internlm/Lean-Github huggingface dataset:
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h0x
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ 0 ≤ t
Tactic:
linarith
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t ≤ Real.pi
Tactic:
linarith [Real.two_le_pi]
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t.sin > 0
Tactic:
apply Real.sin_pos_of_pos_of_lt_pi
Proof state after:
case h0x
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ 0 < t

case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t < Real.pi
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h0x
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ 0 < t

case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t < Real.pi
Tactic:
. linarith
Proof state after:
case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t < Real.pi
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t < Real.pi
Tactic:
. linarith [Real.two_le_pi]
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case h0x
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ 0 < t
Tactic:
linarith
Proof state after:
no goals
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hxp
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
t : ℝ
ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y'
ht' : 0 < t ∧ t ≤ 1
⊢ t < Real.pi
Tactic:
linarith [Real.two_le_pi]
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
⊢ ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t
Tactic:
intro t ht
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
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
t : ℝ
ht : t ∈ Set.uIcc y' y
⊢ HasDerivAt f (f' t) t
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
t : ℝ
ht : t ∈ Set.uIcc y' y
⊢ HasDerivAt f (f' t) t
Tactic:
have : f = fun t ↦ c t / d t := by simp
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
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this : f = fun t => c t / d t
⊢ HasDerivAt f (f' t) t
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this : f = fun t => c t / d t
⊢ HasDerivAt f (f' t) t
Tactic:
rw [this]
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
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this : f = fun t => c t / d t
⊢ HasDerivAt (fun t => c t / d t) (f' t) t
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this : f = fun t => c t / d t
⊢ HasDerivAt (fun t => c t / d t) (f' t) t
Tactic:
have : f' = fun t ↦ ((c' t * d t - c t * d' t) / d t ^ 2) := by
  ext t
  rw [f'def, cdef, c'def, ddef, d'def]
  simp
  congr
  ring
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
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this✝ : f = fun t => c t / d t
this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2
⊢ HasDerivAt (fun t => c t / d t) (f' t) t
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this✝ : f = fun t => c t / d t
this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2
⊢ HasDerivAt (fun t => c t / d t) (f' t) t
Tactic:
rw [this]
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
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this✝ : f = fun t => c t / d t
this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2
⊢ HasDerivAt (fun t => c t / d t) ((fun t => (c' t * d t - c t * d' t) / d t ^ 2) t) t
Theorem: Hilbert_kernel_regularity_main_part
Proof state before:
case hx
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this✝ : f = fun t => c t / d t
this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2
⊢ d t ≠ 0
Tactic:
. exact d_nonzero ht
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
t : ℝ
ht : t ∈ Set.uIcc y' y
⊢ f = fun t => c t / d t
Tactic:
simp
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
t : ℝ
ht : t ∈ Set.uIcc y' y
this : f = fun t => c t / d t
⊢ f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2
Tactic:
ext t
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
t✝ : ℝ
ht : t✝ ∈ Set.uIcc y' y
this : f = fun t => c t / d t
t : ℝ
⊢ f' t = (c' t * d t - c t * d' t) / d t ^ 2