[{"type":"thinking","thinking":"`intervalIntegral.integral_smul` EXISTS (used at TaylorIntegral.lean:137, CauchyIntegral.lean:215 as `← intervalIntegral.integral_smul`) — so `intervalIntegral.integral_smul (c) (f) : ∫ x in a..b, c • f x = c • ∫ x in a..b, f x` ✓.\n\nNow the full fix list for the remaining errors:\n\n1. hd (H-side): `convert hcomp using 1; simp` — replace with `simpa [Function.comp_apply] using hcomp`.\n2. H-side hftc convert block: `convert hftc using 1; · simp; · simp` — the \"No goals\" at 86 — simplify to `simpa using hftc` or just `exact hftc` after checking the exact form. The goal: ∫ x in z.re..w.re, deriv G (↑x+↑y*I) = G (↑w.re+↑y*I) − G (↑z.re+↑y*I); hftc : ∫ (fun x => deriv G (x+y*I)) = f b − f a = G (w.re+y*I) − G (z.re+y*I) — the explicit f/f' I passed: (f := fun x : ℝ => G (x + y * I)) (f' := fun x : ℝ => deriv G (x + y * I)) — the FTC's conclusion: ∫ y in a..b, f' y = f b − f a with a = z.re, b = w.re ✓ = G (w.re+y*I) − G (z.re+y*I) ✓ — but is the integral's integrand displayed as (fun x => deriv G (x+y*I)) x vs deriv G (↑x+↑y*I) — beta reduction — `exact hftc` should work if everything matches; use `simpa using hftc` to be safe.\n3. V-side hd: same simpa pattern.\n4. V-side final: `rw [← hftc, integral_smul]; field_simp` — hftc : ∫ (fun y => I * deriv G (x+y*I)) = G(x+w.im*I) − G(x+z.im*I). I want ∫ deriv G = −I·ΔG:\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) = I⁻¹ * (G (x + w.im * I) - G (x + z.im * I)) := by\n    rw [← h6]\n    field_simp [Complex.I_ne_zero]\n  rw [h7, Complex.inv_I]\n```\nWait — after rw [← intervalIntegral.integral_smul], the goal becomes ∫ (fun y => I • deriv G (x+y*I)) — hftc's integrand is `I * deriv G (x+y*I)` — • vs * : for ℂ, • is smul — `smul_eq_mul` simp. Let me write: rw [← intervalIntegral.integral_smul, smul_eq_mul]; exact hftc. Hmm — hftc has `I * deriv G (x+y*I)` (mul form) and integral_smul produces `I • ...` — smul_eq_mul converts ✓.\n\nThen h7's proof: after rw [← h6], goal ∫ = I⁻¹·ΔG — from I·∫ = ΔG: ∫ = I⁻¹·ΔG ⟺ I·∫ = I·(I⁻¹ΔG) = ΔG ✓ — the proof `rw [← h6]; field_simp [Complex.I_ne_zero]` — rw [← h6] rewrites ΔG to I·∫: goal ∫ = I⁻¹·(I·∫) — field_simp with I≠0: I⁻¹·I = 1 ✓ closes. Alternatively: `rw [← h6]; simp [Complex.I_ne_zero]`... let me do:\n```\n  have h7 : ∫ y in z.im..w.im, deriv G (x + y * I) = I⁻¹ * (G (x + w.im * I) - G (x + z.im * I)) := by\n    have h8 : I * ∫ y in z.im..w.im, deriv G (x + y * I) = I * (I⁻¹ * (G (x + w.im * I) - G (x + z.im * I))) := by\n      rw [← h6]\n      field_simp [Complex.I_ne_zero]\n    exact (mul_right_cancel₀ Complex.I_ne_zero h8)\n```\nHmm — simpler: `eq_of_mul_eq_mul_left Complex.I_ne_zero`? The cleanest: from I·∫ = ΔG: ∫ = ΔG/I: `eq_div_of_mul_eq Complex.I_ne_zero (by rw [mul_comm]; exact h6)`? `eq_div_of_mul_eq (hc : c ≠ 0) (h : a * c = b) : a = b / c` ✓: h6 : I·∫ = ΔG: mul_comm: ∫·I = ΔG: eq_div_of_mul_eq I_ne_zero: ∫ = ΔG/I ✓. Then ΔG/I = ΔG·I⁻¹ = ΔG·(−I) = −I·ΔG:\n```\n  rw [h7, Complex.inv_I]\n  ring\n```\nh7 : ∫ = ΔG/I; rw [div_eq_mul_inv, Complex.inv_I]: = ΔG·(−I): ring: −I·ΔG ✓ matches target −I·ΔG ✓.\n\nLet me write the final form directly:\n```\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 :=\n    eq_div_of_mul_eq Complex.I_ne_zero (by rw [mul_comm]; exact h6)\n  rw [h7, div_eq_mul_inv, Complex.inv_I]\n  ring\n```\n\n5. rectInt_deriv_eq_zero's `rw [HIntegral_deriv ...]` — the pattern mismatch: the goal has `HIntegral (fun s => deriv G s) z.re w.re z.im` and HIntegral_deriv's LHS is `∫ x in z.re..w.re, deriv G (x + y * I)` — the HIntegral def unfolds to the interval integral — HIntegral_deriv's statement is about the ∫ form — the rw needs HIntegral unfolded first: `rw [RectangleIntegral, HIntegral, HIntegral, VIntegral, VIntegral]` — unfold all four, then the goal has the ∫ forms, then rw [HIntegral_deriv ...] matches. Let me restructure:\n```\n  rw [RectangleIntegral]\n  rw [HIntegral, HIntegral, VIntegral, VIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\n```\nHmm — VIntegral unfolds to I • ∫ — after rw [VIntegral, VIntegral] the goal has I • ∫ ... — then rw [hVr, hVl] rewrites the ∫ parts ✓ then the algebra: [G(b)−G(a)] − [G(d)−G(c)] + I•(−I·(G(d)−G(b))) − I•(−I·(G(c)−G(a))): I•(−I·X) = X: simp [smul_eq_mul, Complex.I_mul_I] then ring.\n\nWait — the smul: I • (−I * X) — smul_eq_mul: I * (−I * X) = −(I*I)·X = −(−1)·X = X ✓ simp [smul_eq_mul, Complex.I_mul_I]; ring ✓.\n\nLet me apply all these fixes.","signature":""},{"type":"text","text":"Applying the final round of fixes to LiPrincipal."},{"type":"tool_use","id":"Bash_248","name":"Bash","input":{"command":"cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. H-side hd: simpa\ns = s.replace(\"\"\"    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\",\n\"\"\"    have hcomp := h1.comp x h2\n    simpa [Function.comp_apply] using hcomp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\")\n\n# 2. H-side hftc: simpa\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun x : ℝ => G (x + y * I)) (f' := fun x : ℝ => deriv G (x + y * I)) hd hint\n  convert hftc using 1\n  · simp\n  · simp\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun x : ℝ => G (x + y * I)) (f' := fun x : ℝ => deriv G (x + y * I)) hd hint\n  simpa using hftc\"\"\")\n\n# 3. V-side hd: simpa\ns = s.replace(\"\"\"    have hcomp := h1.comp y h2\n    convert hcomp using 1\n    ring\"\"\",\n\"\"\"    have hcomp := h1.comp y h2\n    simpa [Function.comp_apply] using hcomp\"\"\")\n\n# 4. V-side final computation\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun y : ℝ => G (x + y * I)) (f' := fun y : ℝ => I * deriv G (x + y * I)) hd hint\n  rw [← hftc, integral_smul]\n  field_simp [Complex.I_ne_zero]\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun y : ℝ => G (x + y * I)) (f' := fun y : ℝ => I * deriv G (x + y * I)) hd hint\n  have h6 : I * ∫ y in z.im..w.im, deriv G (x + y * I) =\n      G (x + w.im * I) - G (x + z.im * I) := by\n    rw [← intervalIntegral.integral_smul, smul_eq_mul]\n    exact hftc\n  have h7 : ∫ y in z.im..w.im, deriv G (x + y * I) =\n      (G (x + w.im * I) - G (x + z.im * I)) / I :=\n    eq_div_of_mul_eq Complex.I_ne_zero (by rw [mul_comm]; exact h6)\n  rw [h7, div_eq_mul_inv, Complex.inv_I]\n  ring\"\"\")\n\n# 5. rectInt_deriv_eq_zero: unfold first\ns = s.replace(\"\"\"  rw [RectangleIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [VIntegral, VIntegral, hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\"\"\",\n\"\"\"  rw [RectangleIntegral, HIntegral, HIntegral, VIntegral, VIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [hVr, hVl]\n  simp [smul_eq_mul, Complex.I_mul_I]\n  ring\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40","description":"Apply final fixes and recompile LiPrincipal","timeout":420000},"extra_content":null}]