[{"type":"thinking","thinking":"§1 compiles clean. Now §2: the IBP identity. This is the longest piece. Let me write it.\n\nComponents:\n1. `b1_eq_fract_sub_half` : for 0 ≤ x: B1 x = Int.fract x − 1/2 (Nat vs Int floor via Nat.natCast_floor_eq_intCast_floor).\n\nWait — what's B1's exact reference? `B1` at root? The EulerMaclaurin.lean def: `noncomputable def B1 (x : ℝ) : ℝ := x - ⌊x⌋₊ - 1 / 2` — no namespace visible. But wait — the file might have a namespace I missed. Let me check for namespace in EulerMaclaurin.lean.\n\n2. The unit-interval IBP:\n```lean\ntheorem intervalIntegral_b1_mul_cpow (k : ℕ) (s : ℂ) :\n    ∫ x in (k:ℝ)..(k+1 : ℝ), (B1 x : ℂ) * (x : ℂ) ^ (-s - 1)\n      = (s + 1) * ∫ x in (k:ℝ)..(k+1 : ℝ), (b2Fun x : ℂ) * (x : ℂ) ^ (-s - 2)\n```\nHmm wait — for k ≥ 1? The identity holds for any interval where things are defined — x ∈ [k, k+1], k ≥ 1 gives x ≥ 1 > 0 for the cpow derivatives ✓. Actually x ≥ 1 needed; k ≥ 1 ✓.\n\nProof via integral_mul_deriv_eq_deriv_mul_of_hasDerivAt (u := fun x => (b2Fun x : ℂ), u' := fun x => (Int.fract x - 1/2 : ℂ), v := fun x => (x:ℂ)^(-s-1), v' := fun x => -(s+1) * (x:ℂ)^(-s-2)):\n∫ u' v + u v' = u b v b − u a v a. b2Fun at k+1, k (integers) = 0 ⟹ RHS = 0. So ∫ u' v = −∫ u v' = −∫ b2Fun·(−(s+1)x^{−s−2}) = (s+1)∫ b2Fun x^{−s−2} ✓.\nAnd ∫ u' v = ∫ ↑(fract−1/2)·x^{−s−1} = ∫ ↑B̃₁·x^{−s−1} via b1_eq_fract_sub_half on [k, k+1] (x ≥ 1 > 0 ✓) — integral_congr.\n\nSignature check: integral_mul_deriv_eq_deriv_mul_of_hasDerivAt (hu : ContinuousOn u [[a,b]]) (hv : ContinuousOn v [[a,b]]) (huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt u (u' x) x) (hvv' : ...) (hu' : IntervalIntegrable u' volume a b) (hv' : IntervalIntegrable v' ...) : ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, u' x * v x.\n\nSo it gives ∫ u·v' = boundary − ∫ u'·v. With u = ↑b2Fun: ∫ ↑b2Fun·v' = 0 − ∫ ↑(fract−1/2)·v ⟹ ∫ ↑(fract−1/2)·v = −∫↑b2Fun·v' ✓.\n\nu' HasDerivAt on Ioo(k, k+1): cast of hasDerivAt_b2Fun: HasDerivAt (fun x => ((b2Fun x : ℝ) : ℂ)) — from HasDerivAt b2Fun (fract x − 1/2) x (real) — ofReal: `HasDerivAt.ofReal`? The map ℝ→ℂ: `HasDerivAt.ofReal_comp`? Hmm — `(hf : HasDerivAt f f' x)` to `HasDerivAt (fun x => (f x : ℂ)) (f' : ℂ) x`: there's `HasDerivAt.ofReal`?? or use ContinuousLinearMap: the coercion ℝ → ℂ is `Complex.ofRealCLM`: `HasDerivAt.comp x (hasDerivAt_ofRealCLM?) hf` — there's `Complex.hasDerivAt_ofReal`? Simpler: `HasDerivAt.ofReal (hf)`?? Let me check: `Complex.hasDerivAt_ofReal (hf : HasDerivAt f f' x) : HasDerivAt (fun x => (f x : ℂ)) (f' : ℂ) x`? I believe something like this exists (maybe `HasDerivAt.ofReal`). I'll find at compile — could also use `(hf.ofReal)`? Let me search quickly before writing. Also for v's HasDerivAt: (x:ℂ)^(−s−1) as function of real x: `Complex.hasDerivAt_cpow_const`? — for real arg: need HasDerivAt ℝ→ℂ: (fun x : ℝ => (x:ℂ)^c): this is cpow with real base — `Real.differentiableAt_cpow_const_of_ne` (Zeta23) gives DifferentiableAt — but I need HasDerivAt with the explicit derivative c·x^(c−1). Mathlib: `hasDerivAt_cpow_const`? Let me grep both.","signature":""},{"type":"text","text":"§1 green. Before writing §2 (the IBP), checking two conversion lemmas: ofReal HasDerivAt and the real-base cpow derivative."},{"type":"tool_use","id":"Bash_199","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem hasDerivAt_ofReal\\|lemma hasDerivAt_ofReal\\|theorem.*ofReal.*HasDerivAt\\|HasDerivAt.*ofReal\" Mathlib/Analysis/ -r --include=*.lean | grep \"theorem\\|lemma\" | head -5; grep -rn \"theorem hasDerivAt_cpow_const\\|lemma hasDerivAt_cpow_const\" Mathlib/Analysis/ -r --include=*.lean | head -3","description":"Find ofReal and cpow HasDerivAt lemmas"},"extra_content":null}]