internlm/Lean-Github huggingface dataset:
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:
norm_cast
Proof state after:
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)
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab.xAbs
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 ≤ |(-t)|
Tactic:
. 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:
case h.hbd.hab.xIcc
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
⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 2)
Tactic:
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]
Proof state after:
case h.hbd.hab.xIcc
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 + t ≤ 2 * Real.pi ∧ y / 2 ≤ 2 * Real.pi + t
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab.xIcc
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 + t ≤ 2 * Real.pi ∧ y / 2 ≤ 2 * Real.pi + t
Tactic:
constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos]
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab.xAbs
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 ≤ |(-t)|
Tactic:
rw [abs_neg, le_abs]
Proof state after:
case h.hbd.hab.xAbs
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 ≤ t ∨ y / 2 ≤ -t
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab.xAbs
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 ≤ t ∨ y / 2 ≤ -t
Tactic:
left
Proof state after:
case h.hbd.hab.xAbs.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
⊢ y / 2 ≤ t
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.hbd.hab.xAbs.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
⊢ y / 2 ≤ t
Tactic:
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) in Ι y' y, 3 / (y / 2 / 2) ^ 2 = (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2)
Tactic:
apply MeasureTheory.setIntegral_const
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)
⊢ (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2) = |y - y'| * (3 / (y / 2 / 2) ^ 2)
Tactic:
congr
Proof state after:
case e_a
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)
⊢ (MeasureTheory.volume (Ι y' y)).toReal = |y - y'|
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case e_a
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)
⊢ (MeasureTheory.volume (Ι y' y)).toReal = |y - y'|
Tactic:
rw [Real.volume_uIoc, ENNReal.toReal_ofReal (abs_nonneg (y - y'))]
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)
⊢ |y - y'| * (3 / (y / 2 / 2) ^ 2) = 3 * (2 * 2) ^ 2 * (1 / y) * (|y - y'| / y)
Tactic:
ring
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)
⊢ 3 * (2 * 2) ^ 2 * (1 / y) * (|y - y'| / y) ≤ 2 ^ 6 * (1 / y) * (|y - y'| / y)
Tactic:
gcongr
Proof state after:
case h.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)
⊢ 3 * (2 * 2) ^ 2 ≤ 2 ^ 6
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h.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)
⊢ 3 * (2 * 2) ^ 2 ≤ 2 ^ 6
Tactic:
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
⊢ |(-y')| ≤ 1
Tactic:
rw [abs_neg, abs_of_nonneg yy'nonneg.2]
Proof state after:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y' ≤ 1
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
⊢ y' ≤ 1
Tactic:
assumption
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
⊢ |(-y)| ≤ 1
Tactic:
rw [abs_neg, abs_of_nonneg yy'nonneg.1]
Proof state after:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y ≤ 1
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
⊢ y ≤ 1
Tactic:
assumption
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity
Proof state before:
x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖K x y - K x y'‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Tactic:
rw [K, K]
Proof state after:
x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Theorem: Hilbert_kernel_regularity
Proof state before:
x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Tactic:
wlog x_eq_zero : x = 0 generalizing x y y'
Proof state after:
case inr
x y y' : ℝ
this :
  ∀ {x y y' : ℝ},
    x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)

x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Theorem: Hilbert_kernel_regularity
Proof state before:
case inr
x y y' : ℝ
this :
  ∀ {x y y' : ℝ},
    x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)

x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Tactic:
. intro h
  set x_ := (0 : ℝ) with x_def
  set y_ := y - x with y_def
  set y'_ := y' - x with y'_def
  have h_ : 2 * |y_ - y'_| ≤ |x_ - y_| := by
    rw [x_def, y_def, y'_def]
    simpa
  have := this x_def h_
  rw [x_def, y_def, y'_def] at this
  simpa
Proof state after:
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Theorem: Hilbert_kernel_regularity
Proof state before:
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
Tactic:
rw [x_eq_zero]
Proof state after:
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |0 - y| → ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|)