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

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf : ∀ (n : ι) (x : E), HasFDerivAt (f n) (f' n x) x
hfg : ∀ (x : E), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : E
⊢ HasFDerivAt g (g' x) x

Tactic:
have hf : ∀ n : ι, ∀ x : E, x ∈ Set.univ → HasFDerivAt (f n) (f' n x) x := by simp [hf]

Goal state after:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf✝ : ∀ (n : ι) (x : E), HasFDerivAt (f n) (f' n x) x
hfg : ∀ (x : E), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : E
hf : ∀ (n : ι), ∀ x ∈ Set.univ, HasFDerivAt (f n) (f' n x) x
⊢ HasFDerivAt g (g' x) x
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
F' : 𝕜 → α → E
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, HasDerivAt (fun x ↦ F x a) (F' x a) x
⊢ Integrable (F' x₀) μ ∧ HasDerivAt (fun n ↦ ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀

Tactic:
by
  rcases Metric.mem_nhds_iff.1 hs with ⟨ε, ε_pos, hε⟩
  have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
  have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' x₀ a) x₀ :=
    h_diff.mono fun a ha ↦ ha x₀ (hε x₀_in)
  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 ↦ F x a) (ball x₀ ε) := by
    apply (h_diff.and h_bound).mono
    rintro a ⟨ha_deriv, ha_bound⟩
    refine (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
      (fun x x_in ↦ (ha_deriv x (hε x_in)).hasDerivWithinAt) fun x x_in ↦ ?_
    rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
    exact (ha_bound x (hε x_in)).trans (le_abs_self _)
  exact hasDerivAt_integral_of_dominated_loc_of_lip (ball_mem_nhds x₀ ε_pos) hF_meas hF_int
    hF'_meas this bound_integrable diff_x₀

Goal state after:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
A : ∀ (n : ℕ), μ ((x * z⁻¹) • k \ k) ≤ u n
B : μ ((x * z⁻¹) • k \ k) = 0
this : μ ((z * x⁻¹) • ((x * z⁻¹) • k \ k)) = 0
⊢ μ (k \ (z * x⁻¹) • k) = μ ((z * x⁻¹) • ((x * z⁻¹) • k \ k))

Tactic:
smul_set_sdiff,

Goal state after:
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
A : ∀ (n : ℕ), μ ((x * z⁻¹) • k \ k) ≤ u n
B : μ ((x * z⁻¹) • k \ k) = 0
this : μ ((z * x⁻¹) • ((x * z⁻¹) • k \ k)) = 0
⊢ μ (k \ (z * x⁻¹) • k) = μ ((z * x⁻¹) • (x * z⁻¹) • k \ (z * x⁻¹) • k)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ w D x

Tactic:
apply mul_nonneg _ (u_nonneg _)

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = ↑i
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color ↑j}
color_j : p.color jy = sInf (univ \ A)
⊢ Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) (p.index jx) (p.index jy)

Tactic:
have h : p.color jy ∈ univ \ A := by
      rw [color_j]
      apply csInf_mem
      refine ⟨N, ?_⟩
      simp only [A, not_exists, true_and, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
        mem_univ, Set.mem_sdiff, Subtype.exists]
      intro k hk _
      exact (p.color_lt (hk.trans jy_lt) hN).ne'

Goal state after:
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = ↑i
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color ↑j}
color_j : p.color jy = sInf (univ \ A)
h : p.color jy ∈ univ \ A
⊢ Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) (p.index jx) (p.index jy)
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
hi : ‖a.c i‖ ≤ 2
hj : 2 < ‖a.c j‖
ah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
hcrj : ‖a.c j‖ ≤ a.r j + 1
H : last N ≤ i
⊢ a.r i ≤ 2

Tactic:
· have : i = last N := top_le_iff.1 H
      rw [this, lastr]
      exact one_le_two

Goal state after:
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
i j : Fin N.succ
inej : i ≠ j
ah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j
δnonneg : 0 ≤ δ
hcrj : ‖a.c j‖ ≤ a.r j + 1
hj : 2 < ‖a.c j‖
s : ℝ := ‖a.c i‖
hi : 2 < s
hij : s ≤ ‖a.c j‖
A : a.r i ≤ s
spos : 0 < s
d : E := (s / ‖a.c j‖) • a.c j
hd : d = (s / ‖a.c j‖) • a.c j
⊢ 0 ≤ 1 - s / ‖a.c j‖

Tactic:
rwa [sub_nonneg, div_le_iff₀ (zero_lt_two.trans hj), one_mul]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: 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
⊢ (fun x ↦
      (ftaylorSeries ℝ g (f x)).taylorComp (ftaylorSeries ℝ f x) k -
        (ftaylorSeries ℝ g (f a)).taylorComp (ftaylorSeries ℝ f a) k) =O[𝓝 a]
    fun x ↦ ‖x - a‖ ^ ↑α

Tactic:
apply FormalMultilinearSeries.taylorComp_sub_taylorComp_isBigO <;> intro i hi
      · exact ((hg.contDiffAt.continuousAt_iteratedFDeriv (mod_cast hi)).comp hf.continuousAt)
          |>.norm.isBoundedUnder_le
      · by_cases hfd : DifferentiableAt ℝ f a
        · refine ((hg.of_order_le hi).isBigO.comp_tendsto hf.continuousAt).trans ?_
          refine .rpow α.2.1 (.of_forall fun _ ↦ norm_nonneg _) <| .norm_norm ?_
          exact hfd.isBigO_sub
        · obtain rfl : k = 0 := by
            contrapose! hfd
            exact hf.differentiableAt hfd
          obtain rfl : i = 0 := by rwa [nonpos_iff_eq_zero] at hi
          refine .of_norm_left ?_
          simp only [ftaylorSeries, iteratedFDeriv_zero_eq_comp, Function.comp_apply, ← map_sub,
            LinearIsometryEquiv.norm_map, isBigO_norm_left]
          refine ((hd.resolve_right hfd).isBigO_sub.comp_tendsto hf.continuousAt).trans ?_
          exact (zero_order_iff.mp hf).2
      · exact (hf.contDiffAt.continuousAt_iteratedFDeriv (mod_cast hi)).norm.isBoundedUnder_le
      · exact isBoundedUnder_const
      · exact (hf.of_order_le hi).isBigO

Goal state after:
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)
⊢ ∃ t r,
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧ t.PairwiseDisjoint fun x ↦ closedBall x (r x)

Tactic:
have hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := fun x hx δ δpos ↦ by
    rcases hf x hx (min δ (R x)) (lt_min δpos (hR x hx)) with ⟨r, hr⟩
    exact ⟨r, ⟨⟨hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)⟩,
      ⟨hr.2.1, hr.2.2.trans_le (min_le_left _ _)⟩⟩⟩

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
⊢ ∃ t r,
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧ t.PairwiseDisjoint fun x ↦ closedBall x (r x)
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F :=
  fun t ↦ f (x + h • v + (t * h) • w) - (t * h) • (f' x) w - (t * h ^ 2) • (f'' v) w - ((t * h) ^ 2 / 2) • (f'' w) w
g' : ℝ → F :=
  fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
g_deriv : ∀ t ∈ Icc 0 1, HasDerivWithinAt g (g' t) (Icc 0 1) t
t : ℝ
ht : t ∈ Ico 0 1
I : ‖h • v + (t * h) • w‖ ≤ h * (‖v‖ + ‖w‖)
⊢ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖

Tactic:
gcongr
        have H : x + h • v + (t * h) • w ∈ Metric.ball x δ ∩ interior s := by
          refine ⟨?_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
          rw [add_assoc, add_mem_ball_iff_norm]
          exact I.trans_lt hδ
        simpa only [mem_ofPred_eq, add_assoc x, add_sub_cancel_left] using sδ H

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
F : Type u_4
G : Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)
H₁ :
  HasStrictFDerivAt (fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (ContinuousMultilinearMap.compContinuousLinearMapL (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G) +
      (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).1.fderivCompContinuousLinearMap
          (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G))
    (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2)
H₂ : HasStrictFDerivAt (⇑(toContinuousMultilinearMapCLM 𝕜)) (toContinuousMultilinearMapCLM 𝕜) fg.1
⊢ HasStrictFDerivAt (toContinuousMultilinearMap ∘ fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (toContinuousMultilinearMapCLM 𝕜 ∘SL
      (compContinuousLinearMapCLM fg.2 ∘SL ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G) +
        fg.1.fderivCompContinuousLinearMap fg.2 ∘SL ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)))
    fg

Tactic:
have H₃ := hasStrictFDerivAt_pi.mpr fun i : ι ↦ hasStrictFDerivAt_id (𝕜 := 𝕜) fg.2

Goal state after:
𝕜 : Type u_1
ι : Type u_2
F : Type u_4
G : Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)
H₁ :
  HasStrictFDerivAt (fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (ContinuousMultilinearMap.compContinuousLinearMapL (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G) +
      (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).1.fderivCompContinuousLinearMap
          (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2).2 ∘SL
        ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 (fun x ↦ G) H) (ι → F →L[𝕜] G))
    (fg.1.toContinuousMultilinearMap, fun x ↦ fg.2)
H₂ : HasStrictFDerivAt (⇑(toContinuousMultilinearMapCLM 𝕜)) (toContinuousMultilinearMapCLM 𝕜) fg.1
H₃ : HasStrictFDerivAt (fun x i ↦ id x) (ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 (F →L[𝕜] G)) fg.2
⊢ HasStrictFDerivAt (toContinuousMultilinearMap ∘ fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (toContinuousMultilinearMapCLM 𝕜 ∘SL
      (compContinuousLinearMapCLM fg.2 ∘SL ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G) +
        fg.1.fderivCompContinuousLinearMap fg.2 ∘SL ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G)))
    fg
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : α → Set ℝ := fun x ↦ {r | 0 < r ∧ closedBall x r ∈ f x}
⊢ ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty

Tactic:
intro x xs δ δpos

Goal state after:
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : α → Set ℝ := fun x ↦ {r | 0 < r ∧ closedBall x r ∈ f x}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
⊢ (g x ∩ Ioo 0 δ).Nonempty
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : StrongDual ℝ E
x : E
t : ℝ
ht : 0 < t
h : HasFDerivAt (fun x ↦ ‖x‖) f x
⊢ HasFDerivAt (fun x ↦ ‖x‖) f (t • x)

Tactic:
by
  simpa [ht] using h.hasFDerivAt_norm_smul ht.ne'

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
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ lineDeriv ℝ (fun x ↦ ‖x‖) x x = ‖x‖

Tactic:
lineDeriv

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ deriv (fun t ↦ ‖x + t • x‖) 0 = ‖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
x : F
frechet : StrongDual 𝕜 F
s : Set F
⊢ HasFDerivWithinAt f frechet s x ↔ HasGradientWithinAt f ((toDual 𝕜 F).symm frechet) s x

Tactic:
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, (toDual 𝕜 F).apply_symm_apply frechet]

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
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
this : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖
| fun t ↦ |1 + t|

Tactic:
change _root_.abs ∘ (fun t ↦ 1 + t)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
this : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖
| _root_.abs ∘ fun t ↦ 1 + t
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
g f : 𝕜 → 𝕜
x : 𝕜
n : ℕ∞ω
i : ℕ
hg : ContDiffAt 𝕜 n g (f x)
hf : ContDiffAt 𝕜 n f x
hi : ↑i ≤ n
⊢ iteratedDeriv i (g ∘ f) x = ∑ c, iteratedDeriv c.length g (f x) * ∏ j, iteratedDeriv (c.partSize j) f x

Tactic:
iteratedDeriv_scomp_eq_sum_orderedFinpartition hg hf hi

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
g f : 𝕜 → 𝕜
x : 𝕜
n : ℕ∞ω
i : ℕ
hg : ContDiffAt 𝕜 n g (f x)
hf : ContDiffAt 𝕜 n f x
hi : ↑i ≤ n
⊢ ∑ c, (∏ j, iteratedDeriv (c.partSize j) f x) • iteratedDeriv c.length g (f x) =
    ∑ c, iteratedDeriv c.length g (f x) * ∏ j, iteratedDeriv (c.partSize j) f x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
g f : 𝕜 → 𝕜
s t : Set 𝕜
x : 𝕜
hg : ContDiffWithinAt 𝕜 2 g t (f x)
hf : ContDiffWithinAt 𝕜 2 f s x
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
⊢ iteratedDerivWithin 2 (g ∘ f) s x =
    iteratedDerivWithin 2 g t (f x) * derivWithin f s x ^ 2 + derivWithin g t (f x) * iteratedDerivWithin 2 f s x

Tactic:
rw [iteratedDerivWithin_scomp_two hg hf ht hs hx hst]
  simp only [smul_eq_mul, mul_comm]

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E → F
φ : E → ℝ
x₀ : E
f' : E →L[ℝ] F
φ' : StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀
Λ' : F × ℝ →ₗ[ℝ] ℝ
h0 : Λ' ≠ 0
hΛ' : (↑(f'.prod φ')).range ≤ Λ'.ker
e : ((F →ₗ[ℝ] ℝ) × ℝ) ≃ₗ[ℝ] F × ℝ →ₗ[ℝ] ℝ :=
  (LinearEquiv.refl ℝ (F →ₗ[ℝ] ℝ)).prodCongr (LinearMap.ringLmapEquivSelf ℝ ℝ ℝ).symm ≪≫ₗ LinearMap.coprodEquiv ℝ
⊢ ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∀ (x : E), Λ (f' x) + Λ₀ • φ' x = 0

Tactic:
rcases e.surjective Λ' with ⟨⟨Λ, Λ₀⟩, rfl⟩

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E → F
φ : E → ℝ
x₀ : E
f' : E →L[ℝ] F
φ' : StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀
e : ((F →ₗ[ℝ] ℝ) × ℝ) ≃ₗ[ℝ] F × ℝ →ₗ[ℝ] ℝ :=
  (LinearEquiv.refl ℝ (F →ₗ[ℝ] ℝ)).prodCongr (LinearMap.ringLmapEquivSelf ℝ ℝ ℝ).symm ≪≫ₗ LinearMap.coprodEquiv ℝ
Λ : F →ₗ[ℝ] ℝ
Λ₀ : ℝ
h0 : e (Λ, Λ₀) ≠ 0
hΛ' : (↑(f'.prod φ')).range ≤ (e (Λ, Λ₀)).ker
⊢ ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∀ (x : E), Λ (f' x) + Λ₀ • φ' x = 0
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 + ({f x} + closedBall (r • 0) (‖r‖ * ε))

Tactic:
smul_zero,

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) = r • ⇑A '' closedBall 0 1 + ({f x} + closedBall 0 (‖r‖ * ε))