1) : ℝ) / 2 = t from by\n    rw [show (2:ℕ) - 1 = 1 from rfl, pow_one]; push_cast; ring] at h\n  exact h\n```\nThat's the pattern from antitoneOn_log_div_sq (hsimp). Derivatives: y²/2 → t; y³/3 → t²; y⁴/4 → t³. Then the assembly (ha.sub hb).add hc).sub hd has derivative ((1 − t) + t²) − t³ ✓ EXACTLY the target form. \n\n3. ContinuousOn.intervalIntegrable expects uIcc not Icc: `ContinuousOn.intervalIntegrable : ContinuousOn f (Set.uIcc a b) → IntervalIntegrable f μ a b`. Fix: prove hcont1 on uIcc: `Set.uIcc 0 x` — with uIcc_of_le hx rewrite to Icc... just state hcont1 : ContinuousOn f (Set.uIcc 0 x) := by rw [Set.uIcc_of_le hx]; <Icc proof>.\n\n4. hx.le: `Real.le.le` invalid — hx : (0:ℝ) ≤ x and `hx.le`?? The error: \"Real.le✝ 0 x\" — hx is being elaborated weirdly? `intervalIntegral.integral_mono_on hx.le` — integral_mono_on's first arg is (h : a ≤ b) — just pass hx. The `.le` was wrong.\n\n5. Line 896: neg of composition: `-fun x => Real.log (1-x)` vs `fun x => -Real.log (1-x)` — Pi.neg_apply issue. Fix: \n```\nhave h2 := ((Real.hasDerivAt_log ...).comp t h1).neg\n-- h2 : HasDerivAt (-(Real.log ∘ fun x => 1-x)) (-(↑...)) t\nrw [show (-(Real.log ∘ fun x : ℝ => 1 - x)) = (fun x => -Real.log (1 - x)) from by\n  ext y; simp [Function.comp_def, Pi.neg_apply]] at h2\n```\nAnd the derivative: comp gives (↑(1−t)⁻¹ * −1)? The chain: Real.hasDerivAt_log gives (1−t)⁻¹; h1's derivative is −1; comp multiplies: the value form — HasDerivAt.comp for ℝ: (g' • f') = f' * g'? For ℝ→ℝ: (h₂.comp x h₁) : HasDerivAt (f₂∘f₁) (h₁' * h₂'(f₁ x)) — actually it's `HasDerivAt.comp : HasDerivAt f₂ f₂' (f₁ x) → HasDerivAt f₁ f₁' x → HasDerivAt (f₂ ∘ f₁) (f₁' * f₂') x` — derivative f₁' * f₂' = −1 * (1−t)⁻¹. Then .neg: −(−1 * (1−t)⁻¹) = (1−t)⁻¹. My target (after rw one_div): (1−t)⁻¹. So:\n```\nhave h2 := ((Real.hasDerivAt_log (by linarith : (1:ℝ) - t ≠ 0)).comp t h1).neg\nrw [show (-1 : ℝ) * (1 - t)⁻¹ = -((1 - t)⁻¹) from by ring] at h2  -- hmm the neg wraps everything\n```\nMessy. Cleaner: \n```\nhave h2 : HasDerivAt (fun x : ℝ => -Real.log (1 - x)) ((1 - t)⁻¹) t := by\n  have h3 := (Real.hasDerivAt_log (by linarith : (1:ℝ) - t ≠ 0)).comp t h1\n  -- h3 : HasDerivAt (Real.log ∘ fun x => 1-x) (-1 * (1-t)⁻¹) t\n  have h4 := h3.neg\n  rw [show (-(Real.log ∘ fun x : ℝ => 1 - x)) = (fun x => -Real.log (1 - x)) from by\n    ext y; simp only [Pi.neg_apply, Function.comp_apply]] at h4\n  rw [show (-(-1 * (1 - t)⁻¹)) = (1 - t)⁻¹ from by ring] at h4\n  exact h4\n```\nThe exact form of the comp derivative: unsure whether it's `-1 * (1-t)⁻¹` or `(1-t)⁻¹ * -1` or `(-1) • (1-t)⁻¹`. For ℝ: HasDerivAt.comp states (h₂.comp) derivative as `h₁' * h₂'`? Let me check the actual form used elsewhere in this file: in hasDerivAt_neg_log_add_one_div they used `.add_const/.div` not comp. In log_diff_le: `(Complex.hasDerivAt_log ...).comp ... ((hasDerivAt_id' _).mul_const ...).const_add ...` then `simpa [hz, Function.comp_def, mul_comm, one_mul] using h2` — the simpa with mul_comm fixed the derivative product. So for h3: simpa [Function.comp_def] should normalize. Plan:\n```\nhave h2 : HasDerivAt (fun x : ℝ => -Real.log (1 - x)) ((1 - t)⁻¹) t := by\n  have h3 := ((Real.hasDerivAt_log (by linarith : (1:ℝ) - t ≠ 0)).comp t h1).neg\n  rw [show (-(Real.log ∘ fun x : ℝ => 1 - x)) = (fun x => -Real.log (1 - x)) from by\n    ext y; simp only [Pi.neg_apply, Function.comp_apply]] at h3\n  rw [show (-1 : ℝ) * (1 - t)⁻¹ = -(1 - t)⁻¹ from by ring] at h3\n  -- now h3 : HasDerivAt (fun x => -log(1-x)) (-(-((1-t)⁻¹))) t? \n```\nI'm guessing at forms. SAFER: use `convert` — NO (banked: NEVER convert on HasDerivAt). Alternative safest: build via differentiation rules that produce clean forms: -log(1−x): write as the COMPOSITION-free chain: `HasDerivAt.neg` of a clean comp. You know what — simplest: use the value form via `.deriv`? No.\n\nCleanest robust approach: state hF with derivative `-(−1 * (1−t)⁻¹)`... ugh. Alternative: build hF from scratch via `HasDerivAt.comp` and let `simpa` normalize everything:\n```\nintro t ht\nrw [Set.uIcc_of_le hy0, Set.mem_Icc] at ht\nhave h1 : HasDerivAt (fun x : ℝ => 1 - x) (-1 : ℝ) t := by\n  simpa using (hasDerivAt_id' t).const_sub 1\nhave h2 := ((Real.hasDerivAt_log (by linarith : (1:ℝ) - t ≠ 0)).comp t h1).neg\nrw [one_div]\nsimpa [Function.comp_def, Pi.neg_apply] using h2\n```\n— mirror the log_diff_le simpa approach: `simpa [Function.comp_def, Pi.neg_apply] using h2` — the simp normalizes h3's function to fun x => -log(1-x)?? Function.comp_def gives Real.log (1-x); Pi.neg_apply turns -(fun x => ...) into fun x => -(...). The derivative: whatever product form × neg — simp [Pi.neg_apply, mul_neg, neg_neg, one_mul, one_div...] — add more simp lemmas: `simpa [Function.comp_def, Pi.neg_apply, mul_neg, neg_neg, one_mul, one_div] using h2`. The derivative of comp for ℝ: I believe the form is `(1-t)⁻¹ * -1` (f₂'∘f₁ * f₁'? order?) then neg of that: -((1-t)⁻¹ * -1) = (1-t)⁻¹ ✓ via mul_neg/neg_neg. If it's smul: one_mul/neg_smul. Add `neg_smul, smul_eq_mul` to the set. Overkill simp sets are fine.\n\nSimilarly for hF in log_one_add_ge: comp derivative (1+t)⁻¹ * 1 or 1 • (1+t)⁻¹: my current `rw [one_div]; simpa [Function.comp_def] using h2` — the target derivative (1+t)⁻¹; h2's derivative maybe (1+t)⁻¹ * 1 = (1+t)⁻¹ ✓ via mul_one. Probably fine — error was only at hcont1/hx.le lines, hF itself passed (no error at 836-843? The errors at 845+ are hG's div_const). Let me fix everything:\n\nFix list:\n(a) hcont1: state on uIcc (rw uIcc_of_le then Icc proof with Pi.add_apply simp for the side goal).\n(b) hF: keep but ensure the side conditions — looks like it compiled (no error reported at hF's lines except via hcont1).\n(c) hG: rebuild the polynomial pieces with normalized derivatives.\n(d) integral_eq_sub_of_hasDerivAt hF (ContinuousOn.intervalIntegrable hcont1) — the uIcc fix handles.\n(e) integral_mono_on hx.le → integral_mono_on hx; also the integrability args — ContinuousOn.intervalIntegrable hcont1 (uIcc-fixed ✓); the other `by fun_prop` ones — `ContinuousOn.intervalIntegrable (by fun_prop)` — fun_prop proves ContinuousOn? The pattern `Continuous.intervalIntegrable (by fun_prop)` for global. For ContinuousOn: (by fun_prop : ContinuousOn _ _) should work.\n(f) neg_log_one_sub_le's hF: the simpa approach above.\n(g) line 865: hx.le → hx.\n\nAlso line 953/1033/1074/1089/1199/1221 errors — in the log_ten_ge / log_thirteen_le / log_fourteen_ge / log_23_13_le / log_le_div_exp lemmas. Let me look at those after fixing the first batch. Line 953: log_ten_ge's hlog25 : `rw [hy]; ring_nf` — the goal Real.log (2.5/exp 1) = Real.log (1 − (1 − 2.5/exp 1)): after rw hy the RHS is Real.log (1 − (1 − 2.5/exp 1)) — need to show 2.5/exp 1 = 1 − (1 − 2.5/exp 1) — `ring_nf` should close... error \"unsolved goals\" at 953:76. And 1033: log_fourteen_ge's hlog7: `rw [hy]` — goal Real.log (7/(e·e)) = Real.log (1 − (1 − 7/(e·e))) — same ring_nf issue: ring_nf doesn't know e·e ≠ 0 to field_simp. Just: `rw [hy]; congr 1; ring`. 1074: log_thirteen_le's `rw [← Real.exp_nat_mul]` — pattern mismatch. Let me check Real.exp_nat_mul's form: probably `Real.exp_nat_mul (x : ℝ) (n : ℕ) : Real.exp (↑n * x) = Real.exp x ^ n`? Hmm — my usage: (exp a * exp a)*(exp a * exp a) = exp (a*4): simpler: rw [show (0.64125*4 : ℝ) = 0.64125+0.64125+0.64125+0.64125 from by norm_num, Real.exp_add, Real.exp_add, Real.exp_add]. Or: the goal is (ea*ea)*(ea*ea) = exp(a*4): rw [← Real.exp_add, ← Real.exp_add, ← Real.exp_add] gives exp(a+a+(a+a)) then congr. Let me do that.\n\nLine 1089: log_23_13_le unsolved goals — the Finset sum unfold: Σ_{k<6}: my rw chain has 5 sum_range_succ + sum_range_zero for n = 6: sum_range 6 = 5 succs + zero ✓ — but error at 1089:76: maybe norm_num at h left something. Actually \"unsolved goals\" at the `rw` line — the rw chain produced extra goals? Hmm — 1089:76 is `Finset.sum_range_zero] at h` col 76 — error \"unsolved goals\" probably means after rw the h didn't match?? Let me see the actual errors after the first fixes.\n\nLine 1199: log_le_div_exp's `rw [div_eq_mul_one_div] at h2` — unsolved goals? and 1221: type mismatch in log_log_le_div_exp.\n\nLet me fix the first batch and recompile to see remaining errors.","signature":""},{"type":"text","text":"Fixing the batch: uIcc forms, pow-derivative normalization, Pi.neg forms, and `hx.le` misuse."},{"type":"tool_use","id":"Edit_44","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean","old_string":"  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","new_string":"  have hcont1 : ContinuousOn (fun t : ℝ => 1 / (1 + t)) (Set.uIcc 0 x) := by\n    rw [Set.uIcc_of_le hx]\n    apply ContinuousOn.div continuousOn_const (continuousOn_const.add continuousOn_id)\n    intro t ht\n    rw [Set.mem_Icc] at ht\n    simp only [Pi.add_apply, Pi.one_apply, id]\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) t t := by\n      have h := (hasDerivAt_pow 2 t).div_const (2 : ℝ)\n      rw [show ((2 : ℕ) : ℝ) * t ^ (2 - 1) / 2 = t from by\n        rw [show (2 : ℕ) - 1 = 1 from rfl, pow_one]\n        push_cast\n        ring] at h\n      exact h\n    have hc : HasDerivAt (fun y : ℝ => y ^ 3 / 3) (t ^ 2) t := by\n      have h := (hasDerivAt_pow 3 t).div_const (3 : ℝ)\n      rw [show ((3 : ℕ) : ℝ) * t ^ (3 - 1) / 3 = t ^ 2 from by\n        rw [show (3 : ℕ) - 1 = 2 from rfl]\n        push_cast\n        ring] at h\n      exact h\n    have hd : HasDerivAt (fun y : ℝ => y ^ 4 / 4) (t ^ 3) t := by\n      have h := (hasDerivAt_pow 4 t).div_const (4 : ℝ)\n      rw [show ((4 : ℕ) : ℝ) * t ^ (4 - 1) / 4 = t ^ 3 from by\n        rw [show (4 : ℕ) - 1 = 3 from rfl]\n        push_cast\n        ring] at h\n      exact h\n    exact ((ha.sub hb).add hc).sub hd"},"extra_content":null}]