mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
V W : E → E
x : E
f : E → F
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
hW : DifferentiableAt 𝕜 W x
hV : DifferentiableAt 𝕜 V x
⊢ (fderiv 𝕜 f x) (lieBracket 𝕜 V W x) =
    (fderiv 𝕜 (fun x ↦ (fderiv 𝕜 f x) (W x)) x) (V x) - (fderiv 𝕜 (fun x ↦ (fderiv 𝕜 f x) (V x)) x) (W x)

Tactic:
apply fderiv_apply_lieBracket_of_isSymmSndFDerivAt

Goal state after:
case hf
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
V W : E → E
x : E
f : E → F
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
hW : DifferentiableAt 𝕜 W x
hV : DifferentiableAt 𝕜 V x
⊢ ContDiffAt 𝕜 2 f x

case hsymm
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
V W : E → E
x : E
f : E → F
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
hW : DifferentiableAt 𝕜 W x
hV : DifferentiableAt 𝕜 V x
⊢ IsSymmSndFDerivAt 𝕜 f x

case hW
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
V W : E → E
x : E
f : E → F
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
hW : DifferentiableAt 𝕜 W x
hV : DifferentiableAt 𝕜 V x
⊢ DifferentiableAt 𝕜 W x

case hV
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
V W : E → E
x : E
f : E → F
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
hW : DifferentiableAt 𝕜 W x
hV : DifferentiableAt 𝕜 V x
⊢ DifferentiableAt 𝕜 V x
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: case e'_3
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F :=
  fun t ↦ f (x + h • v + (t * h) • w) - (t * h) • (f' x) w - (t * h ^ 2) • (f'' v) w - ((t * h) ^ 2 / 2) • (f'' w) w
g' : ℝ → F :=
  fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
g_deriv : ∀ t ∈ Icc 0 1, HasDerivWithinAt g (g' t) (Icc 0 1) t
g'_bound : ∀ t ∈ Ico 0 1, ‖g' t‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2
I : ‖g 1 - g 0‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2
⊢ ‖f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w‖ = ‖g 1 - g 0‖

Tactic:
congr 1
    simp only [g, add_zero, one_mul, zero_div, zero_mul, sub_zero,
      zero_smul, Ne, not_false_iff, zero_pow, reduceCtorEq]
    abel

Goal state after:
Module: Mathlib.Analysis.Calculus.Gradient.Basic

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x y : F
s : Set F
h : HasGradientWithinAt f f' s x
hs : UniqueDiffWithinAt 𝕜 s x
⊢ ((toDual 𝕜 F) f') y = ⟪f', y⟫

Tactic:
toDual_apply_apply

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x y : F
s : Set F
h : HasGradientWithinAt f f' s x
hs : UniqueDiffWithinAt 𝕜 s x
⊢ ⟪f', y⟫ = ⟪f', y⟫
Module: Mathlib.Analysis.Calculus.Gradient.Basic

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x y : F
h : HasGradientAt f f' x
⊢ ((toDual 𝕜 F) f') y = ⟪f', y⟫

Tactic:
toDual_apply_apply

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x y : F
h : HasGradientAt f f' x
⊢ ⟪f', y⟫ = ⟪f', y⟫
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case pos.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h✝ : Nontrivial E
hd : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ fderiv ℝ (fun x ↦ ‖x‖) 0 = 0

Tactic:
exact fderiv_zero_of_not_differentiableAt <| not_differentiableAt_norm_zero E

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
g : 𝕜 → E
f : 𝕜 → 𝕜
x : 𝕜
hg : ContDiffAt 𝕜 3 g (f x)
hf : ContDiffAt 𝕜 3 f x
⊢ iteratedDeriv 3 (g ∘ f) x =
    deriv f x ^ 3 • iteratedDeriv 3 g (f x) + 3 • iteratedDeriv 2 f x • deriv f x • iteratedDeriv 2 g (f x) +
      iteratedDeriv 3 f x • deriv g (f x)

Tactic:
by
  simp only [← iteratedDerivWithin_univ, ← derivWithin_univ]
  exact iteratedDerivWithin_scomp_three hg hf uniqueDiffOn_univ uniqueDiffOn_univ (mem_univ _)
    (mapsTo_univ _ _)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
g f : 𝕜 → 𝕜
x : 𝕜
hg : ContDiffAt 𝕜 3 g (f x)
hf : ContDiffAt 𝕜 3 f x
⊢ iteratedDeriv 3 (g ∘ f) x =
    iteratedDeriv 3 g (f x) * deriv f x ^ 3 + 3 * iteratedDeriv 2 g (f x) * iteratedDeriv 2 f x * deriv f x +
      deriv g (f x) * iteratedDeriv 3 f x

Tactic:
simp only [← iteratedDerivWithin_univ, ← derivWithin_univ]
  exact iteratedDerivWithin_comp_three hg hf uniqueDiffOn_univ uniqueDiffOn_univ (mem_univ _)
    (mapsTo_univ _ _)

Goal state after:
Module: Mathlib.Analysis.Calculus.LagrangeMultipliers

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E → F
φ : E → ℝ
x₀ : E
f' : E →L[ℝ] F
φ' : StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀
htop : (↑(f'.prod φ')).range = ⊤
⊢ False

Tactic:
set fφ := fun x => (f x, φ x)

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E → F
φ : E → ℝ
x₀ : E
f' : E →L[ℝ] F
φ' : StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀
htop : (↑(f'.prod φ')).range = ⊤
fφ : E → F × ℝ := fun x ↦ (f x, φ x)
⊢ False
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hcf : ContinuousOn f (Ioc a b)
hcg : ContinuousOn g (Ioc a b)
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : f b = 0
hgb : g b = 0
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] b) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] b) l

Tactic:
refine lhopital_zero_left_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv
  · rw [← hfb, ← nhdsWithin_Ioo_eq_nhdsLT hab]
    exact ((hcf b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto
  · rw [← hgb, ← nhdsWithin_Ioo_eq_nhdsLT hab]
    exact ((hcg b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto

Goal state after:
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
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
⊢ ∃ ε, 0 < ε ∧ ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))

Tactic:
refine ⟨r (f' z) - ‖f' x - f' z‖, ?_, le_of_eq (by abel)⟩

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
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
⊢ 0 < ↑(r (f' ↑z)) - ‖f' x - f' ↑z‖
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case inl
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : s ∩ Iio a = ∅
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l

Tactic:
rcases eq_empty_or_nonempty (s ∩ Ioi a) with hs_Ioi | hs_Ioi

Goal state after:
case inl.inl
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : s ∩ Iio a = ∅
hs_Ioi : s ∩ Ioi a = ∅
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l

case inl.inr
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : s ∩ Iio a = ∅
hs_Ioi : (s ∩ Ioi a).Nonempty
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case inr
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : (s ∩ Iio a).Nonempty
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l

Tactic:
rcases eq_empty_or_nonempty (s ∩ Ioi a) with hs_Ioi | hs_Ioi

Goal state after:
case inr.inl
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : (s ∩ Iio a).Nonempty
hs_Ioi : s ∩ Ioi a = ∅
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l

case inr.inr
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, g' x ≠ 0
hfa : Tendsto f (𝓝[s \ {a}] a) (𝓝 0)
hga : Tendsto g (𝓝[s \ {a}] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s \ {a}] a) l
has : a ∈ closure s
h : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, s \ {a} ∈ 𝓝 x
hff' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝[s \ {a}] a, HasDerivAt g (g' x) x
hs_Iio : (s ∩ Iio a).Nonempty
hs_Ioi : (s ∩ Ioi a).Nonempty
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[s \ {a}] a) l
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
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
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∃ t A,
    Pairwise (Disjoint on t) ∧
      (∀ (n : ℕ), MeasurableSet (t n)) ∧
        s ⊆ ⋃ n, t n ∧
          (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Tactic:
refine
    ⟨disjointed t, A, disjoint_disjointed _,
      MeasurableSet.disjointed fun n => (t_closed n).measurableSet, ?_, ?_, ht⟩

Goal state after:
case refine_1
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
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
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ s ⊆ ⋃ n, disjointed t n

case refine_2
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
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
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ disjointed t n) (r (A n))
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (x : ℝ) in Icc a b, f' x • g (f x) = ∫ (u : ℝ) in f '' Ioo a b, g u

Tactic:
integral_Icc_eq_integral_Ioo,

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, 0 ≤ f' x
hab : a ≤ b
M : MonotoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f a) (f b), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (t : ℝ) in Ioo a b, f' t • g (f t) = ∫ (u : ℝ) in f '' Ioo a b, g u
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (x : ℝ) in Icc a b, f' x • g (f x) = -∫ (u : ℝ) in f '' Ioo a b, g u

Tactic:
integral_Icc_eq_integral_Ioo,

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
g : ℝ → F
hf : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo a b, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (t : ℝ) in Ioo a b, f' t • g (f t) = -∫ (u : ℝ) in f '' Ioo a b, g u
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
⊢ ∀ (a : E), Tendsto (fun r ↦ μ (s ∩ closedBall a r) / μ (closedBall a r)) (𝓝[>] 0) (𝓝 1) → a ∈ s → ‖f' a - A‖₊ ≤ δ

Tactic:
intro x hx xs

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
x : E
hx : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
xs : x ∈ s
⊢ ‖f' x - A‖₊ ≤ δ
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
a : ℝ
⊢ IntegrableOn (fun x ↦ rexp x • g (rexp x)) (Ioi a) volume ↔ IntegrableOn g (Ioi (rexp a)) volume

Tactic:
by
  symm; rw [← image_exp_Ioi]
  simpa [abs_of_pos (exp_pos _)] using integrableOn_image_iff_integrableOn_abs_deriv_smul
      (measurableSet_Ioi (a := a)) (fun x _ ↦ (hasDerivAt_exp x).hasDerivWithinAt)
      (fun x _ y _ hxy ↦ exp_injective hxy) g

Goal state after:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hM : 0 ≤ M
w : ℂ
hw : ‖w - z‖ = r
hf : ∀ (w : ℂ), ‖w - z‖ = r → ‖f w‖ ≤ M
⊢ ‖f w‖ / r ^ 2 ≤ M / r ^ 2

Tactic:
exact div_le_div₀ hM (hf w hw) (sq_pos_of_pos hr) le_rfl

Goal state after:
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
⊢ IsOpen[inst✝¹] s ↔ ∀ (x : M), IsOpen[inst✝²] (↑(chartAt H x) '' ((chartAt H x).source ∩ s))

Tactic:
rw [isOpen_iff_of_cover (fun i ↦ (chartAt H i).open_source) (iUnion_source_chartAt H M)]
  simp only [(chartAt H _).isOpen_image_iff_of_subset_source inter_subset_left]

Goal state after:
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: case inr
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
x : M'
⊢ Sum.inr x ∈ Sum.inr '' (ChartedSpace.chartAt x).source

Tactic:
← OpenPartialHomeomorph.lift_openEmbedding_source _ IsOpenEmbedding.inr

Goal state after:
case inr
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
x : M'
⊢ Sum.inr x ∈ ((ChartedSpace.chartAt x).lift_openEmbedding ⋯).source
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e e' : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas M G
he' : e' ∈ maximalAtlas M G
x : H
hx : x ∈ (e.symm ≫ₕ e').source
f : OpenPartialHomeomorph M H := chartAt H (↑e.symm x)
s : Set H := e.target ∩ ↑e.symm ⁻¹' f.source
hs : IsOpen[inst✝²] s
xs : x ∈ s
A : e.symm ≫ₕ f ∈ G
B : f.symm ≫ₕ e' ∈ G
C : (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' ∈ G
⊢ e.symm ≫ₕ ofSet f.source ⋯ ≫ₕ e' ≈ (e.symm ≫ₕ ofSet f.source ⋯) ≫ₕ e'

Tactic:
trans_assoc

Goal state after:
H : Type u
M : Type u_2
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
G : StructureGroupoid H
e e' : OpenPartialHomeomorph M H
he : e ∈ maximalAtlas M G
he' : e' ∈ maximalAtlas M G
x : H
hx : x ∈ (e.symm ≫ₕ e').source
f : OpenPartialHomeomorph M H := chartAt H (↑e.symm x)
s : Set H := e.target ∩ ↑e.symm ⁻¹' f.source
hs : IsOpen[inst✝²] s
xs : x ∈ s
A : e.symm ≫ₕ f ∈ G
B : f.symm ≫ₕ e' ∈ G
C : (e.symm ≫ₕ f) ≫ₕ f.symm ≫ₕ e' ∈ G
⊢ e.symm ≫ₕ ofSet f.source ⋯ ≫ₕ e' ≈ e.symm ≫ₕ ofSet f.source ⋯ ≫ₕ e'
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
c : OpenPartialHomeomorph M' H
c' : OpenPartialHomeomorph M H
hc : c ∈ atlas H M'
hc' : c' ∈ atlas H M
this : (c.symm ≫ₕ e.toOpenPartialHomeomorph.symm) ≫ₕ c' ∈ G
⊢ c.symm ≫ₕ e.symm.toOpenPartialHomeomorph ≫ₕ c' ∈ G

Tactic:
trans_assoc

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
c : OpenPartialHomeomorph M' H
c' : OpenPartialHomeomorph M H
hc : c ∈ atlas H M'
hc' : c' ∈ atlas H M
this : c.symm ≫ₕ e.toOpenPartialHomeomorph.symm ≫ₕ c' ∈ G
⊢ c.symm ≫ₕ e.symm.toOpenPartialHomeomorph ≫ₕ c' ∈ G
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
H : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ range ↑I.toPartialEquiv = I.target

Tactic:
by
  rw [← I.image_source_eq_target, I.source_eq, image_univ.symm]

Goal state after: