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

tactic: by_cases h₀ : R = 0
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)
state_after: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : R = 0
⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)

case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)

tactic: refine ⟨fun h => h.out, fun h => ?_⟩
state_before: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)
state_after: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)
⊢ CircleIntegrable f c R

tactic: simp only [CircleIntegrable, intervalIntegrable_iff, deriv_circleMap] at h ⊢
state_before: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)
⊢ CircleIntegrable f c R
state_after: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume

tactic: refine (h.norm.const_mul |R|⁻¹).mono' ?_ ?_
state_before: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume
state_after: case neg.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))

case neg.refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)),
    ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖(circleMap 0 R a * I) • f (circleMap c R a)‖

tactic: simp (config := { unfoldPartialApp := true }) [h₀, const]
state_before: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : R = 0
⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)
state_after: no goals

tactic: have H : ∀ {θ}, circleMap 0 R θ * I ≠ 0 := fun {θ} => by simp [h₀, I_ne_zero]
state_before: case neg.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))
state_after: case neg.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
H : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0
⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))

tactic: simpa only [inv_smul_smul₀ H]
  using ((continuous_circleMap 0 R).aestronglyMeasurable.mul_const
    I).aemeasurable.inv.aestronglyMeasurable.smul h.aestronglyMeasurable
state_before: case neg.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
H : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0
⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (volume.restrict (Ι 0 (2 * π)))
state_after: no goals

tactic: simp [h₀, I_ne_zero]
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
θ : ℝ
⊢ circleMap 0 R θ * I ≠ 0
state_after: no goals

tactic: simp [norm_smul, h₀]
state_before: case neg.refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
h₀ : ¬R = 0
h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ι 0 (2 * π)),
    ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖(circleMap 0 R a * I) • f (circleMap c R a)‖
state_after: no goals