[{"type":"thinking","thinking":"Down to a handful of errors:\n\n1. Line 72 (HIntegral_deriv hd): `convert hcomp using 1; simp` — \"simp made no progress\". The goal after convert: HasDerivAt (fun x : ℝ => G (x+y*I)) (deriv G (x+y*I)) x vs hcomp : HasDerivAt (G ∘ (fun x : ℝ => x+y*I)) (deriv G (x+y*I) * 1) x. convert using 1 left some goal that simp can't close. Fix: `convert hcomp using 1` produces goals: (fun x : ℝ => G (x+y*I)) = G ∘ (fun x : ℝ => x+y*I) and deriv G (x+y*I) = deriv G (x+y*I)*1. First: funext/Function.comp_apply rfl-ish: `simp [Function.comp_apply]`? or the goal order flipped. Let me instead do: \n```\n    have hcomp := h1.comp x h2\n    rw [Function.comp_apply, mul_one] at hcomp\n    exact hcomp\n```\nhcomp : HasDerivAt (G ∘ path) (deriv G (x+y*I) * 1) x — Function.comp_apply : (g ∘ f) x = g (f x) — rewrites the function application; but hcomp's function is (G ∘ path) as a whole — rw [Function.comp_apply] at hcomp needs an application point... the function in HasDerivAt is (G ∘ path) — comp_apply applies to (G∘path) x but hcomp has it unapplied. Use `simp only [Function.comp_apply, mul_one] at hcomp` — comp_apply as simp lemma rewrites (G∘path) ?x → G (path ?x) — should work on the unapplied... Actually HasDerivAt's first arg is the function — simp only [Function.comp_apply] normalizes `(G ∘ path) x` occurrences but the bare function... Alternative: `simpa using hcomp` (simpa with comp_apply + mul_one): `simpa [Function.comp_apply] using hcomp` — simpa normalizes hcomp's type (G∘path → G∘path applications) and matches the goal. Let me use:\n```\n    have hcomp := h1.comp x h2\n    simpa [Function.comp_apply] using hcomp\n```\nSame for V-side with mul I: `simpa [Function.comp_apply] using hcomp` — hcomp : HasDerivAt (G∘path) (deriv G (x+y*I) * I) y — goal: HasDerivAt (fun y : ℝ => G (x+y*I)) (I * deriv G (x+y*I)) y — the derivative value: deriv G * I vs I * deriv G — need mul_comm! The comp gives h₂' * h' = deriv G (x+y*I) * I; goal has I * deriv G (x+y*I) — `simpa [Function.comp_apply, mul_comm] using hcomp` or restructure the goal to deriv G * I. Let me change the V-side's hd target to `(deriv G (x + y * I)) * I` and adjust the final assembly (the FTC then gives ∫ deriv G * I directly matching hftc's I * ... hmm — cleaner: keep the comp result and change hint/final to match: hftc gives ∫ (fun y => I * deriv G (x+y*I)) — wait hftc's f' = fun y => I * deriv G (x+y*I) — the comp gives derivative value deriv G * I — for the FTC's f' I wrote fun y => I * deriv G (x+y*I) — MISMATCH (I*X vs X*I — equal by mul_comm but syntactically different — the explicit f' arg fixes it: f' := fun y : ℝ => I * deriv G (x + y * I) ✓ and hd must have derivative values I * deriv G (x+y*I) — from comp with mul_comm: \n```\n    have hcomp := h1.comp y h2\n    rw [mul_comm] at hcomp  -- deriv G (x+y*I) * I → I * deriv G (x+y*I)\n    simpa [Function.comp_apply] using hcomp\n```\n\n2. Line 86 \"No goals to be solved\" — line 86 is in HIntegral_deriv's hftc convert block: `convert hftc using 1; · simp; · simp` — the FIRST simp closed both?? or the convert produced 0 goals and the bulleted simps error. hftc : ∫ (fun x => deriv G (x+y*I)) = G (w.re+y*I) − G (z.re+y*I) — wait the FTC gives f b − f a = G (w.re + y*I) − G (z.re + y*I) ✓ and the goal: ∫ x in z.re..w.re, deriv G (x+y*I) = G (w.re+y*I) − G (z.re+y*I) — these should match exactly after the explicit f/f' — the integrals: ∫ x in z.re..w.re, (fun x => deriv G (x+y*I)) x vs ∫ x in z.re..w.re, deriv G (↑x+↑y*I) — beta: the FTC's statement has f' y applied — should be defeq-identical. The convert using 1 with two simps — the second simp has no goals. Just use `exact hftc` (after checking the statement matches) or `convert hftc using 1` alone. Let me simplify to:\n```\n  convert hftc using 2\n```\nhmm — or just `simpa using hftc`. Let me use simpa for both sides' final steps.\n\n3. Line 105 `ring_nf made no progress` — V-side hcomp's `convert hcomp using 1; ring` — replaced by simpa approach.\n\n4. Line 122: `integral_smul` unknown — for interval integrals it's `intervalIntegral.integral_smul`? or `MeasureTheory.integral_smul`? For ∫ over sets/intervals: `intervalIntegral.integral_smul (c) (f) : ∫ x in a..b, c • f x = c • ∫ x in a..b, f x`? or `integral_smul` in MeasureTheory — the generic `MeasureTheory.integral_smul`? The name for ∫ (c • f) = c • ∫ f: `integral_smul` should exist in MeasureTheory.Integral.Bochner — maybe namespaced: `MeasureTheory.integral_smul`? or for interval: `intervalIntegral.integral_smul`? Let me handle the constant differently: ∫ I * deriv G = I * ∫ deriv G — use `intervalIntegral.integral_smul` if exists, else `MeasureTheory.integral_smul`. Actually — since I know the goal: ∫ (I * deriv G) with I a constant complex: `intervalIntegral.integral_const_mul`? hmm — `integral_const_mul (c : ℂ) (f : ℝ → ℂ) : ∫ x, c * f x = c * ∫ x, f x` — for Bochner: `integral_const_mul` ✓ exists in MeasureTheory.Integral.Bochner. For interval integral: reduce to the set integral or use `intervalIntegral.integral_smul`. Let me grep quickly... actually simpler: the step is ∫_{z.im}^{w.im} deriv G (x+y*I) dy = −I * (G(x+w.im*I) − G(x+z.im*I)). From hftc: ∫ I·deriv G = ΔG. So ∫ deriv G = (1/I)·ΔG = −I·ΔG: \n```\n  have h4 : ∫ y in z.im..w.im, deriv G (x + y * I) =\n      -I * (G (x + w.im * I) - G (x + z.im * I)) := by\n    have h5 : ∫ y in z.im..w.im, I * deriv G (x + y * I) = G (x + w.im * I) - G (x + z.im * I) := hftc\n    have h6 : ∫ y in z.im..w.im, I * deriv G (x + y * I) = I * ∫ y in z.im..w.im, deriv G (x + y * I) := by\n      rw [intervalIntegral.integral_smul]\n    rw [h6] at h5\n    field_simp [Complex.I_ne_zero] at h5 ⊢\n    -- h5 : I * ∫ = ΔG ⟹ ∫ = ΔG / I = -I·ΔG\n```\nintervalIntegral.integral_smul — check name: for interval integrals, `intervalIntegral.integral_smul (c) (f) : ∫ x in a..b, c • f x ∂μ = c • ∫ x in a..b, f x ∂μ` — I think exists. If not: `MeasureTheory.integral_smul`. Then from I * ∫ = ΔG: ∫ = ΔG/I = −I·ΔG: field_simp with I ≠ 0. Actually simpler: ∫ deriv G = ∫ (1/I)·(I·deriv G)... let me do:\n```\n    have hI : I ≠ 0 := Complex.I_ne_zero\n    calc ∫ y in z.im..w.im, deriv G (x + y * I)\n        = (1 / I) * (G (x + w.im * I) - G (x + z.im * I)) := by\n          have h6 : I * ∫ y in z.im..w.im, deriv G (x + y * I) = G (x + w.im * I) - G (x + z.im * I) := by\n            rw [← intervalIntegral.integral_smul]\n            exact hftc\n          field_simp [hI] at *\n          linarith\n      _ = -I * (G (x + w.im * I) - G (x + z.im * I)) := by\n          field_simp [hI]\n```\nHmm messy. Cleanest: from I * ∫ = ΔG get ∫ = ΔG/I by `eq_div_of_mul_eq`? and 1/I = −I via `Complex.inv_I : I⁻¹ = -I` ✓ exists I believe. Let me write:\n```\n    have h6 : I * ∫ y in z.im..w.im, deriv G (x + y * I) = G (x + w.im * I) - G (x + z.im * I) := by\n      rw [← intervalIntegral.integral_smul]\n      exact hftc\n    have h7 : ∫ y in z.im..w.im, deriv G (x + y * I) = (G (x + w.im * I) - G (x + z.im * I)) / I := by\n      rw [← h6]\n      field_simp [Complex.I_ne_zero]\n    rw [h7, Complex.div_I]\n```\n`Complex.div_I : z / I = -I * z`? There's `Complex.inv_I : I⁻¹ = -I` ✓ and z/I = z·I⁻¹. div_I: `Complex.div_I (z : ℂ) : z / I = -(z * I)`? Let me use inv_I: z/I = z·(1/I) = z·(−I) = −I·z ✓.\n\nLet me check intervalIntegral.integral_smul and Complex.inv_I quickly, then apply all fixes.","signature":""},{"type":"text","text":"Let me verify the remaining lemma names before fixing."},{"type":"tool_use","id":"Bash_245","name":"Bash","input":{"command":"cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem intervalIntegral.integral_smul\\b\" Mathlib/ --include=*.lean | head -2; grep -rn \"theorem Complex.inv_I\\|lemma Complex.inv_I\" Mathlib/Data/Complex/Basic.lean | head -2; grep -rn \"theorem integral_smul\\b\" Mathlib/MeasureTheory/Integral/Bochner.lean Mathlib/MeasureTheory/Integral/Bochner/*.lean 2>/dev/null | head -2","description":"Verify integral_smul and Complex.inv_I names"},"extra_content":null}]