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) ((fun t => (c' t * d t - c t * d' t) / d t ^ 2) t) t
Tactic:
apply HasDerivAt.div
Proof state after:
case hc
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 y => c y) (c' t) t

case hd
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 y => d y) (d' t) t

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