Theorem: Complex.circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable
File: Mathlib/Analysis/Complex/CauchyIntegral.lean

Proof steps:
state_before: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
r R : ℝ
h0 : 0 < r
hle : r ≤ R
f : ℂ → E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ ball c r)
hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z
state_after: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
r R : ℝ
h0 : 0 < r
hle : r ≤ R
f : ℂ → E
s : Set ℂ
hs : s.Countable
hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c R \ ball c r
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z
tactic: set A := closedBall c R \ ball c r
state_before: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
r R : ℝ
h0 : 0 < r
hle : r ≤ R
f : ℂ → E
s : Set ℂ
hs : s.Countable
hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c R \ ball c r
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z
state_after: case intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
hle : rexp a ≤ R
hd : ∀ z ∈ (ball c R \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c R \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
tactic: obtain ⟨a, rfl⟩ : ∃ a, Real.exp a = r := ⟨Real.log r, Real.exp_log h0⟩
state_before: case intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
hle : rexp a ≤ R
hd : ∀ z ∈ (ball c R \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c R \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : rexp a ≤ rexp b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, rexp b), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
tactic: obtain ⟨b, rfl⟩ : ∃ b, Real.exp b = R := ⟨Real.log R, Real.exp_log (h0.trans_le hle)⟩
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : rexp a ≤ rexp b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, rexp b), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, rexp b), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
tactic: rw [Real.exp_le_exp] at hle
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ (∮ (z : ℂ) in C(c, rexp b), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, rexp a), (z - c)⁻¹ • f z
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: suffices
    (∫ θ in (0)..2 * π, I • f (circleMap c (Real.exp b) θ)) =
      ∫ θ in (0)..2 * π, I • f (circleMap c (Real.exp a) θ) by
    simpa only [circleIntegral, add_sub_cancel_left, ofReal_exp, ← exp_add, smul_smul, ←
      div_eq_mul_inv, mul_div_cancel_left₀ _ (circleMap_ne_center (Real.exp_pos _).ne'),
      circleMap_sub_center, deriv_circleMap]
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: set R := [[a, b]] ×ℂ [[0, 2 * π]]
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: have hdg : Differentiable ℂ g := differentiable_exp.const_add _
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: replace hs : (g ⁻¹' s).Countable := (hs.preimage (add_right_injective c)).preimage_cexp
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: have h_maps : MapsTo g R A := by rintro z ⟨h, -⟩; simpa [g, A, dist_eq, abs_exp, hle] using h.symm
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
hc : ContinuousOn f A
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: replace hc : ContinuousOn (f ∘ g) R := hc.comp hdg.continuous.continuousOn h_maps
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
hd : ∀ z ∈ (ball c (rexp b) \ closedBall c (rexp a)) \ s, DifferentiableAt ℂ f z
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
hd : ∀ z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s, DifferentiableAt ℂ (f ∘ g) z
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
tactic: replace hd : ∀ z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s,
      DifferentiableAt ℂ (f ∘ g) z := by
    refine fun z hz => (hd (g z) ⟨?_, hz.2⟩).comp z (hdg _)
    simpa [g, dist_eq, abs_exp, hle, and_comm] using hz.1.1
state_before: case intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
f : ℂ → E
s : Set ℂ
a : ℝ
h0 : 0 < rexp a
b : ℝ
hle : a ≤ b
A : Set ℂ := closedBall c (rexp b) \ ball c (rexp a)
R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]
g : ℂ → ℂ := fun x => c + cexp x
hdg : Differentiable ℂ g
hs : (g ⁻¹' s).Countable
h_maps : MapsTo g R A
hc : ContinuousOn (f ∘ g) R
hd : ∀ z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s, DifferentiableAt ℂ (f ∘ g) z
⊢ ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)
state_after: no goals
tactic: simpa [g, circleMap, exp_periodic _, sub_eq_zero, ← exp_add] using
    integral_boundary_rect_eq_zero_of_differentiable_on_off_countable _ ⟨a, 0⟩ ⟨b, 2 * π⟩ _ hs hc hd