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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B ∘SL ↑↑isoE).flip ∘SL ↑↑isoF).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu := ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
hBu : Bu = ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Tactic:
have Bu_eq : (fun y => Bu (fu y) (gu y)) = isoG.symm ∘ (fun y => B (f y) (g y)) ∘ isoD := by
    ext1 y
    simp [Du, Eu, Fu, Gu, hBu, Bu₀, fu, gu]

Goal state after:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B ∘SL ↑↑isoE).flip ∘SL ↑↑isoF).flip
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu := ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
hBu : Bu = ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑↑isoG.symm)) Bu₀
Bu_eq : (fun y ↦ (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y ↦ (B (f y)) (g y)) ∘ ⇑isoD
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ : ℝ≥0
A : Set X
hf : HolderOnWith C₁ r f A
hC : C₁ ≤ C₂
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑C₂ * edist x y ^ ↑r

Tactic:
grw [← hC]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ : ℝ≥0
A : Set X
hf : HolderOnWith C₁ r f A
hC : C₁ ≤ C₂
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑C₁ * edist x y ^ ↑r
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: case hpf
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hi : i ≤ k
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
by_cases hfd : DifferentiableAt ℝ f a

Goal state after:
case pos
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hi : i ≤ k
hfd : DifferentiableAt ℝ f a
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hi : i ≤ k
hfd : ¬DifferentiableAt ℝ f a
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F → G
hg : ContDiffPointwiseHolderAt k α g (f a)
hf : ContDiffPointwiseHolderAt k α f a
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hi : i ≤ k
hfd : ¬DifferentiableAt ℝ f a
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
obtain rfl : k = 0 := by
            contrapose! hfd
            exact hf.differentiableAt hfd

Goal state after:
case neg
E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
α : ↑I
f : E → F
a : E
g : F → G
hd : DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a
i : ℕ
hfd : ¬DifferentiableAt ℝ f a
hg : ContDiffPointwiseHolderAt 0 α g (f a)
hf : ContDiffPointwiseHolderAt 0 α f a
hi : i ≤ 0
⊢ (fun a_1 ↦ ftaylorSeries ℝ g (f a_1) i - ftaylorSeries ℝ g (f a) i) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ ↑α
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)
x : α
hx : x ∈ s
δ : ℝ
δpos : δ > 0
⊢ (g x ∩ Ioo 0 δ).Nonempty

Tactic:
rcases hf x hx (min δ (R x)) (lt_min δpos (hR x hx)) with ⟨r, hr⟩

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)
x : α
hx : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
hr : r ∈ f x ∩ Ioo 0 (min δ (R x))
⊢ (g x ∩ Ioo 0 δ).Nonempty
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
x v : E
⊢ lineDerivWithin 𝕜 f univ x v = lineDeriv 𝕜 f x v

Tactic:
by
  simp [lineDerivWithin, lineDeriv]

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

Goal state before:
pp: 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 : HasFDerivAt (fun x ↦ ‖x‖) f x
h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasFDerivAt (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)

Tactic:
· ext y
    simp only [smul_apply, smul_eq_mul, comp_smulₛₗ, map_inv₀, RingHom.id_apply, comp_id]
    rw [eq_inv_mul_iff_mul_eq₀ ht, ← mul_assoc, self_mul_sign]

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

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
g : E → F
f : 𝕜 → E
x : 𝕜
hg : ContDiffAt 𝕜 2 g (f x)
hf : ContDiffAt 𝕜 2 f x
⊢ iteratedDeriv 2 (g ∘ f) x =
    ((iteratedFDeriv 𝕜 2 g (f x)) fun x_1 ↦ deriv f x) + (fderiv 𝕜 g (f x)) (iteratedDeriv 2 f x)

Tactic:
simp only [← iteratedDerivWithin_univ, ← iteratedFDerivWithin_univ,
    ← derivWithin_univ, ← fderivWithin_univ]

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
g : E → F
f : 𝕜 → E
x : 𝕜
hg : ContDiffAt 𝕜 2 g (f x)
hf : ContDiffAt 𝕜 2 f x
⊢ iteratedDerivWithin 2 (g ∘ f) univ x =
    ((iteratedFDerivWithin 𝕜 2 g univ (f x)) fun x_1 ↦ derivWithin f univ x) +
      (fderivWithin 𝕜 g univ (f x)) (iteratedDerivWithin 2 f univ x)
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
g'inv : G →L[𝕜] E
hg'inv : φ.rightDeriv ∘SL g'inv = ContinuousLinearMap.id 𝕜 G
hg'invf : φ.leftDeriv ∘SL g'inv = 0
x : G
⊢ g'inv x = (fderiv 𝕜 (φ.implicitFunction (φ.leftFun φ.pt)) (φ.rightFun φ.pt)) x

Tactic:
eq_comm,

Goal state after:
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
g'inv : G →L[𝕜] E
hg'inv : φ.rightDeriv ∘SL g'inv = ContinuousLinearMap.id 𝕜 G
hg'invf : φ.leftDeriv ∘SL g'inv = 0
x : G
⊢ (fderiv 𝕜 (φ.implicitFunction (φ.leftFun φ.pt)) (φ.rightFun φ.pt)) x = g'inv x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

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
g : E → F
f : 𝕜 → E
x : 𝕜
hg : ContDiffAt 𝕜 3 g (f x)
hf : ContDiffAt 𝕜 3 f x
⊢ iteratedDeriv 3 (g ∘ f) x =
    ((iteratedFDeriv 𝕜 3 g (f x)) fun x_1 ↦ deriv f x) +
          (iteratedFDeriv 𝕜 2 g (f x)) ![iteratedDeriv 2 f x, deriv f x] +
        2 • (iteratedFDeriv 𝕜 2 g (f x)) ![deriv f x, iteratedDeriv 2 f x] +
      (fderiv 𝕜 g (f x)) (iteratedDeriv 3 f x)

Tactic:
simp only [← iteratedDerivWithin_univ, ← iteratedFDerivWithin_univ,
    ← derivWithin_univ, ← fderivWithin_univ]

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
g : E → F
f : 𝕜 → E
x : 𝕜
hg : ContDiffAt 𝕜 3 g (f x)
hf : ContDiffAt 𝕜 3 f x
⊢ iteratedDerivWithin 3 (g ∘ f) univ x =
    ((iteratedFDerivWithin 𝕜 3 g univ (f x)) fun x_1 ↦ derivWithin f univ x) +
          (iteratedFDerivWithin 𝕜 2 g univ (f x)) ![iteratedDerivWithin 2 f univ x, derivWithin f univ x] +
        2 • (iteratedFDerivWithin 𝕜 2 g univ (f x)) ![derivWithin f univ x, iteratedDerivWithin 2 f univ x] +
      (fderivWithin 𝕜 g univ (f x)) (iteratedDerivWithin 3 f univ x)
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
hg' : ∀ x ∈ Ioo a b, 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
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
hg : ∀ x ∈ Ioo a b, g x ≠ 0
x : ℝ
hx : x ∈ Ioo a b
⊢ ∃ c ∈ Ioo a x, f x * g' c = g x * f' c

Tactic:
rw [← sub_zero (f x), ← sub_zero (g x)]

Goal state after:
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
hg' : ∀ x ∈ Ioo a b, 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
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
hg : ∀ x ∈ Ioo a b, g x ≠ 0
x : ℝ
hx : x ∈ Ioo a b
⊢ ∃ c ∈ Ioo a x, (f x - 0) * g' c = (g x - 0) * f' c
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 : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
⊢ ∃ n z, x ∈ M n z

Tactic:
obtain ⟨δ, δpos, hδ⟩ :
      ∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
      Metric.mem_nhdsWithin_iff.1 ((hf' x xs).isLittleO.def εpos)

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))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
⊢ ∃ n z, x ∈ M n z
Module: Mathlib.MeasureTheory.Function.JacobianOneDim

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

Tactic:
have : HasDerivWithinAt (fun y ↦ f x) 0 (s ∩ Ioo p x) x :=
          hasDerivWithinAt_const x (s ∩ Ioo p x) (f x)

Goal state after:
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf : MonotoneOn f s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}
a_count : a.Countable
s₁ : Set ℝ := s \ a
hs₁ : MeasurableSet s₁
u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y ∧ f x = c ∧ f y = c}
hu : u.Countable
b : Set ℝ := s₁ ∩ f ⁻¹' u
hb : MeasurableSet b
c : Set ℝ := s₁ \ b
hc : MeasurableSet c
x : ℝ
hx : x ∈ b
p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
E : ∀ y ∈ s ∩ Ioo p x, f y = f x
this : HasDerivWithinAt (fun y ↦ f x) 0 (s ∩ Ioo p x) x
⊢ HasDerivWithinAt f 0 (s ∩ Ioo p x) x
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 := ⋯
s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z
n : ℕ
z : ↑T
x : E
xs : x ∈ s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (M n z)
y : E
hy : y ∈ s ∩ ball x (u n)
a : ℕ → E
aM : ∀ (k : ℕ), a k ∈ M n z
a_lim : Tendsto a atTop (𝓝 x)
L : Tendsto (fun k ↦ f (a k)) atTop (𝓝 (f x))
⊢ Tendsto (fun x ↦ (f' ↑↑z) (y - a x)) atTop (𝓝 ((f' ↑↑z) (y - x)))

Tactic:
exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)

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

Goal state before:
pp: case hf'_nonneg
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
⊢ ∀ x ∈ interior [[a, b]], 0 ≤ deriv f x

Tactic:
· rw [uIcc, interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz

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

Goal state before:
pp: case hf'_nonpos
a b : ℝ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → ℝ
g : ℝ → E
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0
⊢ ∀ x ∈ interior [[a, b]], deriv f x ≤ 0

Tactic:
· rw [uIcc, interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz

Goal state after:
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
⊢ IntegrableOn (fun x ↦ f' x • g (f x)) (Icc a b) volume ↔ IntegrableOn g (Icc (f a) (f b)) volume

Tactic:
have M : MonotoneOn f (Icc a b) := by
    apply monotoneOn_of_deriv_nonneg (convex_Icc a b) hf
    · rw [interior_Icc]
      exact fun z hz ↦ (hff' z hz).differentiableAt.differentiableWithinAt
    · rw [interior_Icc]
      intro z hz
      simpa [(hff' z hz).deriv] using hf' z hz
  have A : IntegrableOn g (Icc (f a) (f b)) ↔ IntegrableOn g (f '' (Ioo a b)) := by
    apply integrableOn_congr_set_ae
    rw [← hf.image_Icc_of_monotoneOn hab M]
    refine ae_eq_set.2 ⟨?_, by simp [show f '' Ioo a b \ f '' Icc a b = ∅ by grind]⟩
    have : f '' (Icc a b) \ f '' Ioo a b ⊆ {f a, f b} := by grind
    apply measure_mono_null this
    apply Finite.measure_zero (by simp)
  rw [A, integrableOn_Icc_iff_integrableOn_Ioo,
    integrableOn_image_iff_integrableOn_deriv_smul_of_monotoneOn measurableSet_Ioo]
  · exact fun z hz ↦ (hff' z hz).hasDerivWithinAt
  · exact M.mono Ioo_subset_Icc_self

Goal state after:
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), -∫ (t : ℝ), (B (f' (x, t))) (g (x, t)) ∂μ = -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ.prod volume

Tactic:
rw [integral_neg, integral_prod _ hf'g]

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
F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume
A : Tendsto (⇑F ∘ f) atBot (𝓝 (atBot.limUnder (⇑F ∘ f)))
⊢ Tendsto f atBot (𝓝 0)

Tactic:
have B : limUnder atBot (F ∘ f) = F 0 := by
    have : IntegrableAtFilter (F ∘ f) atBot := by exact ⟨Iic a, Iic_mem_atBot _, Fint⟩
    apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A
    intro s hs
    rcases mem_atBot_sets.1 hs with ⟨b, hb⟩
    apply le_antisymm (le_top)
    rw [← volume_Iic (a := b)]
    exact measure_mono hb

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
A : Tendsto (⇑F ∘ f) atBot (𝓝 (atBot.limUnder (⇑F ∘ f)))
B : atBot.limUnder (⇑F ∘ f) = F 0
⊢ Tendsto f atBot (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
p : ℝ
hp : 0 < p
⊢ ∫ (x : ℝ) in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Ioi 0, g y

Tactic:
simpa [abs_of_nonneg hp.le] using integral_comp_rpow_Ioi g hp.ne'

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
this : DifferentiableOn ℂ F s
⊢ DifferentiableOn ℂ (update f c ((𝓝[≠] c).limUnder f)) s

Tactic:
← differentiableOn_dslope hc,

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
this : DifferentiableOn ℂ (dslope F c) s
⊢ DifferentiableOn ℂ (update f c ((𝓝[≠] c).limUnder f)) s