mathlib-initiative/mathlib-tactics huggingface dataset:
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
hδ2 : δ ≤ 1
⊢ ∃ c', (∀ (n : Fin N.succ), ‖c' n‖ ≤ 2) ∧ Pairwise fun i j ↦ 1 - δ ≤ ‖c' i - c' j‖

Tactic:
by
  let c' : Fin N.succ → E := fun i => if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
  have norm_c'_le : ∀ i, ‖c' i‖ ≤ 2 := by
    intro i
    simp only [c']
    split_ifs with h; · exact h
    by_cases hi : ‖a.c i‖ = 0 <;> simp [norm_smul, hi]
  refine ⟨c', fun n => norm_c'_le n, fun i j inej => ?_⟩
  -- up to exchanging `i` and `j`, one can assume `‖c i‖ ≤ ‖c j‖`.
  wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j
  · rw [norm_sub_rev]; exact this j i inej.symm (le_of_not_ge hij)
  rcases le_or_gt ‖a.c j‖ 2 with (Hj | Hj)
  -- case `‖c j‖ ≤ 2` (and therefore also `‖c i‖ ≤ 2`)
  · simp_rw [c', Hj, hij.trans Hj, if_true]
    exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
  -- case `2 < ‖c j‖`
  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false] using Hj
    rcases le_or_gt ‖a.c i‖ 2 with (Hi | Hi)
    · -- case `‖c i‖ ≤ 2`
      simp_rw [c', Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
    · -- case `2 < ‖c i‖`
      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false] using Hi
      simp_rw [c', H'i, if_false, H'j, if_false]
      exact exists_normalized_aux3 a lastc lastr hτ δ hδ1 i j inej Hi hij

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

Goal state before:
pp: case inr
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✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B.Nonempty
r✝ : ℝ := infDist x B
this : 0 < min r✝ 1
r : ℝ
hr : r ∈ f x
h'r : r ∈ Ioo 0 (min r✝ 1)
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

Tactic:
refine ⟨r, ⟨hr, ⟨h'r.1, h'r.2.trans_le (min_le_right _ _)⟩⟩, ?_⟩

Goal state after:
case inr
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✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B.Nonempty
r✝ : ℝ := infDist x B
this : 0 < min r✝ 1
r : ℝ
hr : r ∈ f x
h'r : r ∈ Ioo 0 (min r✝ 1)
⊢ Disjoint B (closedBall x r)
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
⊢ ∀ ⦃a b : ℝ≥0⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • r + b • s ∈ {r | HolderOnWith C r f A}

Tactic:
_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝ : ℝ≥0
⊢ ∀ ⦃b : ℝ≥0⦄, 0 ≤ a✝ → 0 ≤ b → a✝ + b = 1 → a✝ • r + b • s ∈ {r | HolderOnWith C r f A}
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝ : ℝ≥0
⊢ ∀ ⦃b : ℝ≥0⦄, 0 ≤ a✝ → 0 ≤ b → a✝ + b = 1 → a✝ • r + b • s ∈ {r | HolderOnWith C r f A}

Tactic:
_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝ b✝ : ℝ≥0
⊢ 0 ≤ a✝ → 0 ≤ b✝ → a✝ + b✝ = 1 → a✝ • r + b✝ • s ∈ {r | HolderOnWith C r f A}
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝ b✝ : ℝ≥0
⊢ 0 ≤ a✝ → 0 ≤ b✝ → a✝ + b✝ = 1 → a✝ • r + b✝ • s ∈ {r | HolderOnWith C r f A}

Tactic:
_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝¹ b✝ : ℝ≥0
a✝ : 0 ≤ a✝¹
⊢ 0 ≤ b✝ → a✝¹ + b✝ = 1 → a✝¹ • r + b✝ • s ∈ {r | HolderOnWith C r f A}
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝¹ b✝ : ℝ≥0
a✝ : 0 ≤ a✝¹
⊢ 0 ≤ b✝ → a✝¹ + b✝ = 1 → a✝¹ • r + b✝ • s ∈ {r | HolderOnWith C r f A}

Tactic:
_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
A : Set X
r : ℝ≥0
hr : r ∈ {r | HolderOnWith C r f A}
s : ℝ≥0
hs : s ∈ {r | HolderOnWith C r f A}
a✝² b✝ : ℝ≥0
a✝¹ : 0 ≤ a✝²
a✝ : 0 ≤ b✝
⊢ a✝² + b✝ = 1 → a✝² • r + b✝ • s ∈ {r | HolderOnWith C r f A}
Module: Mathlib.Analysis.Calculus.Darboux

Goal state before:
pp: case inr.inl
a b : ℝ
f f' : ℝ → ℝ
hab : a ≤ b
hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x
m : ℝ
hma : f' a < m
hmb : m < f' b
hab' : a < b
g : ℝ → ℝ := fun x ↦ f x - m * x
hg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x
cmem : b ∈ Icc a b
hc : IsMinOn g (Icc a b) b
hac : a < b
⊢ b ∈ Ioo a b

Tactic:
refine absurd (sub_nonpos.1 <| nonpos_of_mul_nonneg_right ?_ (sub_lt_zero.2 hab'))
        (not_le_of_gt hmb)

Goal state after:
case inr.inl
a b : ℝ
f f' : ℝ → ℝ
hab : a ≤ b
hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x
m : ℝ
hma : f' a < m
hmb : m < f' b
hab' : a < b
g : ℝ → ℝ := fun x ↦ f x - m * x
hg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x
cmem : b ∈ Icc a b
hc : IsMinOn g (Icc a b) b
hac : a < b
⊢ 0 ≤ (a - b) * (f' b - m)
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_1
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✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : α → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : ↑v ⊆ s'
hμv : μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s'
hv : (↑v).PairwiseDisjoint fun x ↦ closedBall x (r x)
⊢ (↑(t ∪ Finset.image (fun x ↦ (x, r x)) v)).PairwiseDisjoint fun p ↦ closedBall p.1 p.2

Tactic:
simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and, Finset.coe_image]
      constructor
      · intro p hp q hq hpq
        rcases (mem_image _ _ _).1 hp with ⟨p', p'v, rfl⟩
        rcases (mem_image _ _ _).1 hq with ⟨q', q'v, rfl⟩
        refine hv p'v q'v fun hp'q' => ?_
        rw [hp'q'] at hpq
        exact hpq rfl
      · intro p hp q hq hpq
        rcases (mem_image _ _ _).1 hq with ⟨q', q'v, rfl⟩
        apply disjoint_of_subset_left _ (hr q' (vs' q'v)).2
        rw [hB, ← Finset.set_biUnion_coe]
        exact subset_biUnion_of_mem (u := fun x : α × ℝ => closedBall x.1 x.2) hp

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
f : 𝕜 → F
f' : F
inst✝ : TrivialStar 𝕜
L : Filter (𝕜 × 𝕜)
h : HasDerivAtFilter f f' L
⊢ HasDerivAtFilter (fun x ↦ star (f x)) (star f') L

Tactic:
simpa using h.hasFDerivAtFilter.star.hasDerivAtFilter

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

Goal state before:
pp: f : ℝ → ℝ
b : ℝ
h : ContinuousAt f b
hd₀ : ∀ᶠ (x : ℝ) in 𝓝[<] b, DifferentiableAt ℝ f x
hd₁ : ∀ᶠ (x : ℝ) in 𝓝[>] b, DifferentiableAt ℝ f x
h₀ : ∀ᶠ (x : ℝ) in 𝓝[<] b, deriv f x ≤ 0
h₁ : ∀ᶠ (x : ℝ) in 𝓝[>] b, deriv f x ≥ 0
⊢ IsLocalMin f b

Tactic:
obtain ⟨a, ha⟩ := (nhdsLT_basis b).eventually_iff.mp <| hd₀.and h₀
  obtain ⟨c, hc⟩ := (nhdsGT_basis b).eventually_iff.mp <| hd₁.and h₁
  exact isLocalMin_of_deriv_Ioo ha.1 hc.1 h
    (fun _ hx => (ha.2 hx).1.differentiableWithinAt)
    (fun _ hx => (hc.2 hx).1.differentiableWithinAt)
    (fun _ hx => (ha.2 hx).2) (fun x hx => (hc.2 hx).2)

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:
by
  filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)
  rwa [← sub_neg, ← sign_eq_neg_one_iff, ← hx', sign_eq_neg_one_iff] at hx

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
⊢ ∑ i ∈ Finset.range (n + 1), ↑n ! * 1 * C * D ^ (n + 1) * (↑(n - i)!)⁻¹ ≤
    ∑ i ∈ Finset.range (n + 1), ↑n ! * 1 * C * D ^ (n + 1) * 1

Tactic:
gcongr with i
      apply inv_le_one_of_one_le₀
      simpa only [Nat.one_le_cast] using! (n - i).factorial_pos

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

Goal state before:
pp: case refine_2
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
i : Fin N
x : { x // x ∈ s' }
⊢ x ∈ S i → x ∈ Subtype.val ⁻¹' s

Tactic:
_

Goal state after:
case refine_2
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
i : Fin N
x : { x // x ∈ s' }
a✝ : x ∈ S i
⊢ x ∈ Subtype.val ⁻¹' s
Module: Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap

Goal state before:
pp: case intro.mp
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
G : Type u_5
H : Type u_6
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
f : E → G [⋀^ι]→L[𝕜] H
f' : E →L[𝕜] G [⋀^ι]→L[𝕜] H
x : E
inst✝ : Finite ι
val✝ : Fintype ι
h : HasStrictFDerivAt (toContinuousMultilinearMap ∘ f) (toContinuousMultilinearMapCLM 𝕜 ∘SL f') x
⊢ HasStrictFDerivAt f f' x

Tactic:
hasStrictFDerivAt_iff_isLittleOTVS

Goal state after:
case intro.mp
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
G : Type u_5
H : Type u_6
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
f : E → G [⋀^ι]→L[𝕜] H
f' : E →L[𝕜] G [⋀^ι]→L[𝕜] H
x : E
inst✝ : Finite ι
val✝ : Fintype ι
h :
  (fun p ↦
      (toContinuousMultilinearMap ∘ f) p.1 - (toContinuousMultilinearMap ∘ f) p.2 -
        (toContinuousMultilinearMapCLM 𝕜 ∘SL f') (p.1 - p.2)) =o[𝕜;
    𝓝 (x, x)] fun p ↦ p.1 - p.2
⊢ (fun p ↦ f p.1 - f p.2 - f' (p.1 - p.2)) =o[𝕜; 𝓝 (x, x)] fun p ↦ p.1 - p.2
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)
⊢ HasStrictFDerivAt (fun fg ↦ fg.1.compContinuousLinearMap fg.2)
    (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:
rw [← hasStrictFDerivAt_toContinuousMultilinearMap_comp_iff]
  have H₁ := ContinuousMultilinearMap.hasStrictFDerivAt_compContinuousLinearMap
    (fg.1.1, fun _ : ι ↦ fg.2)
  have H₂ := ((toContinuousMultilinearMapCLM 𝕜).hasStrictFDerivAt (x := fg.1))
  have H₃ := hasStrictFDerivAt_pi.mpr fun i : ι ↦ hasStrictFDerivAt_id (𝕜 := 𝕜) fg.2
  exact H₁.comp fg (H₂.prodMap fg H₃)

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
U : Set E := {y | ∃ N, ↑N = fderivWithin 𝕜 f s y}
hU : U ∈ 𝓝[s] x
N : E → E ≃L[𝕜] F := fun x ↦ if h : x ∈ U then Exists.choose h else M
eN : (fun y ↦ ↑(N y)) =ᶠ[𝓝[s] x] fun y ↦ fderivWithin 𝕜 f s y
e'N : ↑(N x) = fderivWithin 𝕜 f s x
hN : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x
hN' : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x
v : E
A' : ∀ (y : E), ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm = ContinuousLinearMap.id 𝕜 F
this : (fderivWithin 𝕜 (fun y ↦ ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm) s x) v = 0
⊢ (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:
have I : (N x : E →L[𝕜] F) ∘L (fderivWithin 𝕜 (fun y ↦ ((N y).symm : F →L[𝕜] E)) s x v) =
      - (fderivWithin 𝕜 (fun y ↦ (N y : E →L[𝕜] F)) s x v) ∘L ((N x).symm : F →L[𝕜] E) := by
    rw [ContinuousLinearMap.fderivWithin_of_bilinear _ (hN.differentiableWithinAt one_ne_zero)
      (hN'.differentiableWithinAt one_ne_zero) (hs x hx)] at this
    simpa [eq_neg_iff_add_eq_zero] using this

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}
hU : U ∈ 𝓝[s] x
N : E → E ≃L[𝕜] F := fun x ↦ if h : x ∈ U then Exists.choose h else M
eN : (fun y ↦ ↑(N y)) =ᶠ[𝓝[s] x] fun y ↦ fderivWithin 𝕜 f s y
e'N : ↑(N x) = fderivWithin 𝕜 f s x
hN : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y)) s x
hN' : ContDiffWithinAt 𝕜 1 (fun y ↦ ↑(N y).symm) s x
v : E
A' : ∀ (y : E), ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm = ContinuousLinearMap.id 𝕜 F
this : (fderivWithin 𝕜 (fun y ↦ ((ContinuousLinearMap.compL 𝕜 F E F) ↑(N y)) ↑(N y).symm) s x) v = 0
I : ↑(N x) ∘SL (fderivWithin 𝕜 (fun y ↦ ↑(N y).symm) s x) v = -(fderivWithin 𝕜 (fun y ↦ ↑(N y)) s x) v ∘SL ↑(N x).symm
⊢ (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.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: 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
A : ∑' (x : ↑t0), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε / 2
i : Fin N
x : α
xs' : x ∈ s'
⊢ ⟨x, xs'⟩ ∈ S i → closedBall x (r1 x) ⊆ v

Tactic:
_

Goal state after:
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
A : ∑' (x : ↑t0), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε / 2
i : Fin N
x : α
xs' : x ∈ s'
h✝ : ⟨x, xs'⟩ ∈ S i
⊢ closedBall x (r1 x) ⊆ v
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 : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₀ f₁ : E → F
f' : F
s : Set E
x v : E
h : f₀ =ᶠ[𝓝[s] x] f₁
hx : f₀ x = f₁ x
⊢ HasLineDerivWithinAt 𝕜 f₀ f' s x v ↔ HasLineDerivWithinAt 𝕜 f₁ f' s x v

Tactic:
apply hasDerivWithinAt_iff

Goal state after:
case h₁
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 : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₀ f₁ : E → F
f' : F
s : Set E
x v : E
h : f₀ =ᶠ[𝓝[s] x] f₁
hx : f₀ x = f₁ x
⊢ (fun t ↦ f₀ (x + t • v)) =ᶠ[𝓝[(fun t ↦ x + t • v) ⁻¹' s] 0] fun t ↦ f₁ (x + t • v)

case hx
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 : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₀ f₁ : E → F
f' : F
s : Set E
x v : E
h : f₀ =ᶠ[𝓝[s] x] f₁
hx : f₀ x = f₁ x
⊢ f₀ (x + 0 • v) = f₁ (x + 0 • v)