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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set E
x : E
ω : E → E [⋀^Fin (n + 1)]→L[𝕜] F
V : Fin (n + 2) → E → E
hω : DifferentiableWithinAt 𝕜 ω s x
hV : ∀ (i : Fin (n + 2)), DifferentiableWithinAt 𝕜 (V i) s x
hsx : UniqueDiffWithinAt 𝕜 s x
H₀ : ∀ (i : Fin (n + 2)) (j : Fin (n + 1)), DifferentiableWithinAt 𝕜 (fun y ↦ i.removeNth (fun x ↦ V x y) j) s x
i j : Fin (n + 1)
hj : j ∈ Ici i
⊢ (-1) ^ ↑i.castSucc •
        (ω x)
          (Function.update (fun x_1 ↦ i.castSucc.removeNth (fun x_2 ↦ V x_2 x) x_1) j
            ((fderivWithin 𝕜 (fun y ↦ i.castSucc.removeNth (fun x ↦ V x y) j) s x) (V i.castSucc x))) +
      (-1) ^ ↑j.succ •
        (ω x)
          (Function.update (fun x_1 ↦ j.succ.removeNth (fun x_2 ↦ V x_2 x) x_1) i
            ((fderivWithin 𝕜 (fun y ↦ j.succ.removeNth (fun x ↦ V x y) i) s x) (V j.succ x))) =
    (-1) ^ (↑i + ↑j) •
      (ω x)
        (Matrix.vecCons (lieBracketWithin 𝕜 (V i.castSucc) (V j.succ) s x)
          (j.removeNth (i.castSucc.removeNth fun x_1 ↦ V x_1 x)))

Tactic:
mem_Ici

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
s : Set E
x : E
ω : E → E [⋀^Fin (n + 1)]→L[𝕜] F
V : Fin (n + 2) → E → E
hω : DifferentiableWithinAt 𝕜 ω s x
hV : ∀ (i : Fin (n + 2)), DifferentiableWithinAt 𝕜 (V i) s x
hsx : UniqueDiffWithinAt 𝕜 s x
H₀ : ∀ (i : Fin (n + 2)) (j : Fin (n + 1)), DifferentiableWithinAt 𝕜 (fun y ↦ i.removeNth (fun x ↦ V x y) j) s x
i j : Fin (n + 1)
hj : i ≤ j
⊢ (-1) ^ ↑i.castSucc •
        (ω x)
          (Function.update (fun x_1 ↦ i.castSucc.removeNth (fun x_2 ↦ V x_2 x) x_1) j
            ((fderivWithin 𝕜 (fun y ↦ i.castSucc.removeNth (fun x ↦ V x y) j) s x) (V i.castSucc x))) +
      (-1) ^ ↑j.succ •
        (ω x)
          (Function.update (fun x_1 ↦ j.succ.removeNth (fun x_2 ↦ V x_2 x) x_1) i
            ((fderivWithin 𝕜 (fun y ↦ j.succ.removeNth (fun x ↦ V x y) i) s x) (V j.succ x))) =
    (-1) ^ (↑i + ↑j) •
      (ω x)
        (Matrix.vecCons (lieBracketWithin 𝕜 (V i.castSucc) (V j.succ) s x)
          (j.removeNth (i.castSucc.removeNth fun x_1 ↦ V x_1 x)))
Module: Mathlib.Analysis.Calculus.FDeriv.Partial

Goal state before:
pp: 𝕜 : Type u_1
E₁ : Type u_2
E₂ : Type u_3
F : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E₁
inst✝⁵ : NormedSpace 𝕜 E₁
inst✝⁴ : NormedAddCommGroup E₂
inst✝³ : NormedSpace 𝕜 E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsRCLikeNormedField 𝕜
u : E₁ × E₂
f : E₁ → E₂ → F
f₁ : E₁ → E₂ → E₁ →L[𝕜] F
f₂ : E₁ → E₂ → E₂ →L[𝕜] F
df₁ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f x v.2) (↿f₁ v) v.1
df₂ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f v.1 x) (↿f₂ v) v.2
cf₁ : ContinuousAt (↿f₁) u
cf₂ : ContinuousAt (↿f₂) u
⊢ HasStrictFDerivAt (↿f) ((↿f₁ u).coprod (↿f₂ u)) u

Tactic:
rw [hasStrictFDerivAt_iff_isLittleO]

Goal state after:
𝕜 : Type u_1
E₁ : Type u_2
E₂ : Type u_3
F : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E₁
inst✝⁵ : NormedSpace 𝕜 E₁
inst✝⁴ : NormedAddCommGroup E₂
inst✝³ : NormedSpace 𝕜 E₂
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsRCLikeNormedField 𝕜
u : E₁ × E₂
f : E₁ → E₂ → F
f₁ : E₁ → E₂ → E₁ →L[𝕜] F
f₂ : E₁ → E₂ → E₂ →L[𝕜] F
df₁ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f x v.2) (↿f₁ v) v.1
df₂ : ∀ᶠ (v : E₁ × E₂) in 𝓝 u, HasFDerivAt (fun x ↦ f v.1 x) (↿f₂ v) v.2
cf₁ : ContinuousAt (↿f₁) u
cf₂ : ContinuousAt (↿f₂) u
⊢ (fun p ↦ ↿f p.1 - ↿f p.2 - ((↿f₁ u).coprod (↿f₂ u)) (p.1 - p.2)) =o[𝓝 (u, u)] fun p ↦ p.1 - p.2
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
⊢ HasStrictFDerivAt (fun x ↦ ‖x‖) (↑(SignType.sign t) • f) (t • x)

Tactic:
convert! h2.comp (t • x) h1 with y

Goal state after:
case e'_11
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
x✝ : E
⊢ ‖x✝‖ = |t| * ‖t⁻¹ • x✝‖

case e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : Semiring.toModule ≍ RCLike.toInnerProductSpaceReal.toModule
⊢ ↑(SignType.sign t) • f = (|t| • f) ∘SL (t⁻¹ • ContinuousLinearMap.id ℝ E)
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ (fderiv ℝ (fun x ↦ ‖x‖) x) x = ‖x‖

Tactic:
rw [← h.lineDeriv_eq_fderiv, lineDeriv]
  have (t : ℝ) : ‖x + t • x‖ = |1 + t| * ‖x‖ := by
    rw [← norm_eq_abs, ← norm_smul, add_smul, one_smul]
  simp_rw [this]
  rw [deriv_mul_const]
  · conv_lhs => enter [1, 1]; change _root_.abs ∘ (fun t ↦ 1 + t)
    rw [deriv_comp, deriv_abs, deriv_const_add_id]
    · simp
    · exact differentiableAt_abs (by simp)
    · exact differentiableAt_id.const_add _
  · exact (differentiableAt_abs (by simp)).comp _ (differentiableAt_id.const_add _)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
m : ℤ
k : ℕ
hs : IsOpen s
⊢ Set.EqOn (iteratedDerivWithin k (fun y ↦ y ^ m) s) (fun y ↦ (∏ i ∈ Finset.range k, (↑m - ↑i)) * y ^ (m - ↑k)) s

Tactic:
apply Set.EqOn.trans (iteratedDerivWithin_of_isOpen_eq_iterate hs)

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
m : ℤ
k : ℕ
hs : IsOpen s
⊢ Set.EqOn (deriv^[k] fun y ↦ y ^ m) (fun y ↦ (∏ i ∈ Finset.range k, (↑m - ↑i)) * y ^ (m - ↑k)) s
Module: Mathlib.Analysis.Calculus.IteratedDeriv.WithinZpow

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
k : ℕ
hs : IsOpen s
⊢ Set.EqOn (iteratedDerivWithin k (fun y ↦ 1 / y) s) (fun y ↦ (-1) ^ k * ↑k ! * y ^ (-1 - ↑k)) s

Tactic:
apply Set.EqOn.trans (iteratedDerivWithin_of_isOpen_eq_iterate hs)

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
k : ℕ
hs : IsOpen s
⊢ Set.EqOn (deriv^[k] fun y ↦ 1 / y) (fun y ↦ (-1) ^ k * ↑k ! * y ^ (-1 - ↑k)) s
Module: Mathlib.Analysis.Calculus.LagrangeMultipliers

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
φ : E → ℝ
x₀ : E
φ' : StrongDual ℝ E
ι : Type u_3
inst✝ : Finite ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
⊢ ¬LinearIndependent ℝ (Option.elim' φ' f')

Tactic:
cases nonempty_fintype ι
  rw [Fintype.linearIndependent_iff]; push Not
  rcases hextr.exists_multipliers_of_hasStrictFDerivAt hf' hφ' with ⟨Λ, Λ₀, hΛ, hΛf⟩
  refine ⟨Option.elim' Λ₀ Λ, ?_, ?_⟩
  · simpa [add_comm] using hΛf
  · simpa only [funext_iff, not_and_or, or_comm, Option.exists, Prod.mk_eq_zero, Ne,
      not_forall] using! hΛ

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

Goal state before:
pp: 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

Tactic:
simp_rw [hs.nhdsWithin_sdiff_eq_nhdsGT has hs_Iio hs_Ioi] at *
    exact lhopital_zero_nhdsGT hff' hgg' hg' hfa hga hdiv

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

Goal state before:
pp: case hfa
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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 f (𝓝[≠] a) (𝓝 0)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hfa
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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 f (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hga
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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 g (𝓝[≠] a) (𝓝 0)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hga
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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 g (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hdiv
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hdiv
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x
hgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 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.Function.JacobianOneDim

Goal state before:
pp: s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c

Tactic:
have B : (f '' b ∪ f '' c : Set ℝ) =ᵐ[volume] f '' c :=
      union_ae_eq_right_of_ae_eq_empty (ae_eq_empty.mpr (fb_count.measure_zero _))

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
u : ℝ → ℝ≥0∞
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
I : ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) * u (f x) = ∫⁻ (x : ℝ) in c, ENNReal.ofReal (f' x) * u (f x)
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
B : f '' b ∪ f '' c =ᵐ[volume] f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
g : ℝ → F
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
I : IntegrableOn (fun x ↦ f' x • g (f x)) s volume ↔ IntegrableOn (fun x ↦ f' x • g (f x)) c volume
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c

Tactic:
have B : (f '' b ∪ f '' c : Set ℝ) =ᵐ[volume] f '' c :=
      union_ae_eq_right_of_ae_eq_empty (ae_eq_empty.mpr (fb_count.measure_zero _))

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
g : ℝ → F
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
I : IntegrableOn (fun x ↦ f' x • g (f x)) s volume ↔ IntegrableOn (fun x ↦ f' x • g (f x)) c volume
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
B : f '' b ∪ f '' c =ᵐ[volume] f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hfa
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto f (𝓝[≠] a) (𝓝 0)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hfa
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto f (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hga
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto g (𝓝[≠] a) (𝓝 0)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hga
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto g (𝓝 a) (𝓝 0)
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: case hdiv
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[≠] a) l

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds

Goal state after:
case hdiv
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0
hfa : Tendsto f (𝓝 a) (𝓝 0)
hga : Tendsto g (𝓝 a) (𝓝 0)
hdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l
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
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ⋯
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ f z - f x - A (z - x) + f x ∈ closedBall (f x) (ε * r)

pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ⋯
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ A (z - x) + (f z - f x - A (z - x) + f x) = f z

Tactic:
· rw [mem_closedBall_iff_norm, add_sub_cancel_right]
        calc
          ‖f z - f x - A (z - x)‖ ≤ δ * ‖z - x‖ := hf _ zs _ xs
          _ ≤ ε * r := by gcongr

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ⋯
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ A (z - x) + (f z - f x - A (z - x) + f x) = f z
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
g : ℝ → F
H : IntegrableOn g (f '' s) volume
H' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
a_s : a ⊆ s
bc_s : b ∪ c ⊆ s
b_s : b ⊆ s
c_s : c ⊆ s
I : ∫ (x : ℝ) in s, f' x • g (f x) = ∫ (x : ℝ) in c, f' x • g (f x)
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c

Tactic:
have B : (f '' b ∪ f '' c : Set ℝ) =ᵐ[volume] f '' c :=
      union_ae_eq_right_of_ae_eq_empty (ae_eq_empty.mpr (fb_count.measure_zero _))

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : MonotoneOn f s
g : ℝ → F
H : IntegrableOn g (f '' s) volume
H' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume
a b c : Set ℝ
h_union : a ∪ (b ∪ c) = s
ha : MeasurableSet a
hb : MeasurableSet b
hc : MeasurableSet c
h_disj : Disjoint a (b ∪ c)
h_disj' : Disjoint b c
a_count : a.Countable
fb_count : (f '' b).Countable
deriv_b : ∀ x ∈ b, f' x = 0
deriv_c : ∀ x ∈ c, 0 ≤ f' x
inj_c : InjOn f c
a_s : a ⊆ s
bc_s : b ∪ c ⊆ s
b_s : b ⊆ s
c_s : c ⊆ s
I : ∫ (x : ℝ) in s, f' x • g (f x) = ∫ (x : ℝ) in c, f' x • g (f x)
A : f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' b ∪ f '' c
B : f '' b ∪ f '' c =ᵐ[volume] f '' c
⊢ f '' a ∪ (f '' b ∪ f '' c) =ᵐ[volume] f '' c
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ENNReal.ofReal ‖f a‖ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
a : α
| ENNReal.ofReal ‖f a‖

Tactic:
rw [← coe_nnnorm, ENNReal.ofReal_coe_nnreal]

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ENNReal.ofReal ‖f a‖ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
a : α
| ↑‖f a‖₊
Module: Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedAddCommGroup W
inst✝² : NormedSpace ℝ W
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
f f' : E × ℝ → F
g g' : E × ℝ → G
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) (μ.prod volume)
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) (μ.prod volume)
hfg : Integrable (fun x ↦ (B (f x)) (g x)) (μ.prod volume)
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x (0, 1)
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x (0, 1)
x : E
hf'gx : Integrable (fun y ↦ (B (f' (x, y))) (g (x, y))) volume
hfg'x : Integrable (fun y ↦ (B (f (x, y))) (g' (x, y))) volume
hfgx : Integrable (fun y ↦ (B (f (x, y))) (g (x, y))) volume
⊢ ∫ (t : ℝ), (B (f (x, t))) (g' (x, t)) = -∫ (t : ℝ), (B (f' (x, t))) (g (x, t))

Tactic:
apply integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable ?_ ?_ hfg'x hf'gx hfgx

Goal state after:
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedAddCommGroup W
inst✝² : NormedSpace ℝ W
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
f f' : E × ℝ → F
g g' : E × ℝ → G
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) (μ.prod volume)
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) (μ.prod volume)
hfg : Integrable (fun x ↦ (B (f x)) (g x)) (μ.prod volume)
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x (0, 1)
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x (0, 1)
x : E
hf'gx : Integrable (fun y ↦ (B (f' (x, y))) (g (x, y))) volume
hfg'x : Integrable (fun y ↦ (B (f (x, y))) (g' (x, y))) volume
hfgx : Integrable (fun y ↦ (B (f (x, y))) (g (x, y))) volume
⊢ ∀ x_1 ∈ tsupport fun x_2 ↦ g (x, x_2), HasDerivAt (fun x_2 ↦ f (x, x_2)) (f' (x, x_1)) x_1

E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : NormedAddCommGroup W
inst✝² : NormedSpace ℝ W
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
f f' : E × ℝ → F
g g' : E × ℝ → G
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) (μ.prod volume)
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) (μ.prod volume)
hfg : Integrable (fun x ↦ (B (f x)) (g x)) (μ.prod volume)
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x (0, 1)
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x (0, 1)
x : E
hf'gx : Integrable (fun y ↦ (B (f' (x, y))) (g (x, y))) volume
hfg'x : Integrable (fun y ↦ (B (f (x, y))) (g' (x, y))) volume
hfgx : Integrable (fun y ↦ (B (f (x, y))) (g (x, y))) volume
⊢ ∀ x_1 ∈ tsupport fun x_2 ↦ f (x, x_2), HasDerivAt (fun x_2 ↦ g (x, x_2)) (g' (x, x_1)) x_1