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:
rw [mem_Ici] at hj

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.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 : i ≤ j
⊢ (-1) ^ ↑i.castSucc •
        (-1) ^ ↑j •
          (ω x)
            (Matrix.vecCons ((fderivWithin 𝕜 (fun y ↦ i.castSucc.removeNth (fun x ↦ V x y) j) s x) (V i.castSucc x))
              (j.removeNth fun x_1 ↦ i.castSucc.removeNth (fun x_2 ↦ V x_2 x) x_1)) +
      (-1) ^ ↑j.succ •
        (-1) ^ ↑i •
          (ω x)
            (Matrix.vecCons ((fderivWithin 𝕜 (fun y ↦ j.succ.removeNth (fun x ↦ V x y) i) s x) (V j.succ x))
              (i.removeNth fun x_1 ↦ j.succ.removeNth (fun x_2 ↦ V x_2 x) x_1)) =
    (-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:
Fin.removeNth_removeNth_eq_swap

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 •
        (-1) ^ ↑j •
          (ω x)
            (Matrix.vecCons ((fderivWithin 𝕜 (fun y ↦ i.castSucc.removeNth (fun x ↦ V x y) j) s x) (V i.castSucc x))
              ((j.predAbove i.castSucc).removeNth ((i.castSucc.succAbove j).removeNth fun x_1 ↦ V x_1 x))) +
      (-1) ^ ↑j.succ •
        (-1) ^ ↑i •
          (ω x)
            (Matrix.vecCons ((fderivWithin 𝕜 (fun y ↦ j.succ.removeNth (fun x ↦ V x y) i) s x) (V j.succ x))
              (i.removeNth fun x_1 ↦ j.succ.removeNth (fun x_2 ↦ V x_2 x) x_1)) =
    (-1) ^ (↑i + ↑j) •
      (ω x)
        (Matrix.vecCons (lieBracketWithin 𝕜 (V i.castSucc) (V j.succ) s x)
          ((j.predAbove i.castSucc).removeNth ((i.castSucc.succAbove j).removeNth fun x_1 ↦ V x_1 x)))
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
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
⊢ i.castSucc < j.succ

Tactic:
by simpa using hj

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
⊢ IsSymmSndFDerivAt 𝕜 f x

Tactic:
intro v w

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
v w : E
⊢ ((fderiv 𝕜 (fderiv 𝕜 f) x) v) w = ((fderiv 𝕜 (fderiv 𝕜 f) x) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
k : ℕ
hk : y k ∈ u
s_mem : s ∈ 𝓝 (y k)
this : IsSymmSndFDerivAt 𝕜 f (y k)
⊢ IsSymmSndFDerivWithinAt 𝕜 f s (y k)

Tactic:
intro v w

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
k : ℕ
hk : y k ∈ u
s_mem : s ∈ 𝓝 (y k)
this : IsSymmSndFDerivAt 𝕜 f (y k)
v w : E
⊢ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
I : ∀ᶠ (k : ℕ) in atTop, IsSymmSndFDerivWithinAt 𝕜 f s (y k)
A : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (s ∩ u)
B : Tendsto (fun k ↦ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) atTop (𝓝 (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x))
C :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
C' :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
⊢ IsSymmSndFDerivWithinAt 𝕜 f s x

Tactic:
intro v w

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
I : ∀ᶠ (k : ℕ) in atTop, IsSymmSndFDerivWithinAt 𝕜 f s (y k)
A : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (s ∩ u)
B : Tendsto (fun k ↦ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) atTop (𝓝 (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x))
C :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
C' :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
v w : E
⊢ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) w) v
Module: Mathlib.Analysis.Calculus.Implicit

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : CompleteSpace E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : CompleteSpace F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
φ : ImplicitFunctionData 𝕜 E F G
⊢ HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt))
    (fderiv 𝕜 (φ.implicitFunction (φ.leftFun φ.pt)) (φ.rightFun φ.pt)) (φ.rightFun φ.pt)

Tactic:
have := φ.hasStrictFDerivAt.to_localInverse.comp (φ.rightFun φ.pt)
    ((hasStrictFDerivAt_const _ _).prodMk (hasStrictFDerivAt_id _))
  convert! this
  exact this.hasFDerivAt.fderiv

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

Goal state before:
pp: case refine_2
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✝ : Fintype ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
hextr : IsLocalExtrOn φ {x | (fun i ↦ f i x) = fun i ↦ f i x₀} x₀
Λ₀ : ℝ
Λ : ι → ℝ
h0 : ((LinearEquiv.piRing ℝ ℝ ι ℝ).symm Λ, Λ₀) ≠ 0
hsum : ∀ (x : E), ((LinearEquiv.piRing ℝ ℝ ι ℝ).symm Λ) ((ContinuousLinearMap.pi f') x) + Λ₀ • φ' x = 0
x : E
⊢ (∑ i, Λ i • f' i + Λ₀ • φ') x = 0 x

Tactic:
simpa [mul_comm] using hsum x

Goal state after:
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 : InjOn f s
g : ℝ → ℝ≥0∞
⊢ ∫⁻ (x : ℝ) in f '' s, g x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal |f' x| * g (f x)

Tactic:
by
  simpa only [det_toSpanSingleton] using
    lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hs
      (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts

Goal state before:
pp: a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ ∫ (x : ℝ) in a..b, u' x • v x + u x • v' x = u b • v b - u a • v a

Tactic:
simp_rw [add_comm]

Goal state after:
a b : ℝ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : IsScalarTower ℝ 𝕜 E
u u' : ℝ → 𝕜
v v' : ℝ → E
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ ∫ (x : ℝ) in a..b, u x • v' x + u' x • v x = u b • v b - u a • v a
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‖}
s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z
closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (M n z) ⊆ M n z
d : ℕ → E
hd : DenseRange d
K : ℕ → ↑T → ℕ → Set E :=
  fun n z p ↦ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (M n z) ∩ closedBall (d p) (u n / 3)
n : ℕ
z : ↑T
p : ℕ
x : E
hx : x ∈ s ∩ K n z p
y : E
hy : y ∈ s ∩ K n z p
yM : y ∈ M n z
⊢ u n / 3 + u n / 3 < u n

Tactic:
linarith [u_pos n]

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‖}
s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z
closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (M n z) ⊆ M n z
d : ℕ → E
hd : DenseRange d
K : ℕ → ↑T → ℕ → Set E :=
  fun n z p ↦ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (M n z) ∩ closedBall (d p) (u n / 3)
K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))
K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K n z p)
F : ℕ → ℕ × ↑T × ℕ
hF : Function.Surjective F
x : E
xs : x ∈ s
n : ℕ
z : ↑T
hnz : x ∈ M n z
⊢ 0 < u n / 3

Tactic:
linarith [u_pos n]

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 : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
h : A.det = 0
⊢ False

Tactic:
simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume
⊢ Tendsto f atBot (𝓝 0)

Tactic:
have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int

Goal state after:
E : Type u_1
f f' : ℝ → E
a : ℝ
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
fint : IntegrableOn f (Iic a) volume
F : E →L[ℝ] Completion E := Completion.toComplL
Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x
Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume
F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume
⊢ Tendsto f atBot (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f : ℝ → E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hf : ContDiff ℝ 1 f
h2f : HasCompactSupport f
b : ℝ
⊢ ∫ (x : ℝ) in Iic b, deriv f x = f b

Tactic:
by
  have := fun x (_ : x ∈ Iio b) ↦ hf.differentiable one_ne_zero x |>.hasDerivAt
  rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
  · refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
  rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f
  exact h2f.filter_mono _root_.atBot_le_cocompact |>.tendsto

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
f f' : ℝ → E
m n : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : Integrable f' volume
hbot : Tendsto f atBot (𝓝 m)
htop : Tendsto f atTop (𝓝 n)
⊢ ∫ (x : ℝ), f' x = n - m

Tactic:
by
  rw [← setIntegral_univ, ← Set.Iic_union_Ioi (a := 0),
    setIntegral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi hf'.integrableOn hf'.integrableOn,
    integral_Iic_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn hbot,
    integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn htop]
  abel

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

Goal state before:
pp: case inl
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
R : ℝ≥0
hR0 : 0 < ↑R
z : ℂ
hd : ∀ᶠ (z : ℂ) in 𝓝[≠] z, DifferentiableAt ℂ f z
hc : ContinuousAt f z
hRs : closedBall z ↑R ∩ {z}ᶜ ⊆ {x | (fun z ↦ DifferentiableAt ℂ f z) x}
hz : z ∈ closedBall z ↑R
⊢ ContinuousAt f z

pp: case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z
hc : ContinuousAt f c
R : ℝ≥0
hR0 : 0 < ↑R
hRs : closedBall c ↑R ∩ {c}ᶜ ⊆ {x | (fun z ↦ DifferentiableAt ℂ f z) x}
z : ℂ
hz : z ∈ closedBall c ↑R
hne : z ≠ c
⊢ ContinuousAt f z

Tactic:
exacts [hc, (hRs ⟨hz, hne⟩).continuousAt]

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
s : Set ℂ
c : ℂ
hc : s ∈ 𝓝 c
hd : DifferentiableOn ℂ f (s \ {c})
ho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹
F : ℂ → E := fun z ↦ (z - c) • f z
⊢ ContinuousAt F c

Tactic:
rw [← continuousWithinAt_compl_self]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
s : Set ℂ
c : ℂ
hc : s ∈ 𝓝 c
hd : DifferentiableOn ℂ f (s \ {c})
ho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹
F : ℂ → E := fun z ↦ (z - c) • f z
⊢ ContinuousWithinAt F {c}ᶜ c
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
r : ℝ
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
⊢ ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)

Tactic:
refine ((continuous_id'.fun_sub continuous_const).fun_pow 2).continuousOn.inv₀
      fun w hw h => hr.ne ?_

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
r : ℝ
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
w : ℂ
hw : w ∈ sphere z r
h : (w - z) ^ 2 = 0
⊢ 0 = r
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
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : NullMeasurableSet s μ
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
t : Set E
ts : t ⊆ s
ht : MeasurableSet t
t_eq_s : t =ᵐ[μ] s
A : f '' s =ᵐ[μ] f '' t
x : E
hx : x ∈ t
⊢ HasFDerivWithinAt f (f' x) t x

Tactic:
exact (hf' x (ts hx)).mono ts

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

Goal state before:
pp: case left
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'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ c.symm.source ∩ ↑c.symm ⁻¹' ((e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := (c.symm ≫ₕ f₁).source ∩ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
open_s : IsOpen[inst✝⁶] s
⊢ x ∈ c.symm.source

Tactic:
exact hx.1

Goal state after: