mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
s : Set X
Cg rg : ℝ≥0
g : Y → Z
t : Set Y
hg : HolderOnWith Cg rg g t
Cf rf : ℝ≥0
f : X → Y
hf : HolderOnWith Cf rf f s
hst : MapsTo f s t
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((g ∘ f) x) ((g ∘ f) y) ≤ ↑Cg * (↑(Cf ^ ↑rg) * (edist x y ^ ↑rf) ^ ↑rg)

Tactic:
ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg,

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : PseudoEMetricSpace X
inst✝¹ : PseudoEMetricSpace Y
inst✝ : PseudoEMetricSpace Z
s : Set X
Cg rg : ℝ≥0
g : Y → Z
t : Set Y
hg : HolderOnWith Cg rg g t
Cf rf : ℝ≥0
f : X → Y
hf : HolderOnWith Cf rf f s
hst : MapsTo f s t
x : X
hx : x ∈ s
y : X
hy : y ∈ s
⊢ edist ((g ∘ f) x) ((g ∘ f) y) ≤ ↑Cg * (↑Cf ^ ↑rg * (edist x y ^ ↑rf) ^ ↑rg)
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 s t₁ t₂ : ℝ≥0
A : Set X
hf₁ : HolderOnWith C₁ r f A
hf₂ : HolderOnWith C₂ s f A
ht : t₁ + t₂ = 1
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑(C₁ ^ ↑t₁ * C₂ ^ ↑t₂) * edist x y ^ ↑(r * t₁ + s * t₂)

Tactic:
calc edist (f x) (f y)
      = (edist (f x) (f y)) ^ (t₁ : ℝ) * (edist (f x) (f y)) ^ (t₂ : ℝ) := by
        simp [← ENNReal.rpow_add_of_nonneg, ← NNReal.coe_add, ht]
    _ ≤ (C₁ * (edist x y) ^ (r : ℝ)) ^ (t₁ : ℝ) * (C₂ * (edist x y) ^ (s : ℝ)) ^ (t₂ : ℝ) := by
        nth_grw 1 [hf₁ x hx y hy, hf₂ x hx y hy]
    _ = ↑(C₁ ^ (t₁ : ℝ) * C₂ ^ (t₂ : ℝ)) * (edist x y) ^ (↑(r * t₁ + s * t₂) : ℝ) := by
        push_cast
        simp (discharger := positivity) only [ENNReal.mul_rpow_of_nonneg,
          ENNReal.rpow_add_of_nonneg, ENNReal.rpow_mul, ENNReal.coe_rpow_of_nonneg]
        ring

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

Goal state before:
pp: case refine_2
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Ico b c)

Tactic:
apply monotoneOn_of_deriv_nonneg (convex_Ico b c) (continuousOn_Ico hb hd₁) <;> simp_all

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

Goal state before:
pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Ico b c)

Tactic:
apply monotoneOn_of_deriv_nonneg (convex_Ico b c) (continuousOn_Ico hb hd₁) <;> simp_all

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

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x - x₀)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b < 0

Tactic:
filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)

Goal state after:
f : ℝ → ℝ
x₀ : ℝ
h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x - x₀)
x : ℝ
hx' : sign (deriv f x) = sign (x - x₀)
hx : x < x₀
⊢ deriv f x < 0
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x₀ - x)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b > 0

Tactic:
filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)

Goal state after:
f : ℝ → ℝ
x₀ : ℝ
h₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x₀ - x)
x : ℝ
hx' : sign (deriv f x) = sign (x₀ - x)
hx : x < x₀
⊢ deriv f x > 0
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
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ (∀ (v w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) w) v) ↔
    ContinuousMultilinearMap.domDomCongr Fin.revPerm (iteratedFDerivWithin 𝕜 2 f s x) = iteratedFDerivWithin 𝕜 2 f s x

Tactic:
ContinuousMultilinearMap.ext_iff,

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
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ (∀ (v w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) w) v) ↔
    ∀ (x_1 : Fin 2 → E),
      (ContinuousMultilinearMap.domDomCongr Fin.revPerm (iteratedFDerivWithin 𝕜 2 f s x)) x_1 =
        (iteratedFDerivWithin 𝕜 2 f s x) x_1
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 v w : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hf : ContinuousMultilinearMap.domDomCongr Fin.revPerm (iteratedFDerivWithin 𝕜 2 f s x) = iteratedFDerivWithin 𝕜 2 f s x
⊢ (iteratedFDerivWithin 𝕜 2 f s x) ![v, w] = (iteratedFDerivWithin 𝕜 2 f s x) ![w, v]

Tactic:
ContinuousMultilinearMap.ext_iff,

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 v w : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hf :
  ∀ (x_1 : Fin 2 → E),
    (ContinuousMultilinearMap.domDomCongr Fin.revPerm (iteratedFDerivWithin 𝕜 2 f s x)) x_1 =
      (iteratedFDerivWithin 𝕜 2 f s x) x_1
⊢ (iteratedFDerivWithin 𝕜 2 f s x) ![v, w] = (iteratedFDerivWithin 𝕜 2 f s x) ![w, v]
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_4
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : μ v ≤ μ s' + ε / 2 / ↑N
r1 : α → ℝ
hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v
q : BallPackage (↑s') α := { c := fun x ↦ ↑x, r := fun x ↦ r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
S : Fin N → Set ↑s'
S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j ↦ closedBall (q.c j) (q.r j)
hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)
S_count : ∀ (i : Fin N), (S i).Countable
r : α → ℝ := fun x ↦ if x ∈ s' then r1 x else r0 x
r_t0 : ∀ x ∈ t0, r x = r0 x
⊢ s ⊆ ⋃ x ∈ t0 ∪ ⋃ i, Subtype.val '' S i, closedBall x (r x)

Tactic:
intro x hx
    by_cases h'x : x ∈ s'
    · obtain ⟨i, y, ySi, xy⟩ : ∃ (i : Fin N) (y : ↥s'), y ∈ S i ∧ x ∈ ball (y : α) (r1 y) := by
        have A : x ∈ range q.c := by
          simpa only [q, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and,
            not_le, mem_ofPred_eq, Subtype.range_coe_subtype, Set.mem_sdiff] using h'x
        simpa only [mem_iUnion, mem_image, bex_def] using hS A
      refine mem_iUnion₂.2 ⟨y, Or.inr ?_, ?_⟩
      · simp only [mem_iUnion, mem_image]
        exact ⟨i, y, ySi, rfl⟩
      · have : (y : α) ∈ s' := y.2
        simp only [r, if_pos this]
        exact ball_subset_closedBall xy
    · obtain ⟨y, yt0, hxy⟩ : ∃ y : α, y ∈ t0 ∧ x ∈ closedBall y (r0 y) := by
        simpa [s', hx, -mem_closedBall] using h'x
      refine mem_iUnion₂.2 ⟨y, Or.inl yt0, ?_⟩
      rwa [r_t0 _ yt0]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ W₁ : E → E
s t : Set E
x✝ : E
hV : V₁ =ᶠ[𝓝[s] x✝] V
hW : W₁ =ᶠ[𝓝[s] x✝] W
ht : t ⊆ s
x : E
⊢ fderivWithin 𝕜 V₁ t x = fderivWithin 𝕜 V t x →
    fderivWithin 𝕜 W₁ t x = fderivWithin 𝕜 W t x →
      V₁ x = V x → W₁ x = W x → lieBracketWithin 𝕜 V₁ W₁ t x = lieBracketWithin 𝕜 V W t x

Tactic:
hV'

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ W₁ : E → E
s t : Set E
x✝ : E
hV : V₁ =ᶠ[𝓝[s] x✝] V
hW : W₁ =ᶠ[𝓝[s] x✝] W
ht : t ⊆ s
x : E
hV' : fderivWithin 𝕜 V₁ t x = fderivWithin 𝕜 V t x
⊢ fderivWithin 𝕜 W₁ t x = fderivWithin 𝕜 W t x →
    V₁ x = V x → W₁ x = W x → lieBracketWithin 𝕜 V₁ W₁ t x = lieBracketWithin 𝕜 V W t x
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
V : F → F
x : E
h'f : (fderivWithin 𝕜 f s x).IsInvertible
⊢ (fderivWithin 𝕜 f s x) (pullbackWithin 𝕜 f V s x) = V (f x)

Tactic:
rcases h'f with ⟨M, hM⟩
  simp [pullbackWithin_eq_of_fderivWithin_eq hM, ← hM]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
f : E → F
s : Set E
x : E
h'f : ContDiffWithinAt 𝕜 2 f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
M : E ≃L[𝕜] F
hM : ↑M = fderivWithin 𝕜 f s x
⊢ ∃ N,
    ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x ∧
      ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x ∧
        (∀ᶠ (y : E) in 𝓝[s] x, ↑(N y) = fderivWithin 𝕜 f s y) ∧
          ∀ (v : E),
            (fderivWithin 𝕜 (fun y ↦ ↑(N y).symm) s x) v =
              -↑(N x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(N x).symm

Tactic:
let U := {y | ∃ (N : E ≃L[𝕜] F), N = fderivWithin 𝕜 f s y}

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
f : E → F
s : Set E
x : E
h'f : ContDiffWithinAt 𝕜 2 f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
M : E ≃L[𝕜] F
hM : ↑M = fderivWithin 𝕜 f s x
U : Set E := {y | ∃ N, ↑N = fderivWithin 𝕜 f s y}
⊢ ∃ N,
    ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x ∧
      ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x ∧
        (∀ᶠ (y : E) in 𝓝[s] x, ↑(N y) = fderivWithin 𝕜 f s y) ∧
          ∀ (v : E),
            (fderivWithin 𝕜 (fun y ↦ ↑(N y).symm) s x) v =
              -↑(N x).symm ∘SL (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v ∘SL ↑(N x).symm
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)
this : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f (s ∩ u)) (s ∩ u)) (s ∩ u)
⊢ ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (s ∩ u)

Tactic:
apply this.congr

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)
this : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f (s ∩ u)) (s ∩ u)) (s ∩ u)
⊢ EqOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (fderivWithin 𝕜 (fderivWithin 𝕜 f (s ∩ u)) (s ∩ u)) (s ∩ u)
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)
this : Tendsto y atTop (𝓝[s ∩ u] x)
⊢ Tendsto (fun k ↦ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) atTop (𝓝 (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x))

Tactic:
exact (A x ⟨h'x, xu⟩ ).tendsto.comp this

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
x : F
f₁ : F → 𝕜
hL : f₁ =ᶠ[𝓝 x] f
⊢ (toDual 𝕜 F).symm (fderiv 𝕜 f₁ x) = (toDual 𝕜 F).symm (fderiv 𝕜 f x)

Tactic:
Filter.EventuallyEq.fderiv_eq

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
x : F
f₁ : F → 𝕜
hL : f₁ =ᶠ[𝓝 x] f
⊢ (toDual 𝕜 F).symm (fderiv 𝕜 ?m.33 x) = (toDual 𝕜 F).symm (fderiv 𝕜 f x)

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
x : F
f₁ : F → 𝕜
hL : f₁ =ᶠ[𝓝 x] f
⊢ f₁ =ᶠ[𝓝 x] ?m.33

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
x : F
f₁ : F → 𝕜
hL : f₁ =ᶠ[𝓝 x] f
⊢ F → 𝕜
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
z₀ : 𝕜
R R₁ : 𝕜 → 𝕜
hf1 : ∀ (z : 𝕜), AnalyticAt 𝕜 R₁ z
k : ℕ
IH :
  ∀ {t : ℕ},
    (∀ (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)
t : ℕ
hR₁ : ∀ (z : 𝕜), R z = (z - z₀) ^ (k + 1 + t) * R₁ z
R₂ : 𝕜 → 𝕜
hR₂ : ∀ (z : 𝕜), AnalyticAt 𝕜 R₂ z
hR₂_eq : ∀ (z : 𝕜), iteratedDeriv k R z = (z - z₀) ^ (t + 1) * (↑(k + (t + 1))! / ↑(t + 1)! * R₁ z + (z - z₀) * R₂ z)
R₂' : 𝕜 → 𝕜 := fun z ↦ (↑t + 1) * R₂ z + (↑(k + (t + 1))! / ↑(t + 1)! * deriv R₁ z + (R₂ z + (z - z₀) * deriv R₂ z))
z : 𝕜
this : ↑t + 1 ≠ 0
⊢ ↑(t + 1)! = (↑t + 1) * ↑t !

Tactic:
simp [Nat.factorial_succ]

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

Goal state before:
pp: case refine_1
a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hcf : ContinuousOn f (Ioc a b)
hcg : ContinuousOn g (Ioc a b)
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : f b = 0
hgb : g b = 0
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] b) l
⊢ Tendsto f (𝓝[<] b) (𝓝 0)

Tactic:
rw [← hfb, ← nhdsWithin_Ioo_eq_nhdsLT hab]
    exact ((hcf b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto

Goal state after:
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
K : HasDerivWithinAt f 0 (s ∩ Ioo p x) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo p x) x
J1 : (s ∩ Ioo p x) \ {x} = s ∩ Ioo p x
J2 : 𝓝[s ∩ Ioo p x] x = 𝓝[s ∩ Iio x] x
⊢ (𝓝[s ∩ Iio x] x).NeBot

Tactic:
simp only [mem_inter_iff, Set.mem_sdiff, hx.1.1, mem_union, mem_ofPred_eq, true_and, not_or,
        mem_preimage, b, s₁, a] at hx

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 p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : p < x
K : HasDerivWithinAt f 0 (s ∩ Ioo p x) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo p x) x
J1 : (s ∩ Ioo p x) \ {x} = s ∩ Ioo p x
J2 : 𝓝[s ∩ Ioo p x] x = 𝓝[s ∩ Iio x] x
hx : (¬𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬𝓝[s ∩ Iio x] x = ⊥) ∧ f x ∈ u
⊢ (𝓝[s ∩ Iio x] x).NeBot
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 : x < p
K : HasDerivWithinAt f 0 (s ∩ Ioo x p) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo x p) x
J1 : (s ∩ Ioo x p) \ {x} = s ∩ Ioo x p
J2 : 𝓝[s ∩ Ioo x p] x = 𝓝[s ∩ Ioi x] x
⊢ (𝓝[s ∩ Ioi x] x).NeBot

Tactic:
simp only [mem_inter_iff, Set.mem_sdiff, hx.1.1, mem_union, mem_ofPred_eq, true_and, not_or,
        mem_preimage, b, s₁, a] at hx

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 p : ℝ
ps₁ : p ∈ s₁
px✝ : p ≠ x
fpx : f p = f x
px : x < p
K : HasDerivWithinAt f 0 (s ∩ Ioo x p) x
K' : HasDerivWithinAt f (f' x) (s ∩ Ioo x p) x
J1 : (s ∩ Ioo x p) \ {x} = s ∩ Ioo x p
J2 : 𝓝[s ∩ Ioo x p] x = 𝓝[s ∩ Ioi x] x
hx : (¬𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬𝓝[s ∩ Iio x] x = ⊥) ∧ f x ∈ u
⊢ (𝓝[s ∩ Ioi x] x).NeBot
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderClosedTopology α
inst✝ : OpensMeasurableSpace α
b c : ι → α
B : α
hb : Tendsto b l (𝓝 B)
hc : Tendsto c l atBot
_x : α
hx : _x ∈ Iio B
⊢ (∀ᶠ (x : ι) in l, c x ≤ _x) ∧ ∀ᶠ (x : ι) in l, _x < b x

Tactic:
exact ⟨hc.eventually (eventually_le_atBot _x), hb.eventually (eventually_gt_nhds hx)⟩

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : MeasurableSpace α
μ : Measure α
l : Filter ι
β : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
f : α → β
φ : ι → Set α
hφ : AECover μ l φ
hfm : ∀ (i : ι), AEStronglyMeasurable f (μ.restrict (φ i))
⊢ AEStronglyMeasurable f μ

Tactic:
obtain ⟨u, hu⟩ := l.exists_seq_tendsto
  have := aestronglyMeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n)
  rwa [Measure.restrict_eq_self_of_ae_mem] at this
  filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists

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, f' x ≤ 0
hab : a ≤ b
M : AntitoneOn f (Icc a b)
A : ∫ (u : ℝ) in Icc (f b) (f a), g u = ∫ (u : ℝ) in f '' Ioo a b, g u
⊢ ∫ (t : ℝ) in Ioo a b, f' t • g (f t) = -∫ (x : ℝ) in Ioo a b, -f' x • g (f x)

Tactic:
simp [integral_neg]

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

Tactic:
integral_prod _ hf'g

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)
⊢ -∫ (a : E), ∫ (t : ℝ), (B (f' (a, t))) (g (a, t)) ∂μ = -∫ (x : E), ∫ (y : ℝ), (B (f' (x, y))) (g (x, y)) ∂μ
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) in univ, f' x = n - m

Tactic:
← Set.Iic_union_Ioi (a := 0),

Goal state after:
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) in Iic 0 ∪ Ioi 0, f' x = n - m