[{"type":"thinking","thinking":"`open ... MeasureTheory` — that's the pattern. Add MeasureTheory to LiTightR's open line. Then IntegrableOn, integrableOn_Ioi_deriv_of_nonneg', integral_Ioi_of_hasDerivAt_of_tendsto' resolve (they're in MeasureTheory namespace? — integral_Ioi_of_hasDerivAt_of_tendsto' at IntegralEqImproper.lean:813 — let me assume namespace MeasureTheory; if open doesn't resolve, qualify).\n\nActually wait — is integral_Ioi_of_hasDerivAt_of_tendsto' in MeasureTheory namespace? The file's docstring said \"MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto is a version of FTC-2\" — suggesting the namespace. With open MeasureTheory it resolves either way if namespaced there.\n\nNow apply all the fixes:\n1. Add MeasureTheory to open.\n2. dirichletTerm_norm_eq h2 fix.\n3. hasDerivAt_neg_log_add_one_div: the convert goals fix.\n4. Real.tendsto_pow_log_div_mul_add_atTop.\n5. tendsto_inv_atTop_zero for 1/x: `(1:ℝ)/x = x⁻¹` via one_div — heq needs care: hB : Tendsto (fun x => (1:ℝ)/x) atTop (𝓝 0) := by simpa [one_div] using tendsto_inv_atTop_zero? Tendsto (fun x => x⁻¹) vs (fun x => 1/x): one_div: 1/x = x⁻¹ — `by simpa only [one_div] using tendsto_inv_atTop_zero` — simpa only [one_div] rewrites 1/x → x⁻¹ in the goal, matching tendsto_inv_atTop_zero ✓. Or write the hB with x⁻¹ directly and adjust heq. Let me use: \n```lean\n  have hB : Tendsto (fun x => (1 : ℝ) / x) atTop (𝓝 0) := by\n    simpa only [one_div] using tendsto_inv_atTop_zero\n```\nHmm — `simpa only [one_div] using tendsto_inv_atTop_zero`: one_div : 1/a = a⁻¹ — simpa rewrites the GOAL (fun x => 1/x) → (fun x => x⁻¹)? simp rewrites occurrences of 1/?x — in a lambda body ✓ should work.\n\n6. hasDerivAt_neg_log_add_one_div convert:\n```lean\n  convert h2 using 1\n  · ext y\n    simp only [Pi.neg_apply, Pi.div_apply]\n    rw [neg_div]\n  · rw [inv_mul_cancel₀ hx0]\n    ring\n```\nWait — the convert goals from the error: goal 1: `(fun y => -(Real.log y + 1) / y) = -((fun x => Real.log x + 1) / fun x => x)`. ext y: `-(Real.log y + 1) / y = (-((fun x => Real.log x + 1) / fun x => x)) y`: simp only [Pi.neg_apply, Pi.div_apply]: RHS: -( ((fun x => log x+1) / (fun x => x)) y ) = -( (log y+1)/y )·(Pi.div_apply) — wait: Pi.neg_apply: (-f) y = -(f y); Pi.div_apply: (f/g) y = f y / g y. So RHS → -( (log y + 1)/y ). Goal: -(log y+1)/y = -((log y+1)/y) — then rw [neg_div] on the LHS: -(a/b) = (-a)/b... neg_div : -(a/b) = -a/b — LHS −(log y+1)/y parses as (−(log y+1))/y — hmm: `-(Real.log y + 1) / y` = `(-(Real.log y + 1)) / y` (unary minus binds tighter than /) — and the RHS −((log y+1)/y). neg_div : -(a / b) = -a / b — rewriting RHS direction... the goal is LHS = RHS with LHS = (−(log y+1))/y and RHS = −((log y+1)/y): rw [neg_div] turns −((log y+1)/y) into (−(log y+1))/y ✓ closes by rfl (rw auto-closes). But rw [neg_div] rewrites ALL −(_/_) patterns: the LHS (−(log y+1))/y has no −(_/_) form (the minus is on the numerator) ✓ so only RHS rewrites ✓.\n\nWait — but actually h2's function is `-((fun x => Real.log x + 1) / fun x => x)` — Pi.neg of the Pi.div. My convert-goal-1 as displayed: `(fun y => -(Real.log y + 1) / y) = -((fun x => Real.log x + 1) / fun x => x)` ✓. After ext y and simp only [Pi.neg_apply, Pi.div_apply]: RHS becomes -(Real.log y + 1) / y·... hold on: (−(F/G)) y → Pi.neg_apply: −((F/G) y) → Pi.div_apply: −(F y / G y) = −((log y + 1)/y). And LHS: (fun y => −(log y+1)/y) y → beta: −(log y+1)/y = (−(log y+1))/y. So goal: (−(log y+1))/y = −((log y+1)/y): rw [neg_div] rewrites RHS −((log y+1)/y) → (−(log y+1))/y ✓ rfl. \n\nHmm — but does ext y produce the beta-reduced forms? ext y then the goal shows the applications; simp only [Pi.neg_apply, Pi.div_apply] also beta-reduces? simp does beta-reduce ✓.\n\nValue goal: `Real.log x / x ^ 2 = -((x⁻¹ * x - (Real.log x + 1) * 1) / x ^ 2)`: rw [inv_mul_cancel₀ hx0] — x⁻¹ * x = 1: inv_mul_cancel₀ : a⁻¹ * a = 1? The name: `inv_mul_cancel₀ (h : a ≠ 0) : a⁻¹ * a = 1` ✓. Then goal: log x/x² = −((1 − (log x+1)·1)/x²): ring ✓ (field-free: same denominator — ring handles −((1−(L+1))/x²) = L/x² as ring identity with x⁻² atom? ring works in the field with x as an atom — the identity: −((1−(L+1)·1)/x²) = (L)/x² — yes ring proves it treating x as a variable (no division elimination needed since both sides have /x² — ring normalizes division by x² as ·x⁻²... in a field expression, ring handles ✓).\n\n7. In integral_log_div_sq_Ioi_three: `rw [h]` then show. h : ∫ ... = 0 − (fun y => −(log y+1)/y) 3 — the beta-redex. My `show (0:ℝ) − (−(Real.log 3 + 1)/3) = (Real.log 3+1)/3` — defeq to the redex form ✓ then ring. Hmm — actually integral_Ioi_of_hasDerivAt_of_tendsto' gives ∫ = m − f a where f a is (fun y => −(log y+1)/y) 3 — beta: −(log 3+1)/3. show with the reduced form ✓ defeq ✓.\n\n8. In antitoneOn_log_div_sq: `differentiableAt_pow _ _` — signature check: `differentiableAt_pow (n : ℕ) (x : 𝕜)?`: I recall `differentiableAt_pow (n : ℕ) (x : 𝕜) : DifferentiableAt 𝕜 (fun x => x ^ n) x`? or (𝕜) (n)? It's `differentiableAt_pow (n : ℕ) (x : 𝕜)` hmm — actually I think it's `differentiableAt_pow (n : ℕ) (x : R)` with R implicit — let me just write `(differentiableAt_pow 2 x)`? hmm the error-prone part. Alternative robust: `(DifferentiableAt.pow differentiableAt_id 2)`? differentiableAt_id : DifferentiableAt 𝕜 id x; .pow 2 hmm — DifferentiableAt.pow : DifferentiableAt 𝕜 f x → ∀ n, DifferentiableAt 𝕜 (fun x => (f x)^n) x? — `differentiableAt_id.pow 2`? hmm the power of id: (fun x => x^2) vs (id)^2 — defeq fine. Let me use `(differentiableAt_id.pow 2)`. Hmm — DifferentiableAt.pow : DifferentiableAt 𝕜 f x → (n : ℕ) → DifferentiableAt 𝕜 (f ^ ·) x — the function (fun x => (id x)^2) = (fun x => x^2) ✓ defeq — but the .div application needs the exact function fun y => y^2 — (id ·)^2 vs (fun y => y^2): Pi.pow_apply-ish defeq — the DifferentiableAt.div expects DifferentiableAt g with g syntactically = (fun y => y^2)? — no, up to defeq for `exact` but for the `.div` METHOD: `(Real.differentiableAt_log hx0).div (differentiableAt_pow ...) _` produces DifferentiableAt (f/g) with g = whatever the second arg's function is — then the goal's (fun y => log y/y^2) vs (log ·)/(·^2) — Pi.div of the two — defeq; my `exact` needs the final form defeq to the goal ✓ should be OK.\n\nLet me check hasDerivAt_pow's signature: `hasDerivAt_pow (n : ℕ) (x : 𝕜) : HasDerivAt (fun x => x^n) (n*x^(n-1)) x`? I used `(hasDerivAt_pow 2 x)` in the deriv computation ✓ that should be right.\n\n9. The `summable_nat_add_iff` — for ℝ: it exists (the NNReal one is nonrec of the general). \n\n10. In dirichletC2_le_one — the hhead block: my sum_range_succ chain + simp only + the show-rws. Let me simplify: after the range expansion, the goal is ((((0 + h 0) + h 1) + h 2) + h 3) = log 2/4 + log 3/9 — with h = fun n => if n = 0 then 0 else log n/n². rw [hh] turns h 0 into the redex (fun n => ...) 0 — hmm — actually with `set h := ... with hh`, the occurrences in my written goals... the h 0 terms appear after the sum_range expansion — they're `h 0` (the set-var applied). rw [hh] rewrites h → the lambda: (fun n => if n = 0 then ...) 0. Then `simp` should beta-reduce + if_pos + log_one + ... Let me simplify the whole head block:\n```lean\n  have hhead : ∑ n ∈ Finset.range 4, h n = Real.log 2 / 4 + Real.log 3 / 9 := by\n    rw [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,\n      Finset.sum_range_succ, Finset.sum_range_zero]\n    have h0 : h 0 = (0:ℝ) := by rw [hh]; simp\n    have h1' : h 1 = (0:ℝ) := by rw [hh]; simp\n    have h2' : h 2 = Real.log 2 / 4 := by rw [hh]; simp; ring_nf\n    have h3' : h 3 = Real.log 3 / 9 := by rw [hh]; simp; ring_nf\n    rw [h0, h1', h2', h3']\n    ring\n```\nh2': h 2 = (fun n => if n = 0 then 0 else log ↑n/↑n²) 2 — simp: beta + if_neg (2 ≠ 0) + cast: gives log 2/(2:ℝ)² hmm — the cast (↑(2:ℕ) : ℝ) = (2:ℝ): simp's Nat.cast_ofNat ✓ and (2:ℝ)² = 4: simp might not do that — add ring_nf: log 2/(2^2) = log 2/4: ring_nf ✓.\n\nLet me also double check `Real.log_two_lt_d9` — used in two_le_C0hat? no — used in LiContour two_le_C0hat: `Real.exp_one_lt_d9.le` ✓ and log_two: from LiGapCircles/journal: \"log 4150 ≤ 9 (e > 2.71)\"... Real.log_two_lt_d9 : Real.log 2 < 0.6931471808 ✓ exists in Mathlib (Real.log_two_lt_d9). ✓\n\n11. hsplit's sum_add_tsum_nat_add: `sum_add_tsum_nat_add (k : ℕ) (hf : Summable f) : (∑ i ∈ range k, f i) + ∑' i, f (i + k) = ∑' i, f i`? If the name/direction differs, fix at build. Also the h (n+4) shift: the shifted function fun n => h (n + 4) — h applied to ↑(n+4)?? — h : ℕ → ℝ so h (n+4) with n+4 : ℕ ✓.\n\n12. In htail's h1: `show (if (n + 4 : ℕ) = 0 then (0 : ℝ) else Real.log (n + 4) / ((n + 4 : ℕ) : ℝ) ^ 2) = Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2` — after rw [hh]: h (n+4) = (fun n => if...) (n+4) — the show's LHS must be defeq: (if n+4 = 0 then 0 else log ↑(n+4)/↑(n+4)²) — hmm: the lambda's body has Real.log (n+4) — with the n+4 inside log: ℕ or ℝ? The h def: `fun n => if n = 0 then (0 : ℝ) else Real.log n / (n : ℝ) ^ 2` — Real.log n with n : ℕ coerced: Real.log (↑n) — and /(↑n)². So h (n+4) = if (↑(n+4))... if (n+4 : ℕ) = 0 then 0 else log (↑(n+4))/↑(n+4)². My show: `Real.log (n + 4) / ((n + 4 : ℕ) : ℝ) ^ 2` — the `Real.log (n + 4)`: (n+4) here elaborates as ↑(n+4)? — Real.log needs ℝ: (n+4) with n : ℕ — coercion: ↑(n+4)? or ↑n + 4? Ambiguity — Lean elaborates Real.log (n + 4) by coercing... the expected type ℝ makes (n + 4 : ℝ) with n coerced: ↑n + 4 (HAdd ℝ). Hmm — actually for `Real.log (n + 4)` with n : ℕ: unification: Real.log : ℝ → ℝ; the arg n + 4 must be ℝ: n + 4 could elaborate in ℕ then cast (OfNat 4 in ℕ, HAdd ℕ, then ↑) or in ℝ (coe n, OfNat 4 in ℝ). Lean's default: the coercion is inserted around the SMALLEST subterm — for `Real.log (n + 4)`: the elaborator tries to unify (n + 4)'s type with ℝ — n + 4 : ℕ (n : ℕ, 4 : ℕ via OfNat) then coe ↑ — I THINK it does ↑(n + 4). Hmm — actually with the coercion elaboration, `n + 4` at expected type ℝ: the + gets unified at ℝ: (↑n + 4) — Lean 4 coerces by inserting ↑n — I believe it produces ↑n + 4 (the coercion goes on the variable). So my show's LHS has log (↑n + 4)? and h (n+4) has log ↑(n+4) — MISMATCH (↑(n+4) vs ↑n + 4 — defeq via push_cast but not syntactic for `show`). To be safe, write the h-application forms with explicit casts: h (n+4) = if (n+4) = 0 then 0 else Real.log (↑(n+4))/(↑(n+4))² and the tail form Real.log (↑n + 4)/(↑n+4)² — the bridge: ↑(n+4) = ↑n + 4 by push_cast. Ugh. Let me simplify: define the tail function as `fun n : ℕ => Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2` EVERYWHERE consistently (as I did) and prove h (n + 4) = that by: rw [hh]; show (if ...) = ...; rw [if_neg (by omega : (n+4:ℕ) ≠ 0)]; push_cast; ring_nf. The push_cast converts ↑(n+4) → ↑n + 4 ✓ then the sides match — ring_nf/rfl. That's what I wrote ✓.\n\nOK — apply all fixes now. Let me make the edits to the file.","signature":""},{"type":"text","text":"Applying all fixes: MeasureTheory open, the h2 cast, the convert-goal forms, and the tendsto names."},{"type":"tool_use","id":"Bash_174","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1 && grep -n \"^open\" LiTightR.lean && grep -n \"hasDerivAt_neg_log_add_one_div\\|tendsto_neg_log_add_one_div\\|h2 : ‖\\|convert h2\\|have hval\\|have hfn\\|rw \\[← hfn\" LiTightR.lean | head -12","description":"Locate edit targets in the spliced file"},"extra_content":null}]