Theorem: norm_cauchyPowerSeries_le
File: Mathlib/MeasureTheory/Integral/CircleIntegral.lean

tactic: simp [cauchyPowerSeries, norm_smul, Real.pi_pos.le]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ ‖cauchyPowerSeries f c R n‖ = (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖
state_after: no goals

tactic: simp [Real.pi_pos.le]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ 0 ≤ (2 * π)⁻¹
state_after: no goals

tactic: simp [norm_smul, mul_left_comm |R|]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ (2 * π)⁻¹ *
      ∫ (θ : ℝ) in 0 ..2 * π,
        ‖deriv (circleMap c R) θ • (circleMap c R θ - c)⁻¹ ^ n • (circleMap c R θ - c)⁻¹ • f (circleMap c R θ)‖ =
    (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖)))
state_after: no goals

tactic: rcases eq_or_ne R 0 with (rfl | hR)
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n
state_after: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
n : ℕ
⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n

case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n

tactic: cases n <;> simp [-mul_inv_rev]
state_before: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
n : ℕ
⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n
state_after: case inl.zero
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
⊢ 0 ≤ (2 * π)⁻¹ * (2 * π * ‖f c‖)

tactic: rw [← mul_assoc, inv_mul_cancel (Real.two_pi_pos.ne.symm), one_mul]
state_before: case inl.zero
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
⊢ 0 ≤ (2 * π)⁻¹ * (2 * π * ‖f c‖)
state_after: case inl.zero
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
⊢ 0 ≤ ‖f c‖

tactic: apply norm_nonneg
state_before: case inl.zero
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
⊢ 0 ≤ ‖f c‖
state_after: no goals

tactic: rw [mul_inv_cancel_left₀, mul_assoc, mul_comm (|R|⁻¹ ^ n)]
state_before: case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n
state_after: case inr.h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ |R| ≠ 0

tactic: rwa [Ne, _root_.abs_eq_zero]
state_before: case inr.h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ |R| ≠ 0
state_after: no goals