mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
⊢ f' x ∈ ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))

Tactic:
]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
⊢ f' x ∈ ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
⊢ ((fun x ↦ f' x / g' x) ∘ fun x ↦ x⁻¹) =ᶠ[𝓝[>] 0] fun x ↦ f' x⁻¹ * -(x ^ 2)⁻¹ / (g' x⁻¹ * -(x ^ 2)⁻¹)

Tactic:
filter_upwards [self_mem_nhdsWithin] with x (hx : 0 < x)

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : 0 < x
⊢ ((fun x ↦ f' x / g' x) ∘ fun x ↦ x⁻¹) x = f' x⁻¹ * -(x ^ 2)⁻¹ / (g' x⁻¹ * -(x ^ 2)⁻¹)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : 0 < x
⊢ f' x⁻¹ / g' x⁻¹ = f' x⁻¹ / g' x⁻¹

pp: case hc
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : 0 < x
⊢ -(x ^ 2)⁻¹ ≠ 0

Tactic:
]

Goal state after:
case hc
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : 0 < x
⊢ -(x ^ 2)⁻¹ ≠ 0
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
this : ∃ x_1 ∈ T, f' x ∈ ball (f' ↑x_1) ↑(r (f' ↑x_1))
⊢ ∃ z ∈ T, f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))

Tactic:
]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z ↦ {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
this : ∃ x_1 ∈ T, f' x ∈ ball (f' ↑x_1) ↑(r (f' ↑x_1))
⊢ ∃ z ∈ T, f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case pos
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Tactic:
]

Goal state after:
case pos
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case refine_1
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ f a ∈ f '' [[a, b]]

Tactic:
]

Goal state after:
case refine_1
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ f a ∈ f '' [[a, b]]
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: case refine_2
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ MapsTo f [[a, b]] (f '' [[a, b]])

Tactic:
]

Goal state after:
case refine_2
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
⊢ MapsTo f [[a, b]] (f '' [[a, b]])
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x ↦ f x / g x) atBot l

Tactic:
]

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Iio a, g' x ≠ 0
hfbot : Tendsto f atBot (𝓝 0)
hgbot : Tendsto g atBot (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atBot l
hdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioi (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
⊢ Tendsto (fun x ↦ f x / g x) atBot l
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
⊢ Icc c d ⊆ Ioo (min a b) (max a b)

Tactic:
]

Goal state after:
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
⊢ Icc c d ⊆ Ioo (min a b) (max a b)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case refine_1
a b : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hab : a < b
hdf : DifferentiableOn ℝ f (Ioo a b)
hcf : ContinuousOn f (Ico a b)
hcg : ContinuousOn g (Ico a b)
hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0
hfa : f a = 0
hga : g a = 0
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[>] a) l
⊢ Tendsto f (𝓝[Ioo a b] a) (𝓝 (f a))

Tactic:
]

Goal state after:
case refine_1
a b : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hab : a < b
hdf : DifferentiableOn ℝ f (Ioo a b)
hcf : ContinuousOn f (Ico a b)
hcg : ContinuousOn g (Ico a b)
hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0
hfa : f a = 0
hga : g a = 0
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[>] a) l
⊢ Tendsto f (𝓝[Ioo a b] a) (𝓝 (f a))
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
y : ℝ
hy : y ∈ s ∩ Ioo p x
⊢ f p ≤ f y

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
y : ℝ
hy : y ∈ s ∩ Ioo p x
⊢ f p ≤ f y
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case refine_2
a b : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hab : a < b
hdf : DifferentiableOn ℝ f (Ioo a b)
hcf : ContinuousOn f (Ico a b)
hcg : ContinuousOn g (Ico a b)
hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0
hfa : f a = 0
hga : g a = 0
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[>] a) l
⊢ Tendsto g (𝓝[Ioo a b] a) (𝓝 (g a))

Tactic:
]

Goal state after:
case refine_2
a b : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hab : a < b
hdf : DifferentiableOn ℝ f (Ioo a b)
hcf : ContinuousOn f (Ico a b)
hcg : ContinuousOn g (Ico a b)
hg' : ∀ x ∈ Ioo a b, deriv g x ≠ 0
hfa : f a = 0
hga : g a = 0
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[>] a) l
⊢ Tendsto g (𝓝[Ioo a b] a) (𝓝 (g a))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x

Tactic:
]

Goal state after:
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ HasDerivWithinAt (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, g' y ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[>] a) l

Tactic:
]

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt f (f' y) y
hgg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt g (g' y) y
hg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, g' y ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[>] a) l
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ f x ∈ f '' [[c, d]]

Tactic:
]

Goal state after:
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
⊢ f x ∈ f '' [[c, d]]
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
K : HasDerivWithinAt f 0 (s ∩ Ioo p x) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo p x) x
J1 : (s ∩ Ioo p x) \ {x} = s ∩ Ioo p x
J2 : 𝓝[s ∩ Ioo p x] x = 𝓝[s ∩ Iio x] x
⊢ (𝓝[s ∩ Iio x] x).NeBot

Tactic:
]

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
K : HasDerivWithinAt f 0 (s ∩ Ioo p x) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo p x) x
J1 : (s ∩ Ioo p x) \ {x} = s ∩ Ioo p x
J2 : 𝓝[s ∩ Ioo p x] x = 𝓝[s ∩ Iio x] x
⊢ (𝓝[s ∩ Iio x] x).NeBot
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
⊢ [[f a, f x]] ⊆ f '' [[a, b]]

Tactic:
]

Goal state after:
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x ↦ f' x • (g ∘ f) x) [[a, b]] volume
hE : CompleteSpace E
h_cont : ContinuousOn (fun u ↦ ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (min a b) (max a b)
c : ℝ
hc : c ∈ Ioo (min a b) x
d : ℝ
hd : d ∈ Ioo x (max a b)
cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b)
J : Set ℝ := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
hg_cont : ContinuousOn g J
hJ : f '' [[c, d]] = J
h2x : f x ∈ J
⊢ [[f a, f x]] ⊆ f '' [[a, b]]
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[>] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[>] a
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ 𝓝[>] a
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ u ∈ Ioi a, Ioo a u ⊆ s
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[>] a) l

Tactic:
]

Goal state after:
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[>] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[>] a
hgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y
s₃ : Set ℝ
hs₃ : s₃ ∈ 𝓝[>] a
hg' : ∀ y ∈ s₃, g' y ≠ 0
s : Set ℝ := s₁ ∩ s₂ ∩ s₃
hs : ∃ u ∈ Ioi a, Ioo a u ⊆ s
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[>] a) l