mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : α → Set ℝ := fun x ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
v : Set (α × ℝ)
v_count : v.Countable
vs : ∀ p ∈ v, p.1 ∈ s
vg : ∀ p ∈ v, p.2 ∈ g p.1
μv : μ (s \ ⋃ p ∈ v, closedBall p.1 p.2) = 0
v_disj : v.PairwiseDisjoint fun p ↦ closedBall p.1 p.2
⊢ ∃ r t, v = graphOn r t

Tactic:
have I : ∀ p ∈ v, 0 ≤ p.2 := fun p hp => (vg p hp).2.1.le

Goal state after:
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : α → Set ℝ := fun x ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
v : Set (α × ℝ)
v_count : v.Countable
vs : ∀ p ∈ v, p.1 ∈ s
vg : ∀ p ∈ v, p.2 ∈ g p.1
μv : μ (s \ ⋃ p ∈ v, closedBall p.1 p.2) = 0
v_disj : v.PairwiseDisjoint fun p ↦ closedBall p.1 p.2
I : ∀ p ∈ v, 0 ≤ p.2
⊢ ∃ r t, v = graphOn r t
Module: Mathlib.Analysis.Calculus.DifferentialForm.VectorField

Goal state before:
pp: case succ
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
s : Set E
x : E
hsx : UniqueDiffWithinAt 𝕜 s x
n : ℕ
ω : E → E [⋀^Fin (n + 1)]→L[𝕜] F
V : Fin (n + 1 + 1) → E → E
hω : DifferentiableWithinAt 𝕜 ω s x
hV : ∀ (i : Fin (n + 1 + 1)), DifferentiableWithinAt 𝕜 (V i) s x
hcomm : Pairwise fun i j ↦ lieBracketWithin 𝕜 (V i) (V j) s x = 0
i j : Fin (n + 1)
hj : j ∈ Ici i
⊢ (-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))) =
    0

Tactic:
rw [hcomm (ne_of_lt <| by simpa using hj), (ω x).map_coord_zero 0] <;>
      simp

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
hd : ContDiffAt ℝ n (fun x ↦ ‖x‖) (t • x)
⊢ x = 1 • x

Tactic:
one_smul

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : WithTop ℕ∞
x : E
t : ℝ
ht : t ≠ 0
hd : ContDiffAt ℝ n (fun x ↦ ‖x‖) (t • x)
⊢ x = x
Module: Mathlib.Analysis.Calculus.FDeriv.Partial

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
u : E
v w : α → E
l : Filter α
hv : Tendsto v l (𝓝 u)
hw : Tendsto w l (𝓝 u)
s : Set E
seg✝ : ∀ᶠ (χ : α) in l, [w χ -[ℝ] v χ] ⊆ s
f : α → E → F
f' : α → E → E →L[𝕜] F
φ : E →L[𝕜] F
ε : ℝ
hε : 0 < ε
df'✝ : ∀ᶠ (χ : α) in l, ∀ z ∈ [w χ -[ℝ] v χ], HasFDerivWithinAt (f χ) (f' χ z) s z
cf' : ∀ᶠ (χ : α) in l, ∀ z ∈ [w χ -[ℝ] v χ], ‖f' χ z - φ‖ < ε
χ : α
seg : [w χ -[ℝ] v χ] ⊆ s
df' : ∀ z ∈ [w χ -[ℝ] v χ], HasFDerivWithinAt (f χ) (f' χ z) s z
⊢ (∀ z ∈ [w χ -[ℝ] v χ], ‖f' χ z - φ‖ < ε) → ‖f χ (v χ) - f χ (w χ) - φ (v χ - w χ)‖ ≤ ε * ‖v χ - w χ‖

Tactic:
cf'

Goal state after:
α : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
u : E
v w : α → E
l : Filter α
hv : Tendsto v l (𝓝 u)
hw : Tendsto w l (𝓝 u)
s : Set E
seg✝ : ∀ᶠ (χ : α) in l, [w χ -[ℝ] v χ] ⊆ s
f : α → E → F
f' : α → E → E →L[𝕜] F
φ : E →L[𝕜] F
ε : ℝ
hε : 0 < ε
df'✝ : ∀ᶠ (χ : α) in l, ∀ z ∈ [w χ -[ℝ] v χ], HasFDerivWithinAt (f χ) (f' χ z) s z
cf'✝ : ∀ᶠ (χ : α) in l, ∀ z ∈ [w χ -[ℝ] v χ], ‖f' χ z - φ‖ < ε
χ : α
seg : [w χ -[ℝ] v χ] ⊆ s
df' : ∀ z ∈ [w χ -[ℝ] v χ], HasFDerivWithinAt (f χ) (f' χ z) s z
cf' : ∀ z ∈ [w χ -[ℝ] v χ], ‖f' χ z - φ‖ < ε
⊢ ‖f χ (v χ) - f χ (w χ) - φ (v χ - w χ)‖ ≤ ε * ‖v χ - w χ‖
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : ℕ∞ω
⊢ n ≤ minSmoothness 𝕜 n

Tactic:
by
  simp only [minSmoothness]
  split_ifs <;> simp

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n m : ℕ∞ω
⊢ minSmoothness 𝕜 (n + m) = minSmoothness 𝕜 n + m

Tactic:
by
  simp only [minSmoothness]
  split_ifs <;> simp

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
ht : t ≠ 0
hd : DifferentiableAt ℝ (fun x ↦ ‖x‖) (t • x)
⊢ x = 1 • x

Tactic:
one_smul

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
t : ℝ
ht : t ≠ 0
hd : DifferentiableAt ℝ (fun x ↦ ‖x‖) (t • x)
⊢ x = x
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
t : ℝ
⊢ ‖x + t • x‖ = ‖1 • x + t • x‖

Tactic:
one_smul

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
t : ℝ
⊢ ‖x + t • x‖ = ‖x + t • x‖
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 : F
s : Set F
f₁ : F → 𝕜
h : HasGradientWithinAt f f' s x
hs : ∀ x ∈ s, f₁ x = f x
hx : f₁ x = f x
⊢ s ⊆ s

Tactic:
tauto

Goal state after:
Module: Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ × E₂ → F
f'u : E₁ × E₂ →L[𝕜] F
dfu : HasStrictFDerivAt f f'u u
if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible
hψ : Tendsto (dfu.implicitFunctionOfProdDomain if₂u) (𝓝 u.1) (𝓝 u.2)
⊢ ∀ᶠ (x : E₁) in 𝓝 u.1, f (x, dfu.implicitFunctionOfProdDomain if₂u x) = f u

Tactic:
set ψ := dfu.implicitFunctionOfProdDomain if₂u

Goal state after:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ × E₂ → F
f'u : E₁ × E₂ →L[𝕜] F
dfu : HasStrictFDerivAt f f'u u
if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible
ψ : E₁ → E₂ := dfu.implicitFunctionOfProdDomain if₂u
hψ : Tendsto ψ (𝓝 u.1) (𝓝 u.2)
⊢ ∀ᶠ (x : E₁) in 𝓝 u.1, f (x, ψ x) = f u
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.FiniteDimensional

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : FiniteDimensional ℝ F
s : Set E
f : E → F
f' : E ≃L[ℝ] F
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ lipschitzExtensionConstant F * c < ‖↑f'.symm‖₊⁻¹
⊢ ∃ g, EqOn f (⇑g) s

Tactic:
by
  -- the difference `f - f'` is Lipschitz on `s`. It can be extended to a Lipschitz function `u`
  -- on the whole space, with a slightly worse Lipschitz constant. Then `f' + u` will be the
  -- desired homeomorphism.
  obtain ⟨u, hu, uf⟩ :
    ∃ u : E → F, LipschitzWith (lipschitzExtensionConstant F * c) u ∧ EqOn (f - ⇑f') u s :=
    hf.lipschitzOnWith.extend_finite_dimension
  let g : E → F := fun x => f' x + u x
  have fg : EqOn f g s := fun x hx => by simp_rw [g, ← uf hx, Pi.sub_apply, add_sub_cancel]
  have hg : ApproximatesLinearOn g (f' : E →L[ℝ] F) univ (lipschitzExtensionConstant F * c) := by
    apply LipschitzOnWith.approximatesLinearOn
    rw [lipschitzOnWith_univ]
    convert! hu
    ext x
    simp only [g, add_sub_cancel_left, ContinuousLinearEquiv.coe_coe, Pi.sub_apply]
  have : FiniteDimensional ℝ E := f'.symm.finiteDimensional
  exact ⟨hg.toHomeomorph g hc, fg⟩

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace 𝕜
k t : ℕ
z₀ : 𝕜
R R₁ : 𝕜 → 𝕜
hf1 : ∀ (z : 𝕜), AnalyticAt 𝕜 R₁ z
hR₁ : ∀ (z : 𝕜), R z = (z - z₀) ^ (k + t) * R₁ z
⊢ ∃ R₂,
    (∀ (z : 𝕜), AnalyticAt 𝕜 R₂ z) ∧
      ∀ (z : 𝕜), iteratedDeriv k R z = (z - z₀) ^ t * (↑(k + t)! / ↑t ! * R₁ z + (z - z₀) * R₂ z)

Tactic:
by
  induction k generalizing t with
  | zero => exact ⟨0, fun _ ↦ analyticAt_const, by simp [hR₁, Nat.factorial_ne_zero]⟩
  | succ k IH =>
    obtain ⟨R₂, hR₂, hR₂_eq⟩ := IH (t := t + 1) (by grind)
    set R₂' : 𝕜 → 𝕜 := fun z ↦ (t + 1) * R₂ z + ((k + (t + 1))! / (t + 1)! * deriv R₁ z +
      (R₂ z + (z - z₀) * deriv R₂ z))
    refine ⟨R₂', by fun_prop, fun z ↦ ?_⟩
    calc iteratedDeriv (k + 1) R z
      _ = deriv (fun w ↦ (w - z₀) ^ (t + 1)
          * (↑(k + (t + 1))! / ↑(t + 1)! * R₁ w + (w - z₀) * R₂ w)) z := by
        rw [iteratedDeriv_succ, funext hR₂_eq]
      _ = (t + 1) * (z - z₀) ^ t * ((k + (t + 1))! / (t + 1)! * R₁ z + (z - z₀) * R₂ z) +
        (z - z₀) ^ (t + 1) * ((k + (t + 1))! / (t + 1)! * deriv R₁ z +
          (R₂ z + (z - z₀) * deriv R₂ z)) := by
        have hsub : HasDerivAt (· - z₀) 1 z := (hasDerivAt_id z).sub_const z₀
        simpa using! ((hsub.fun_pow (t + 1)).mul
          (((hf1 z).differentiableAt.hasDerivAt.const_mul ((k + (t + 1))! / (t + 1)! : 𝕜)).add
            (hsub.mul (hR₂ z).differentiableAt.hasDerivAt))).deriv
      _ = (z - z₀) ^ t * ((k + 1 + t)! / t ! * R₁ z + (z - z₀) * R₂' z) := by
        have : (t : 𝕜) + 1 ≠ 0 := mod_cast t.succ_ne_zero
        have : ((t + 1)! : 𝕜) = (t + 1) * t ! := by simp [Nat.factorial_succ]
        grind

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

Goal state before:
pp: case e_a.e_6.«_@»._internal._hyg.0.«0»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (Fin.cons 1 (Fin.cons 1 fun x ↦ 1) ((fun i ↦ i) ⟨0, ⋯⟩)) f s x = derivWithin f s x

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_6.«_@»._internal._hyg.0.«1»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (Fin.cons 1 (Fin.cons 1 fun x ↦ 1) ((fun i ↦ i) ⟨1, ⋯⟩)) f s x = derivWithin f s x

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_6.«_@»._internal._hyg.0.«2»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (Fin.cons 1 (Fin.cons 1 fun x ↦ 1) ((fun i ↦ i) ⟨2, ⋯⟩)) f s x = derivWithin f s x

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_6.«_@»._internal._hyg.0.«0»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (Fin.cons 2 (fun x ↦ 1) ((fun i ↦ i) ⟨0, ⋯⟩)) f s x =
    ![iteratedDerivWithin 2 f s x, derivWithin f s x] ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_6.«_@»._internal._hyg.0.«1»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (Fin.cons 2 (fun x ↦ 1) ((fun i ↦ i) ⟨1, ⋯⟩)) f s x =
    ![iteratedDerivWithin 2 f s x, derivWithin f s x] ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_a.e_6.«_@»._internal._hyg.0.«0»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (update (Fin.cons 1 fun x ↦ 1) 1 2 ((fun i ↦ i) ⟨0, ⋯⟩)) f s x =
    ![derivWithin f s x, iteratedDerivWithin 2 f s x] ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_a.e_6.«_@»._internal._hyg.0.«1»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (update (Fin.cons 1 fun x ↦ 1) 1 2 ((fun i ↦ i) ⟨1, ⋯⟩)) f s x =
    ![derivWithin f s x, iteratedDerivWithin 2 f s x] ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_a.e_a.e_6.«_@»._internal._hyg.0.«0»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (update (Fin.cons 1 fun x ↦ 1) 1 2 ((fun i ↦ i) ⟨0, ⋯⟩)) f s x =
    ![derivWithin f s x, iteratedDerivWithin 2 f s x] ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [this]

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

Goal state before:
pp: case e_a.e_a.e_a.e_a.e_6.«_@»._internal._hyg.0.«1»
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
g : E → F
f : 𝕜 → E
s : Set 𝕜
t : Set E
x : 𝕜
hg : ContDiffWithinAt 𝕜 3 g t (f x)
hf : ContDiffWithinAt 𝕜 3 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
this : ∀ (j : Fin (2 + 1)), Fin.cons 1 (Fin.cons 1 fun x ↦ 1) j = 1
⊢ iteratedDerivWithin (update (Fin.cons 1 fun x ↦ 1) 1 2 ((fun i ↦ i) ⟨1, ⋯⟩)) f s x =
    ![derivWithin f s x, iteratedDerivWithin 2 f s x] ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [this]

Goal state after:
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

Goal state before:
pp: case refine_1
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
⊢ x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∨ x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥ → x ∈ s

Tactic:
tauto

Goal state after:
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∞ := ENNReal.ofReal |A.det|
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
hδ : δ ∈ Iio (NNReal.mk ε ⋯)
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
⊢ μ (f '' s) ≤ ↑m * μ s

Tactic:
simp only [mem_Iio, ← NNReal.coe_lt_coe, NNReal.coe_mk] at hδ

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∞ := ENNReal.ofReal |A.det|
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
⊢ μ (f '' s) ≤ ↑m * μ s
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∞ := ENNReal.ofReal |A.det|
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
K : f '' (s ∩ closedBall x r) ⊆ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r)
⊢ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r) = r • ⇑A '' closedBall 0 1 + closedBall (f x) (‖r‖ * ε)

Tactic:
← image_smul_set,

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∞ := ENNReal.ofReal |A.det|
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
K : f '' (s ∩ closedBall x r) ⊆ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r)
⊢ ⇑A '' closedBall 0 r + closedBall (f x) (ε * r) = ⇑A '' (r • closedBall 0 1) + closedBall (f x) (‖r‖ * ε)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover μ l fun i ↦ Ioc (a i) (b i)
i : ι
⊢ a i ≤ 0 → 0 ≤ b i → ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ∂μ ≤ I

Tactic:
hai

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I
hφ : AECover μ l fun i ↦ Ioc (a i) (b i)
i : ι
hai : a i ≤ 0
⊢ 0 ≤ b i → ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ∂μ ≤ I
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a : ι → ℝ
f : ℝ → E
I b : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc (a i) b) μ
ha : Tendsto a l atBot
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Iic b)) l fun i ↦ Ioi (a i)
hfi : ∀ (i : ι), IntegrableOn f (Ioi (a i)) (μ.restrict (Iic b))
i : ι
⊢ a i ≤ b → ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi (a i), ‖f x‖ ∂μ.restrict (Iic b) ≤ I

Tactic:
hai

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a : ι → ℝ
f : ℝ → E
I b : ℝ
hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc (a i) b) μ
ha : Tendsto a l atBot
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I
hφ : AECover (μ.restrict (Iic b)) l fun i ↦ Ioi (a i)
hfi : ∀ (i : ι), IntegrableOn f (Ioi (a i)) (μ.restrict (Iic b))
i : ι
hai : a i ≤ b
⊢ ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi (a i), ‖f x‖ ∂μ.restrict (Iic b) ≤ I
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ι → ℝ
f : ℝ → E
hfi : Integrable f μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
φ : ι → Set ℝ := fun i ↦ Ioc (a i) (b i)
hφ : AECover μ l φ
i : ι
⊢ a i ≤ 0 → 0 ≤ b i → ∫ (x : ℝ) in φ i, f x ∂μ = ∫ (x : ℝ) in a i..b i, f x ∂μ

Tactic:
hai

Goal state after:
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ι → ℝ
f : ℝ → E
hfi : Integrable f μ
ha : Tendsto a l atBot
hb : Tendsto b l atTop
φ : ι → Set ℝ := fun i ↦ Ioc (a i) (b i)
hφ : AECover μ l φ
i : ι
hai : a i ≤ 0
⊢ 0 ≤ b i → ∫ (x : ℝ) in φ i, f x ∂μ = ∫ (x : ℝ) in a i..b i, f x ∂μ