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

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
hR : 1 < R
right✝ : (R, x).2 ∈ univ
A : 0 < (R - 1) / (R + 1)
⊢ (R - 1) / (R + 1) < 1

Tactic:
apply (div_lt_one _).2 <;> linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
hR : 1 < R
A : 0 < (R + 1) / 2
⊢ (R - 1) / (R + 1) < 1

Tactic:
apply (div_lt_one _).2 <;> linarith

Goal state after:
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
x : β
hx : x ∈ s i
y : β
hy : y ∈ s i
x_ne_y : x ≠ y
⊢ ∃ jx < p.lastStep, p.color jx = ↑i ∧ x = p.index jx

Tactic:
by
      simpa only [s, exists_prop, mem_iUnion, mem_singleton_iff] using hx

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case neg
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
h'm : ¬m.Countable
H : ∀ (ρ : Measure G), ρ.IsEverywherePos s → ρ s = ∞
Hν : ν.IsEverywherePos s
Hμ' : μ'.IsEverywherePos s
⊢ μ' s = ν s

Tactic:
H ν Hν,

Goal state after:
case neg
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
h'm : ¬m.Countable
H : ∀ (ρ : Measure G), ρ.IsEverywherePos s → ρ s = ∞
Hν : ν.IsEverywherePos s
Hμ' : μ'.IsEverywherePos s
⊢ μ' s = ∞
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:

Tactic:
μ o = 1 / (N + 1) * μ s + N / (N + 1) * μ s := by
        rw [μo, ← add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul] <;> simp
      _ ≤ μ ((⋃ x ∈ w, closedBall (↑x) (r ↑x)) ∩ o) + N / (N + 1) * μ s := by
        gcongr
        rw [one_div, mul_comm, ← div_eq_mul_inv]
        apply hw.le.trans (le_of_eq _)
        rw [← Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion₂, Finset.set_biUnion_coe,
          measure_biUnion_finset]
        · have : (w : Set (u i)).PairwiseDisjoint
              fun b : u i => closedBall (b : α) (r (b : α)) := by
            intro k _ l _ hkl; exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)
          exact this.mono fun k => inter_subset_right
        · intro b _
          apply omeas.inter measurableSet_closedBall

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : ∀ᶠ (x : E) in 𝓝[s] a, (restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s) x = iteratedFDerivWithin 𝕜 n f s x
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
⊢ a ∈ s → (restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' (n + 1) f s) a = iteratedFDerivWithin 𝕜 (n + 1) f s a

Tactic:
h₄a

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : ∀ᶠ (x : E) in 𝓝[s] a, (restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s) x = iteratedFDerivWithin 𝕜 n f s x
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
h₄a : a ∈ s
⊢ (restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' (n + 1) f s) a = iteratedFDerivWithin 𝕜 (n + 1) f s a
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case inl
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 = ∅
r : ℝ
hr : r ∈ f x
h'r : r ∈ Ioo 0 1
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

Tactic:
exact ⟨r, ⟨hr, h'r⟩, by simp only [hB, empty_disjoint]⟩

Goal state after:
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
C D : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑(C * D ^ ↑r) * edist x y ^ ↑0

Tactic:
simp only [NNReal.coe_zero, ENNReal.rpow_zero, mul_one]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑(C * D ^ ↑r)
Module: Mathlib.Analysis.Calculus.Darboux

Goal state before:
pp: 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
c : ℝ
cmem : c ∈ Icc a b
hc : IsMinOn g (Icc a b) c
cmem' : c ∈ Ioo a b
⊢ Icc a b ∈ 𝓝 c

Tactic:
← mem_interior_iff_mem_nhds,

Goal state after:
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
c : ℝ
cmem : c ∈ Icc a b
hc : IsMinOn g (Icc a b) c
cmem' : c ∈ Ioo a b
⊢ c ∈ interior (Icc a b)
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: case pos
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
x : 𝕜
inst✝ : TrivialStar 𝕜
s : Set 𝕜
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ star (f y)) s x = star (derivWithin f s x)

Tactic:
exact DFunLike.congr_fun (fderivWithin_star hxs) _

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

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] x₀ ⊔ pure x₀, sign (f x) = sign (x - x₀)

Tactic:
eventually_sup

Goal state after:
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
⊢ (∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (f x) = sign (x - x₀)) ∧ ∀ᶠ (x : ℝ) in pure x₀, sign (f x) = sign (x - x₀)
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 Gu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
f : E → Fu
n : ℕ
s : Set E
t : Set Fu
x : E
hg : ContDiffOn 𝕜 (↑n) g t
hf : ContDiffOn 𝕜 (↑n) f s
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n

Tactic:
induction n using Nat.case_strong_induction_on generalizing Gu with

Goal state after:
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 t : Set E
f : E → F
x : E
h : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
this : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜 2 f t y
⊢ ∀ᶠ (y : E) in 𝓝[s] x, fderivWithin 𝕜 f s y = fderivWithin 𝕜 f t y

Tactic:
filter_upwards [self_mem_nhdsWithin, this, eventually_eventually_nhdsWithin.2 h]
      with y hy h'y h''y

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 t : Set E
f : E → F
x : E
h : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
this : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜 2 f t y
y : E
hy : y ∈ s
h'y : ContDiffWithinAt 𝕜 2 f t y
h''y : ∀ᶠ (x : E) in 𝓝[s] y, t x
⊢ fderivWithin 𝕜 f s y = fderivWithin 𝕜 f t y
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 t : Set E
f : E → F
x : E
h : IsSymmSndFDerivWithinAt 𝕜 f t x
hst : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
⊢ IsSymmSndFDerivWithinAt 𝕜 f s x

Tactic:
by
  intro v w
  rw [fderivWithin_fderivWithin_eq_of_mem_nhdsWithin hst hf hs ht hx]
  exact h v w

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
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F →L[𝕜] G
c : F
s : Set E
x : E
N : ℕ∞ω
n : ℕ
hf : ContDiffWithinAt 𝕜 N f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (f y) c) s x‖ ≤ ‖c‖ * ‖iteratedFDerivWithin 𝕜 n f s x‖

Tactic:
apply ((ContinuousLinearMap.apply 𝕜 G c).norm_iteratedFDerivWithin_comp_left hf hs hx hn).trans
  gcongr
  refine (ContinuousLinearMap.apply 𝕜 G c).opNorm_le_bound (norm_nonneg _) fun f => ?_
  rw [ContinuousLinearMap.apply_apply, mul_comm]
  exact f.le_opNorm c

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 : E → E
s : Set E
x : E
h : s ∈ 𝓝 x
⊢ lieBracketWithin 𝕜 V W s x = lieBracketWithin 𝕜 V W univ x

Tactic:
← univ_inter s,

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 : E → E
s : Set E
x : E
h : s ∈ 𝓝 x
⊢ lieBracketWithin 𝕜 V W (univ ∩ s) x = lieBracketWithin 𝕜 V W univ x
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: case pos.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h✝ : Nontrivial E
hd : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ fderiv ℝ (fun x ↦ ‖x‖) (0 • x) = ↑(SignType.sign 0) • fderiv ℝ (fun x ↦ ‖x‖) x

Tactic:
simp only [zero_smul, _root_.sign_zero, SignType.coe_zero]

Goal state after:
case pos.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h✝ : Nontrivial E
hd : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
⊢ fderiv ℝ (fun x ↦ ‖x‖) 0 = 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|
HC : IsCompact (⇑A '' closedBall 0 1)
L0 : Tendsto (fun ε ↦ μ (cthickening ε (⇑A '' closedBall 0 1))) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L1 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L2 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (d * μ (closedBall 0 1)))
⊢ ∃ ε, μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1) ∧ 0 < ε

Tactic:
have I : d * μ (closedBall 0 1) < m * μ (closedBall 0 1) := by
      gcongr; exacts [(measure_closedBall_pos μ _ zero_lt_one).ne', measure_closedBall_lt_top.ne]

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|
HC : IsCompact (⇑A '' closedBall 0 1)
L0 : Tendsto (fun ε ↦ μ (cthickening ε (⇑A '' closedBall 0 1))) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L1 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (μ (⇑A '' closedBall 0 1)))
L2 : Tendsto (fun ε ↦ μ (closedBall 0 ε + ⇑A '' closedBall 0 1)) (𝓝[>] 0) (𝓝 (d * μ (closedBall 0 1)))
I : d * μ (closedBall 0 1) < ↑m * μ (closedBall 0 1)
⊢ ∃ ε, μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1) ∧ 0 < ε
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
φ : ℕ → Set α
hφ : AECover μ atTop φ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
lim₁ : Tendsto (fun i ↦ ∫⁻ (x : α) in ⋂ k ∈ Ici i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
⊢ Tendsto (fun x ↦ ∫⁻ (x : α) in φ x, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))

Tactic:
have lim₂ := lintegral_tendsto_of_monotone_of_nat hφ.biUnion_Iic_aecover
    (fun i j hij => biUnion_subset_biUnion_left (Iic_subset_Iic.mpr hij)) hfm

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
φ : ℕ → Set α
hφ : AECover μ atTop φ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
lim₁ : Tendsto (fun i ↦ ∫⁻ (x : α) in ⋂ k ∈ Ici i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
lim₂ : Tendsto (fun i ↦ ∫⁻ (x : α) in ⋃ k ∈ Iic i, φ k, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
⊢ Tendsto (fun x ↦ ∫⁻ (x : α) in φ x, f x ∂μ) atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
Module: Mathlib.MeasureTheory.Integral.IntegralEqImproper

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ENNReal.ofReal ‖f a‖ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
a : α
| ENNReal.ofReal ‖f a‖

Tactic:
← coe_nnnorm,

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ENNReal.ofReal ‖f a‖ ∂μ).toReal ≤ I
hfm : AEStronglyMeasurable f μ
i : ι
a : α
| ENNReal.ofReal ↑‖f a‖₊
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
⊢ Tendsto (⇑F ∘ f) atBot (𝓝 (atBot.limUnder (⇑F ∘ f)))

Tactic:
apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int

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

Goal state before:
pp: case e'_4
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
this : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖deriv f' y‖ₑ
y : ℝ
⊢ ‖deriv f y‖ₑ = ‖I (deriv f y)‖ₑ

Tactic:
simp [I]

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✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
this : finrank ℝ (E' × ℝ) = n
⊢ ∃ L, L v = (0, 1)

Tactic:
have L₀ : E ≃L[ℝ] (E' × ℝ) := (ContinuousLinearEquiv.ofFinrankEq this).symm

Goal state after:
E : Type u_1
F : Type u_2
G : Type u_3
W : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace ℝ G
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
f f' : E → F
g g' : E → G
v : E
B : F →L[ℝ] G →L[ℝ] W
hf'g : Integrable (fun x ↦ (B (f' x)) (g x)) μ
hfg' : Integrable (fun x ↦ (B (f x)) (g' x)) μ
hfg : Integrable (fun x ↦ (B (f x)) (g x)) μ
hf : ∀ x ∈ tsupport g, HasLineDerivAt ℝ f (f' x) x v
hg : ∀ x ∈ tsupport f, HasLineDerivAt ℝ g (g' x) x v
hW : CompleteSpace W
hv : v ≠ 0
this✝ : Nontrivial E
n : ℕ := finrank ℝ E
E' : Type := Fin (n - 1) → ℝ
this : finrank ℝ (E' × ℝ) = n
L₀ : E ≃L[ℝ] E' × ℝ
⊢ ∃ L, L v = (0, 1)
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), μ (f '' (s ∩ closedBall 0 R)) = 0
⊢ μ (f '' s) = 0

Tactic:
by
    rw [← nonpos_iff_eq_zero, ← iUnion_inter_closedBall_nat s 0]
    calc
      μ (f '' ⋃ n : ℕ, s ∩ closedBall 0 n) ≤ ∑' n : ℕ, μ (f '' (s ∩ closedBall 0 n)) := by
        rw [image_iUnion]; exact measure_iUnion_le _
      _ ≤ 0 := by simp only [H, tsum_zero, nonpos_iff_eq_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ
εpos : ε > 0
⊢ ∃ f, AEMeasurable f (μ.restrict s) ∧ ∀ᵐ (x : E) ∂μ.restrict s, dist (f x) (f' x) ≤ ε

Tactic:
let δ : ℝ≥0 := ⟨ε, le_of_lt εpos⟩

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ
εpos : ε > 0
δ : ℝ≥0 := ⟨ε, ⋯⟩
⊢ ∃ f, AEMeasurable f (μ.restrict s) ∧ ∀ᵐ (x : E) ∂μ.restrict s, dist (f x) (f' x) ≤ ε
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := (c.symm ≫ₕ f₁).source ∩ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
open_s : IsOpen[inst✝⁶] s
this : x ∈ s
F₁ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f₁ ≫ₕ g) ≫ₕ g.symm ≫ₕ f₂ ≫ₕ c'
A : F₁ ∈ G
F₂ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f ≫ₕ c').restr s
⊢ (c.symm ≫ₕ (f₁ ≫ₕ f₂) ≫ₕ c').restr s ≈ F₂

Tactic:
simp only [F₂, feq, _root_.refl]

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

Goal state before:
pp: H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e : OpenPartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ maximalAtlas M G
xe : x ∈ e.source
⊢ ↑(chartAt H x) x ∈ ((chartAt H x).symm ≫ₕ e).source

Tactic:
simp [xe]

Goal state after: