internlm/Lean-Github huggingface dataset:
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
⊢ Summable maj
Tactic:
by_cases Ceq0 : C = 0
Proof state after:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable maj

case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable maj
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable maj

case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable maj
Tactic:
. rw [maj_def, Ceq0]
  simp only [one_div, mul_zero]
  exact summable_zero
Proof state after:
case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable maj
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable maj
Tactic:
. rw [← summable_div_const_iff Ceq0]
  convert Real.summable_one_div_int_pow.mpr one_lt_two using 1
  rw [maj_def]
  ext i
  simp only [one_div]
  rw [mul_div_cancel_right₀]
  exact Ceq0
Proof state after:
no goals
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable maj
Tactic:
rw [maj_def, Ceq0]
Proof state after:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable fun i => 1 / ↑i ^ 2 * 0
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable fun i => 1 / ↑i ^ 2 * 0
Tactic:
simp only [one_div, mul_zero]
Proof state after:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable fun i => 0
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case pos
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : C = 0
⊢ Summable fun i => 0
Tactic:
exact summable_zero
Proof state after:
no goals
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable maj
Tactic:
rw [← summable_div_const_iff Ceq0]
Proof state after:
case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable fun i => maj i / C
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case neg
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ Summable fun i => maj i / C
Tactic:
convert Real.summable_one_div_int_pow.mpr one_lt_two using 1
Proof state after:
case h.e'_5
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ (fun i => maj i / C) = fun n => 1 / ↑n ^ 2
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case h.e'_5
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ (fun i => maj i / C) = fun n => 1 / ↑n ^ 2
Tactic:
rw [maj_def]
Proof state after:
case h.e'_5
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ (fun i => (fun i => 1 / ↑i ^ 2 * C) i / C) = fun n => 1 / ↑n ^ 2
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case h.e'_5
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
⊢ (fun i => (fun i => 1 / ↑i ^ 2 * C) i / C) = fun n => 1 / ↑n ^ 2
Tactic:
ext i
Proof state after:
case h.e'_5.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ (fun i => 1 / ↑i ^ 2 * C) i / C = 1 / ↑i ^ 2
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case h.e'_5.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ (fun i => 1 / ↑i ^ 2 * C) i / C = 1 / ↑i ^ 2
Tactic:
simp only [one_div]
Proof state after:
case h.e'_5.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ (↑i ^ 2)⁻¹ * C / C = (↑i ^ 2)⁻¹
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case h.e'_5.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ (↑i ^ 2)⁻¹ * C / C = (↑i ^ 2)⁻¹
Tactic:
rw [mul_div_cancel_right₀]
Proof state after:
case h.e'_5.h.hb
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ C ≠ 0
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
case h.e'_5.h.hb
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
Ceq0 : ¬C = 0
i : ℤ
⊢ C ≠ 0
Tactic:
exact Ceq0
Proof state after:
no goals
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
⊢ 0 ≤ fun x => ‖fourierCoeffOn ⋯ f x‖
Tactic:
intro i
Proof state after:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
i : ℤ
⊢ 0 i ≤ (fun x => ‖fourierCoeffOn ⋯ f x‖) i
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
i : ℤ
⊢ 0 i ≤ (fun x => ‖fourierCoeffOn ⋯ f x‖) i
Tactic:
simp
Proof state after:
no goals
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
⊢ ∀ (i : ℤ), i ≠ 0 → ‖fourierCoeffOn ⋯ f i‖ ≤ maj i
Tactic:
intro i ine0
Proof state after:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
i : ℤ
ine0 : i ≠ 0
⊢ ‖fourierCoeffOn ⋯ f i‖ ≤ maj i
Theorem: fourierConv_ofTwiceDifferentiable
Proof state before:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
i : ℤ
ine0 : i ≠ 0
⊢ ‖fourierCoeffOn ⋯ f i‖ ≤ maj i
Tactic:
rw [maj_def, Complex.norm_eq_abs]
Proof state after:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : ∀ {i : ℤ}, fourierCoeff (⇑g) i = fourierCoeffOn ⋯ f i
C : ℝ
hC : ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
maj : ℤ → ℝ := fun i => 1 / ↑i ^ 2 * C
maj_def : maj = fun i => 1 / ↑i ^ 2 * C
summable_maj : Summable maj
i : ℤ
ine0 : i ≠ 0
⊢ Complex.abs (fourierCoeffOn ⋯ f i) ≤ (fun i => 1 / ↑i ^ 2 * C) i