internlm/Lean-Github huggingface dataset:
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
⊢ ∀ a ∈ Ι 0 (2 * π), ‖(circleMap 0 r a * I) • f n (circleMap c r a)‖ ≤ r * b n
Tactic:
intro t _
Proof state after:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ ‖(circleMap 0 r t * I) • f n (circleMap c r t)‖ ≤ r * b n
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ ‖(circleMap 0 r t * I) • f n (circleMap c r t)‖ ≤ r * b n
Tactic:
rw [norm_smul, Complex.norm_eq_abs]
Proof state after:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ Complex.abs (circleMap 0 r t * I) * ‖f n (circleMap c r t)‖ ≤ r * b n
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ Complex.abs (circleMap 0 r t * I) * ‖f n (circleMap c r t)‖ ≤ r * b n
Tactic:
simp
Proof state after:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ |r| * ‖f n (circleMap c r t)‖ ≤ r * b n
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ |r| * ‖f n (circleMap c r t)‖ ≤ r * b n
Tactic:
rw [abs_of_pos rp]
Proof state after:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ r * ‖f n (circleMap c r t)‖ ≤ r * b n
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ r * ‖f n (circleMap c r t)‖ ≤ r * b n
Tactic:
refine mul_le_mul_of_nonneg_left ?_ rp.le
Proof state after:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ ‖f n (circleMap c r t)‖ ≤ b n
Theorem: sum_integral_commute
Proof state before:
case h_bound.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
n : ℕ
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ ‖f n (circleMap c r t)‖ ≤ b n
Tactic:
exact fb n (circleMap c r t) (circleMap_mem_sphere _ (by linarith) _)
Proof state after:
no goals
Theorem: sum_integral_commute
Proof state before:
case bound_summable
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ᵐ (t : ℝ) ∂MeasureTheory.volume, t ∈ Ι 0 (2 * π) → Summable fun n => r * b n
Tactic:
apply MeasureTheory.ae_of_all
Proof state after:
case bound_summable.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π), Summable fun n => r * b n
Theorem: sum_integral_commute
Proof state before:
case bound_summable.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π), Summable fun n => r * b n
Tactic:
intro t _
Proof state after:
case bound_summable.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ Summable fun n => r * b n
Theorem: sum_integral_commute
Proof state before:
case bound_summable.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ Summable fun n => r * b n
Tactic:
exact Summable.mul_left _ bs
Proof state after:
no goals
Theorem: sum_integral_commute
Proof state before:
case bound_integrable
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ IntervalIntegrable (fun t => ∑' (n : ℕ), r * b n) MeasureTheory.volume 0 (2 * π)
Tactic:
simp only [_root_.intervalIntegrable_const]
Proof state after:
no goals
Theorem: sum_integral_commute
Proof state before:
case h_lim
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ᵐ (t : ℝ) ∂MeasureTheory.volume,
    t ∈ Ι 0 (2 * π) →
      HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t))
Tactic:
apply MeasureTheory.ae_of_all
Proof state after:
case h_lim.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π),
    HasSum (fun n => (circleMap 0 r a * I) • f n (circleMap c r a)) ((circleMap 0 r a * I) • g (circleMap c r a))
Theorem: sum_integral_commute
Proof state before:
case h_lim.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π),
    HasSum (fun n => (circleMap 0 r a * I) • f n (circleMap c r a)) ((circleMap 0 r a * I) • g (circleMap c r a))
Tactic:
intro t _
Proof state after:
case h_lim.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t))
Theorem: sum_integral_commute
Proof state before:
case h_lim.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t))
Tactic:
apply HasSum.const_smul
Proof state after:
case h_lim.a.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun i => f i (circleMap c r t)) (g (circleMap c r t))
Theorem: sum_integral_commute
Proof state before:
case h_lim.a.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun i => f i (circleMap c r t)) (g (circleMap c r t))
Tactic:
exact h (circleMap c r t) (circleMap_mem_sphere _ (by linarith) _)
Proof state after:
no goals
Theorem: sum_integral_commute
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ 0 ≤ r
Tactic:
linarith
Proof state after:
no goals
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∮ (z : ℂ) in C(c, r), f z‖ ≤ 2 * π * r * b
Tactic:
rw [circleIntegral]
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Tactic:
simp only [deriv_circleMap]
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Tactic:
have nonneg_2π := Real.two_pi_pos.le
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Tactic:
have ib : ‖(∫ t in (0)..(2*π), (circleMap 0 r t * I) • f (circleMap c r t))‖ ≤
    (∫ t in (0)..(2*π), ‖(circleMap 0 r t * I) • f (circleMap c r t)‖) :=
  intervalIntegral.norm_integral_le_integral_norm nonneg_2π
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
  ‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
    ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
  ‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
    ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
Tactic:
refine le_trans ib ?_
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
  ‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
    ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
  ‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
    ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
Tactic:
clear ib
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
Tactic:
simp_rw [norm_smul, Complex.norm_eq_abs]
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, Complex.abs (circleMap 0 r t * I) * ‖f (circleMap c r t)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, Complex.abs (circleMap 0 r t * I) * ‖f (circleMap c r t)‖ ≤ 2 * π * r * b
Tactic:
simp only [map_mul, abs_circleMap_zero, Complex.abs_I, mul_one, integral_const_mul]
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Tactic:
have mo : ∀ t, t ∈ Set.Icc 0 (2 * π) → ‖f (circleMap c r t)‖ ≤ b := fun t _ ↦
  fb (circleMap c r t) (circleMap_mem_sphere c (by linarith) t)
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Tactic:
have i0 : IntervalIntegrable (fun t ↦ ‖f (circleMap c r t)‖) Real.measureSpace.volume
    0 (2*π) := by
  apply ContinuousOn.intervalIntegrable
  have ca : ContinuousOn (norm : E → ℝ) Set.univ := Continuous.continuousOn continuous_norm
  refine ContinuousOn.comp ca ?_ (Set.mapsTo_univ _ _)
  apply ContinuousOn.comp fc
  exact Continuous.continuousOn (continuous_circleMap _ _)
  intro t _; exact circleMap_mem_sphere _ (by linarith) _
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Theorem: bounded_circleIntegral
Proof state before:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
Tactic:
have i1 : IntervalIntegrable (fun _ ↦ b) Real.measureSpace.volume 0 (2 * π) :=
  intervalIntegrable_const
Proof state after:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b