Datasets:

pkuAI4M
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lean_github

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Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [cauchyPowerSeries_apply f c r n w, smul_eq_mul, Complex.abs.map_mul]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d ⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d ⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	generalize hg : (fun z ↦ (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	have gs : ∀ z ∈ sphere c r, ‖g z‖ ≤ wr * r⁻¹ := by intro z; simp only [mem_sphere_iff_norm, Complex.norm_eq_abs]; intro zr simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀, Complex.abs_pow, map_inv₀] have zb : z ∈ closedBall c r := by simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist, Complex.dist_eq, zr, le_refl] have zs := h z zb calc abs w ^ n / ↑r ^ n * (r⁻¹ * abs (f z)) _ = abs w ^ n * (r⁻¹ ^ n : ℝ≥0) * (r⁻¹ * abs (f z)) := by rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv] _ ≤ abs w ^ n * r⁻¹ ^ n * (r⁻¹ * d) := by bound _ = abs w ^ n * r⁻¹ ^ n * d * r⁻¹ := by ring _ = wr * r⁻¹ := rfl	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	have cn := circleIntegral.norm_integral_le_of_norm_le_const (NNReal.coe_nonneg r) gs	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [Complex.norm_eq_abs] at cn	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	simp only [mul_inv_rev, Complex.inv_I, AbsoluteValue.map_neg, AbsoluteValue.map_mul, Complex.abs_I, map_inv₀, Complex.abs_ofReal, Complex.abs_two, one_mul, div_pow, Algebra.id.smul_eq_mul] at hg cn ⊢	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) ⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	have p3 : \|π\| = π := abs_eq_self.mpr (by bound)	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	calc \|π\|⁻¹ * 2⁻¹ * abs (circleIntegral g c ↑r) _ ≤ \|π\|⁻¹ * 2⁻¹ * (2 * π * r * (wr * r⁻¹)) := by bound _ = π * \|π\|⁻¹ * (r * r⁻¹) * wr := by ring _ = π * π⁻¹ * (r * r⁻¹) * wr := by rw [p3] _ = 1 * (r * r⁻¹) * wr := by rw [mul_inv_cancel Real.pi_ne_zero] _ = wr := by rw [NNReal.coe_inv, mul_inv_cancel (NNReal.coe_ne_zero.mpr rp.ne'), one_mul, one_mul]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	intro z	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g ⊢ ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g ⊢ ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	simp only [mem_sphere_iff_norm, Complex.norm_eq_abs]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	intro zr	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ ⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀, Complex.abs_pow, map_inv₀]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	have zb : z ∈ closedBall c r := by simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist, Complex.dist_eq, zr, le_refl]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	have zs := h z zb	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	calc abs w ^ n / ↑r ^ n * (r⁻¹ * abs (f z)) _ = abs w ^ n * (r⁻¹ ^ n : ℝ≥0) * (r⁻¹ * abs (f z)) := by rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv] _ ≤ abs w ^ n * r⁻¹ ^ n * (r⁻¹ * d) := by bound _ = abs w ^ n * r⁻¹ ^ n * d * r⁻¹ := by ring _ = wr * r⁻¹ := rfl	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist, Complex.dist_eq, zr, le_refl]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ z ∈ closedBall c ↑r	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r ⊢ z ∈ closedBall c ↑r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n / ↑r ^ n * (↑r⁻¹ * Complex.abs (f z)) = Complex.abs w ^ n * ↑(r⁻¹ ^ n) * (↑r⁻¹ * Complex.abs (f z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n / ↑r ^ n * (↑r⁻¹ * Complex.abs (f z)) = Complex.abs w ^ n * ↑(r⁻¹ ^ n) * (↑r⁻¹ * Complex.abs (f z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	bound	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n * ↑(r⁻¹ ^ n) * (↑r⁻¹ * Complex.abs (f z)) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * (↑r⁻¹ * ↑d)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n * ↑(r⁻¹ ^ n) * (↑r⁻¹ * Complex.abs (f z)) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * (↑r⁻¹ * ↑d) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	ring	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n * ↑r⁻¹ ^ n * (↑r⁻¹ * ↑d) = Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d * ↑r⁻¹	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g z : ℂ zr : Complex.abs (z - c) = ↑r zb : z ∈ closedBall c ↑r zs : Complex.abs (f z) ≤ ↑d ⊢ Complex.abs w ^ n * ↑r⁻¹ ^ n * (↑r⁻¹ * ↑d) = Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d * ↑r⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	bound	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g ⊢ 0 ≤ π	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g ⊢ 0 ≤ π TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	bound	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ \|π\|⁻¹ * 2⁻¹ * (2 * π * ↑r * (wr * ↑r⁻¹))	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ \|π\|⁻¹ * 2⁻¹ * (2 * π * ↑r * (wr * ↑r⁻¹)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	ring	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * (2 * π * ↑r * (wr * ↑r⁻¹)) = π * \|π\|⁻¹ * (↑r * ↑r⁻¹) * wr	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ \|π\|⁻¹ * 2⁻¹ * (2 * π * ↑r * (wr * ↑r⁻¹)) = π * \|π\|⁻¹ * (↑r * ↑r⁻¹) * wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [p3]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ π * \|π\|⁻¹ * (↑r * ↑r⁻¹) * wr = π * π⁻¹ * (↑r * ↑r⁻¹) * wr	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ π * \|π\|⁻¹ * (↑r * ↑r⁻¹) * wr = π * π⁻¹ * (↑r * ↑r⁻¹) * wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [mul_inv_cancel Real.pi_ne_zero]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ π * π⁻¹ * (↑r * ↑r⁻¹) * wr = 1 * (↑r * ↑r⁻¹) * wr	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ π * π⁻¹ * (↑r * ↑r⁻¹) * wr = 1 * (↑r * ↑r⁻¹) * wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_bound	[89, 1]	[121, 92]	rw [NNReal.coe_inv, mul_inv_cancel (NNReal.coe_ne_zero.mpr rp.ne'), one_mul, one_mul]	f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ 1 * (↑r * ↑r⁻¹) * wr = wr	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r d : ℝ≥0 w : ℂ n : ℕ rp : r > 0 h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d g : ℂ → ℂ gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹) hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) = g p3 : \|π\| = π ⊢ 1 * (↑r * ↑r⁻¹) * wr = wr TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [circleIntegral]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r ⊢ circleIntegral f c r - circleIntegral g c r = circleIntegral (f - g) c r	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r ⊢ (∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)) - circleIntegral g c r = circleIntegral (f - g) c r	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r ⊢ circleIntegral f c r - circleIntegral g c r = circleIntegral (f - g) c r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	generalize hf : (fun θ : ℝ ↦ deriv (circleMap c r) θ • f (circleMap c r θ)) = fc	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r ⊢ (∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)) - circleIntegral g c r = circleIntegral (f - g) c r	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - circleIntegral g c r = circleIntegral (f - g) c r	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r ⊢ (∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)) - circleIntegral g c r = circleIntegral (f - g) c r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [circleIntegral]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - circleIntegral g c r = circleIntegral (f - g) c r	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • g (circleMap c r θ) = circleIntegral (f - g) c r	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - circleIntegral g c r = circleIntegral (f - g) c r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	generalize hg : (fun θ : ℝ ↦ deriv (circleMap c r) θ • g (circleMap c r θ)) = gc	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • g (circleMap c r θ) = circleIntegral (f - g) c r	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = circleIntegral (f - g) c r	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc ⊢ intervalIntegral fc 0 (2 * π) volume - ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • g (circleMap c r θ) = circleIntegral (f - g) c r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [circleIntegral]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = circleIntegral (f - g) c r	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • (f - g) (circleMap c r θ)	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = circleIntegral (f - g) c r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	generalize hfg : (fun θ : ℝ ↦ deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • (f - g) (circleMap c r θ)	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • (f - g) (circleMap c r θ) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	have hs : fc - gc = fgc := by rw [← hf, ← hg, ← hfg]; funext simp only [deriv_circleMap, Algebra.id.smul_eq_mul, Pi.sub_apply, mul_sub_left_distrib]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [← hs]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral fgc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	clear hfg hs fgc	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc hs : fc - gc = fgc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	symm	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = intervalIntegral (fc - gc) 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	have fci := CircleIntegrable.out fi	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [hf] at fci	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	have gci := CircleIntegrable.out gi	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [hg] at gci	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable gc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	exact intervalIntegral.integral_sub fci gci	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable gc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume	no goals	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fci : IntervalIntegrable fc volume 0 (2 * π) gci : IntervalIntegrable gc volume 0 (2 * π) ⊢ intervalIntegral (fc - gc) 0 (2 * π) volume = intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	rw [← hf, ← hg, ← hfg]	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ fc - gc = fgc	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ fc - gc = fgc TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	funext	f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)	case h f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc x✝ : ℝ ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) x✝ = deriv (circleMap c r) x✝ • (f - g) (circleMap c r x✝)	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleIntegral_sub	[123, 1]	[138, 46]	simp only [deriv_circleMap, Algebra.id.smul_eq_mul, Pi.sub_apply, mul_sub_left_distrib]	case h f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc x✝ : ℝ ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) x✝ = deriv (circleMap c r) x✝ • (f - g) (circleMap c r x✝)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h f g : ℂ → ℂ c : ℂ r : ℝ fi : CircleIntegrable f c r gi : CircleIntegrable g c r fc : ℝ → ℂ hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc gc : ℝ → ℂ hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc fgc : ℝ → ℂ hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc x✝ : ℝ ⊢ ((fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) - fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) x✝ = deriv (circleMap c r) x✝ • (f - g) (circleMap c r x✝) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleMap_nz	[140, 1]	[142, 67]	simp only [circleMap_sub_center, Ne, circleMap_eq_center_iff, NNReal.coe_eq_zero]	c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 ⊢ circleMap c (↑r) θ - c ≠ 0	c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 ⊢ ¬r = 0	Please generate a tactic in lean4 to solve the state. STATE: c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 ⊢ circleMap c (↑r) θ - c ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleMap_nz	[140, 1]	[142, 67]	intro h	c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 ⊢ ¬r = 0	c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 h : r = 0 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 ⊢ ¬r = 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleMap_nz	[140, 1]	[142, 67]	rw [h] at rp	c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 h : r = 0 ⊢ False	c : ℂ r : ℝ≥0 θ : ℝ rp : 0 > 0 h : r = 0 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: c : ℂ r : ℝ≥0 θ : ℝ rp : r > 0 h : r = 0 ⊢ False TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	circleMap_nz	[140, 1]	[142, 67]	simp only [gt_iff_lt, not_lt_zero'] at rp	c : ℂ r : ℝ≥0 θ : ℝ rp : 0 > 0 h : r = 0 ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: c : ℂ r : ℝ≥0 θ : ℝ rp : 0 > 0 h : r = 0 ⊢ False TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine ContinuousOn.intervalIntegrable ?_	f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r	f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine ContinuousOn.mul ?_ ?_	f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => w ^ n / (circleMap c (↑r) θ - c) ^ n) (Set.uIcc 0 (2 * π)) case refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹ * f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine ContinuousOn.mul continuousOn_const ?_	case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => w ^ n / (circleMap c (↑r) θ - c) ^ n) (Set.uIcc 0 (2 * π))	case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹) (Set.uIcc 0 (2 * π))	Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => w ^ n / (circleMap c (↑r) θ - c) ^ n) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	apply Continuous.continuousOn	case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹) (Set.uIcc 0 (2 * π))	case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹	Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine Continuous.inv₀ ?_ fun x ↦ pow_ne_zero n (circleMap_nz rp)	case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹	case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c) ^ n	Please generate a tactic in lean4 to solve the state. STATE: case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	apply Continuous.pow	case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c) ^ n	case refine_1.h.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun b => circleMap c (↑r) b - c	Please generate a tactic in lean4 to solve the state. STATE: case refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	continuity	case refine_1.h.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun b => circleMap c (↑r) b - c	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_1.h.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun b => circleMap c (↑r) b - c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine ContinuousOn.mul ?_ ?_	case refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹ * f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	case refine_2.refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹) (Set.uIcc 0 (2 * π)) case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹ * f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	apply Continuous.continuousOn	case refine_2.refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹) (Set.uIcc 0 (2 * π))	case refine_2.refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c)⁻¹	Please generate a tactic in lean4 to solve the state. STATE: case refine_2.refine_1 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine Continuous.inv₀ (by continuity) fun x ↦ circleMap_nz rp	case refine_2.refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c)⁻¹	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_2.refine_1.h f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => (circleMap c (↑r) θ - c)⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	continuity	f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => circleMap c (↑r) θ - c	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => circleMap c (↑r) θ - c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	refine ContinuousOn.comp h (Continuous.continuousOn (by continuity)) ?_	case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π))	case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Set.MapsTo (fun θ => circleMap c (↑r) θ) (Set.uIcc 0 (2 * π)) (closedBall c ↑r)	Please generate a tactic in lean4 to solve the state. STATE: case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ ContinuousOn (fun θ => f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	intro θ _	case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Set.MapsTo (fun θ => circleMap c (↑r) θ) (Set.uIcc 0 (2 * π)) (closedBall c ↑r)	case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) θ : ℝ a✝ : θ ∈ Set.uIcc 0 (2 * π) ⊢ (fun θ => circleMap c (↑r) θ) θ ∈ closedBall c ↑r	Please generate a tactic in lean4 to solve the state. STATE: case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Set.MapsTo (fun θ => circleMap c (↑r) θ) (Set.uIcc 0 (2 * π)) (closedBall c ↑r) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	exact circleMap_mem_closedBall c (NNReal.coe_nonneg r) θ	case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) θ : ℝ a✝ : θ ∈ Set.uIcc 0 (2 * π) ⊢ (fun θ => circleMap c (↑r) θ) θ ∈ closedBall c ↑r	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_2.refine_2 f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) θ : ℝ a✝ : θ ∈ Set.uIcc 0 (2 * π) ⊢ (fun θ => circleMap c (↑r) θ) θ ∈ closedBall c ↑r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_is_circleIntegrable	[144, 1]	[157, 74]	continuity	f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => circleMap c (↑r) θ	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 h : ContinuousOn f (closedBall c ↑r) ⊢ Continuous fun θ => circleMap c (↑r) θ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	rw [cauchyPowerSeries_apply f c r n w]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((cauchyPowerSeries f c (↑r) n) fun x => w) - (cauchyPowerSeries g c (↑r) n) fun x => w) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (cauchyPowerSeries g c (↑r) n) fun x => w) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((cauchyPowerSeries f c (↑r) n) fun x => w) - (cauchyPowerSeries g c (↑r) n) fun x => w) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	rw [cauchyPowerSeries_apply g c r n w]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (cauchyPowerSeries g c (↑r) n) fun x => w) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (cauchyPowerSeries g c (↑r) n) fun x => w) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	rw [cauchyPowerSeries_apply (f - g) c r n w]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (cauchyPowerSeries (f - g) c (↑r) n) fun x => w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	set s : ℂ := (2 * π * I)⁻¹	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	simp only [div_pow, Algebra.id.smul_eq_mul, Pi.sub_apply]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) - s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) = s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • (f - g) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	have fi := cauchy_is_circleIntegrable n w rp cf	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	have gi := cauchy_is_circleIntegrable n w rp cg	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	have cia := circleIntegral_sub fi gi	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	rw [← mul_sub_left_distrib, cia]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	clear cia fi gi cf cg rp	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) s : ℂ := (2 * ↑π * I)⁻¹ fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r cia : ((∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	have flip : ((fun z ↦ w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z ↦ w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z ↦ w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z) := rfl	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ flip : ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z) ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_sub	[159, 1]	[178, 41]	simp only [flip, mul_sub_left_distrib]	f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ flip : ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z) ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r : ℝ≥0 n : ℕ w : ℂ s : ℂ := (2 * ↑π * I)⁻¹ flip : ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z) ⊢ s * circleIntegral ((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r = s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_dist	[180, 1]	[186, 69]	rw [Complex.dist_eq, cauchy_sub n w rp cf cg]	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ dist ((cauchyPowerSeries f c (↑r) n) fun x => w) ((cauchyPowerSeries g c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ Complex.abs ((cauchyPowerSeries (f - g) c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ dist ((cauchyPowerSeries f c (↑r) n) fun x => w) ((cauchyPowerSeries g c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_dist	[180, 1]	[186, 69]	refine cauchy_bound rp ?_	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ Complex.abs ((cauchyPowerSeries (f - g) c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ ∀ w ∈ closedBall c ↑r, Complex.abs ((f - g) w) ≤ ↑d	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ Complex.abs ((cauchyPowerSeries (f - g) c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_dist	[180, 1]	[186, 69]	intro z zr	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ ∀ w ∈ closedBall c ↑r, Complex.abs ((f - g) w) ≤ ↑d	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d z : ℂ zr : z ∈ closedBall c ↑r ⊢ Complex.abs ((f - g) z) ≤ ↑d	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d ⊢ ∀ w ∈ closedBall c ↑r, Complex.abs ((f - g) w) ≤ ↑d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_dist	[180, 1]	[186, 69]	simp at h zr	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d z : ℂ zr : z ∈ closedBall c ↑r ⊢ Complex.abs ((f - g) z) ≤ ↑d	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) z : ℂ h : ∀ (z : ℂ), nndist z c ≤ r → Complex.abs (f z - g z) ≤ ↑d zr : nndist z c ≤ r ⊢ Complex.abs ((f - g) z) ≤ ↑d	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d z : ℂ zr : z ∈ closedBall c ↑r ⊢ Complex.abs ((f - g) z) ≤ ↑d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	cauchy_dist	[180, 1]	[186, 69]	refine h z zr	f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) z : ℂ h : ∀ (z : ℂ), nndist z c ≤ r → Complex.abs (f z - g z) ≤ ↑d zr : nndist z c ≤ r ⊢ Complex.abs ((f - g) z) ≤ ↑d	no goals	Please generate a tactic in lean4 to solve the state. STATE: f g : ℂ → ℂ c : ℂ r d : ℝ≥0 n : ℕ w : ℂ rp : r > 0 cf : ContinuousOn f (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) z : ℂ h : ∀ (z : ℂ), nndist z c ≤ r → Complex.abs (f z - g z) ≤ ↑d zr : nndist z c ≤ r ⊢ Complex.abs ((f - g) z) ≤ ↑d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	intro c hc	I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s ⊢ AnalyticOn ℂ g s	I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s ⊢ AnalyticOn ℂ g s TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	rcases Metric.nhds_basis_closedBall.mem_iff.mp (o.mem_nhds hc) with ⟨r, rp, cb⟩	I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s ⊢ AnalyticAt ℂ g c	case intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ rp : 0 < r cb : closedBall c r ⊆ s ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	lift r to ℝ≥0 using rp.le	case intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ rp : 0 < r cb : closedBall c r ⊆ s ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 rp : 0 < ↑r cb : closedBall c ↑r ⊆ s ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ rp : 0 < r cb : closedBall c r ⊆ s ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	simp only [NNReal.coe_pos] at rp	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 rp : 0 < ↑r cb : closedBall c ↑r ⊆ s ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 rp : 0 < ↑r cb : closedBall c ↑r ⊆ s ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have hb : ∀ n, AnalyticOn ℂ (f n) (closedBall c r) := fun n ↦ (h n).mono cb	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	set pr := fun n ↦ cauchyPowerSeries (f n) c r	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have hpf : ∀ n, HasFPowerSeriesOnBall (f n) (pr n) c r := by intro n have cs := cauchy_on_cball_radius rp (hb n) have pn : pr n = cauchyPowerSeries (f n) c r := rfl rw [← pn] at cs; exact cs	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have cfs : ∀ n, ContinuousOn (f n) s := fun n ↦ AnalyticOn.continuousOn (h n)	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have cf : ∀ n, ContinuousOn (f n) (closedBall c r) := fun n ↦ ContinuousOn.mono (cfs n) cb	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have cg : ContinuousOn g (closedBall c r) := ContinuousOn.mono (TendstoUniformlyOn.continuousOn u (Filter.eventually_of_forall cfs)) cb	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	clear h hb hc o cfs	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ o : IsOpen s h : ∀ (n : I), AnalyticOn ℂ (f n) s u : TendstoUniformlyOn f g atTop s c : ℂ hc : c ∈ s r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r) pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cfs : ∀ (n : I), ContinuousOn (f n) s cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	set p := cauchyPowerSeries g c r	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ AnalyticAt ℂ g c	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	refine HasFPowerSeriesOnBall.analyticAt { r_le := le_radius_cauchyPowerSeries g c r r_pos := ENNReal.coe_pos.mpr rp hasSum := ?_ }	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ AnalyticAt ℂ g c	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ AnalyticAt ℂ g c TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	intro y yb	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r ⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have yr := yb	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb yr : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	simp at yr	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb yr : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb yr : y ∈ EMetric.ball 0 ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	set a := abs y / r	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have a0 : a ≥ 0 := by bound	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have a1 : a < 1 := (div_lt_one (NNReal.coe_pos.mpr rp)).mpr yr	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	have a1p : 1 - a > 0 := by bound	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 a1p : 1 - a > 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Analytic/Uniform.lean	uniform_analytic_lim	[189, 1]	[270, 21]	rw [HasSum, Metric.tendsto_atTop]	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 a1p : 1 - a > 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y))	case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 a1p : 1 - a > 0 ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => (cauchyPowerSeries g c (↑r) b) fun x => y) (g (c + y)) < ε	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I → ℂ → ℂ g : ℂ → ℂ s : Set ℂ u : TendstoUniformlyOn f g atTop s c : ℂ r : ℝ≥0 cb : closedBall c ↑r ⊆ s rp : 0 < r pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : ∀ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries g c ↑r y : ℂ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a ≥ 0 a1 : a < 1 a1p : 1 - a > 0 ⊢ HasSum (fun n => (cauchyPowerSeries g c (↑r) n) fun x => y) (g (c + y)) TACTIC:

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