internlm/Lean-Github huggingface dataset:
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
⊢ ENNReal.ofReal |y - y'| < ⊤
Tactic:
apply ENNReal.ofReal_lt_top
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
⊢ MeasurableSet (Ι y' y)
Tactic:
apply measurableSet_uIoc
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2
Tactic:
intro t ht
Proof state after:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : t ∈ Ι y' y
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : t ∈ Ι y' y
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Tactic:
rw [Set.mem_uIoc] at ht
Proof state after:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Tactic:
have ht' : 0 < t ∧ t ≤ 1 := by
  rcases ht with ht | ht <;> (constructor <;> linarith)
Proof state after:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2
Tactic:
rw [f'def]
Proof state after:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ ‖(fun t =>
          (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) /
            (1 - (-(Complex.I * ↑t)).exp) ^ 2)
        t‖ ≤
    3 / (y / 2 / 2) ^ 2
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ ‖(fun t =>
          (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) /
            (1 - (-(Complex.I * ↑t)).exp) ^ 2)
        t‖ ≤
    3 / (y / 2 / 2) ^ 2
Tactic:
simp only [norm_div, Complex.norm_eq_abs, norm_pow]
Proof state after:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) /
      Complex.abs (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≤
    3 / (y / 2 / 2) ^ 2
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)
Tactic:
. rw [mul_comm, ←neg_mul, mul_comm]
  norm_cast
  apply lower_secant_bound
  . simp only [neg_mul, Set.mem_Icc, neg_add_le_iff_le_add, le_add_neg_iff_add_le,
    neg_le_sub_iff_le_add]
    constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos]
  . rw [abs_neg, le_abs]
    left
    rcases ht with ht | ht <;> linarith [ht.1]
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
⊢ 0 < t ∧ t ≤ 1
Tactic:
rcases ht with ht | ht <;> (constructor <;> linarith)
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case inr
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y < t ∧ t ≤ y'
⊢ 0 < t ∧ t ≤ 1
Tactic:
constructor <;> linarith
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3
Tactic:
calc Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t)) + Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t)))
  _ ≤ Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
    apply Complex.abs.isAbsoluteValue.abv_add
  _ ≤ Complex.abs (-1) + Complex.abs (Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
    gcongr
    apply Complex.abs.isAbsoluteValue.abv_add
  _ ≤ 1 + 1 + 1 := by
    gcongr
    . simp
    . rw [mul_comm, ←neg_mul]
      norm_cast
      apply le_of_eq
      apply Complex.abs_exp_ofReal_mul_I
    . simp
      apply mul_le_one
      norm_cast
      rw [abs_of_nonpos] <;> linarith
      simp
      rw [mul_comm, ←neg_mul]
      norm_cast
      apply le_of_eq
      apply Complex.abs_exp_ofReal_mul_I
  _ = 3 := by norm_num
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤
    Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp)
Tactic:
apply Complex.abs.isAbsoluteValue.abv_add
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤
    Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp +
      Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp)
Tactic:
gcongr
Proof state after:
case bc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case bc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
  fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
  f' = fun t =>
    (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
c : ℝ → ℂ := fun t => 1 - ↑t
cdef : c = fun t => 1 - ↑t
c' : ℝ → ℂ := fun t => -1
c'def : c' = fun t => -1
d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp
ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp
d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp
d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp
d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0
f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y)
t : ℝ
ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp
Tactic:
apply Complex.abs.isAbsoluteValue.abv_add
Proof state after:
no goals