mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case refine_2
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ Filter.map (⇑(↑(WithAbs.congr v w (RingEquiv.refl F))).symm) (𝓝 0)
s : Set (WithAbs w)
hs : s ∈ 𝓝 0
hss : ⇑(↑(WithAbs.congr v w (RingEquiv.refl F))).symm '' s ⊆ U
⊢ U ∈ 𝓝 0

Tactic:
rw [← RingEquiv.coe_toEquiv_symm, WithAbs.congr_symm] at hss

Goal state after:
case refine_2
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ Filter.map (⇑(↑(WithAbs.congr v w (RingEquiv.refl F))).symm) (𝓝 0)
s : Set (WithAbs w)
hs : s ∈ 𝓝 0
hss : ⇑↑(WithAbs.congr w v (RingEquiv.refl F).symm) '' s ⊆ U
⊢ U ∈ 𝓝 0
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: case pos
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : u = w
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 (Pi.single w (z w) u))

Tactic:
rw [← hw, Pi.single_eq_same]

Goal state after:
case pos
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u w : ι
x✝ : w ∈ Finset.univ
hw : u = w
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a u)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v u)) (z u)))
    atTop (𝓝 (z u))
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: X : Truncated 2
C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
φ : X ⟶ (truncation 2).obj (nerve C)
x : X.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
⊢ nerve.homEquiv ((Edge.id x).map φ) =
    𝟙 (nerveEquiv ((ConcreteCategory.hom (φ.app (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 }))) x))

Tactic:
simpa using! nerve.homEquiv_id (φ.app _ x)

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : 𝕜
f : 𝕜' → F
g : 𝕜 → 𝕜'
hg : AnalyticAt 𝕜 g x
r : ℕ
hr : AnalyticAt 𝕜' (fun z ↦ (z - g x) ^ r • f z) (g x)
hg' : analyticOrderAt (fun x_1 ↦ g x_1 - g x) x = ⊤
⊢ (fun x_1 ↦ f (g x)) =ᶠ[𝓝[≠] x] f ∘ g

Tactic:
filter_upwards [nhdsWithin_le_nhds <| analyticOrderAt_eq_top.mp hg'] with z hz

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : 𝕜
f : 𝕜' → F
g : 𝕜 → 𝕜'
hg : AnalyticAt 𝕜 g x
r : ℕ
hr : AnalyticAt 𝕜' (fun z ↦ (z - g x) ^ r • f z) (g x)
hg' : analyticOrderAt (fun x_1 ↦ g x_1 - g x) x = ⊤
z : 𝕜
hz : g z - g x = 0
⊢ f (g x) = (f ∘ g) z
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → E
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
hf : MeromorphicAt (f ∘ g) x
r : 𝕜 → 𝕜 := HasStrictDerivAt.localInverse g (deriv g x) x ⋯ hg'
⊢ MeromorphicAt f (g x)

Tactic:
have hra : AnalyticAt 𝕜 r (g x) := hg.analyticAt_localInverse hg'

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → E
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
hf : MeromorphicAt (f ∘ g) x
r : 𝕜 → 𝕜 := HasStrictDerivAt.localInverse g (deriv g x) x ⋯ hg'
hra : AnalyticAt 𝕜 r (g x)
⊢ MeromorphicAt f (g x)
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: 𝕂 : Type u
inst✝⁵ : Field 𝕂
inst✝⁴ : CharZero 𝕂
𝔸 : Type v
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
a : 𝕂
n : ℕ
v : Fin n → 𝔸
⊢ (binomialSeries 𝔸 a n) v = Ring.choose a n • (List.ofFn v).prod

Tactic:
simp [binomialSeries, FormalMultilinearSeries.ofScalars]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑(n + 1) ≤ p
⊢ ContDiff ℝ ↑(n + 1) fun x ↦ x ^ p

Tactic:
have h1 : 1 ≤ p := le_trans (by simp) h
    rw [Nat.cast_add_one, ← le_sub_iff_add_le] at h
    rw [Nat.cast_add_one, contDiff_succ_iff_deriv, deriv_rpow_const' p]
    simp only [WithTop.natCast_ne_top, analyticOn_univ, IsEmpty.forall_iff, true_and]
    exact ⟨differentiable_rpow_const h1, contDiff_const.mul (ihn h)⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Completion

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F
f : α → E
g : α → F
l : Filter α
⊢ (fun x ↦ ↑(g x)) =O[l] f ↔ g =O[l] f

Tactic:
by
  simp only [isBigO_iff, norm_coe]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Completion

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F
f : α → E
g : α → F
l : Filter α
⊢ (f =O[l] fun x ↦ ↑(g x)) ↔ f =O[l] g

Tactic:
by
  simp only [isBigO_iff, norm_coe]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ERealExp

Goal state before:
pp: case bot
y : EReal
h : ⊥ < y
⊢ 0 < y.exp

Tactic:
pos_iff_ne_zero,

Goal state after:
case bot
y : EReal
h : ⊥ < y
⊢ y.exp ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case convert_2
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : 1 < b
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
all_goals simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case convert_2
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : b < 1
⊢ ⊤ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
all_goals simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case convert_2
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : 1 < b
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
all_goals simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case convert_2
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊥

pp: case convert_3
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ ≠ 0 ∨ b.log ≠ ⊤

Tactic:
all_goals simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ ⊤ = ↑y * ⊤

Tactic:
EReal.mul_top_of_pos

Goal state after:
case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ ⊤ = ⊤

case inr.inr.inr.inl
y : ℝ
y_pos : 0 < y
⊢ 0 < ↑y
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f g : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)
hfg : abs ∘ g ≤ᶠ[l] abs ∘ f
z : ℕ
x : α
hx : (abs ∘ g) x ≤ (abs ∘ f) x
⊢ |k x ^ z| * |g x| ≤ |k x ^ z| * |f x|

Tactic:
gcongr _ * ?_

Goal state after:
case hbc
α : Type u_1
β : Type u_2
l : Filter α
k f g : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)
hfg : abs ∘ g ≤ᶠ[l] abs ∘ f
z : ℕ
x : α
hx : (abs ∘ g) x ≤ (abs ∘ f) x
⊢ |g x| ≤ |f x|
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
⊢ (∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)) ↔ ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z

Tactic:
refine ⟨fun h z => ?_, fun h z => ?_⟩

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
z : ℤ
⊢ f =O[l] fun a ↦ k a ^ z

case refine_2
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
h : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z
z : ℤ
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inr
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m ≠ ⊤
⊢ 0 ≤ linearGrowthInf u

Tactic:
rw [← linearGrowthInf_const hm hm']

Goal state after:
case inr
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m ≠ ⊤
⊢ (linearGrowthInf fun x ↦ u m) ≤ linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
u : ι → ℕ → ℝ≥0∞
⊢ expGrowthInf (⨅ i, u i) = ⨅ i, expGrowthInf (u i)

Tactic:
by
  rw [← iInf_univ, expGrowthInf_biInf u Set.finite_univ, iInf_univ]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
ε : ℝ≥0
⊢ ∀ (N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0

Tactic:
N

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
ε N : ℝ≥0
⊢ ∀ (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ 0 < μ ((support fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) ∩ s)

Tactic:
pos_iff_ne_zero,

Goal state after:
case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ μ ((support fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) ∩ s) ≠ 0
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t ∩ sᶜ) ≤ (ν' + ν + ⊤) (t ∩ sᶜ)

Tactic:
have h_mu_le_add_top : μ ≤ ν' + ν + ⊤ := by simp only [add_top, le_top]

Goal state after:
case a.refine_1.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
h_mu_le_add_top : μ ≤ ν' + ν + ⊤
⊢ μ (t ∩ sᶜ) ≤ (ν' + ν + ⊤) (t ∩ sᶜ)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ✝ ν : Measure α
ι : Type u_2
inst✝¹ : Countable ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν
⊢ Measurable (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).2

Tactic:
dsimp only

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ✝ ν : Measure α
ι : Type u_2
inst✝¹ : Countable ι
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν
⊢ Measurable (∑' (i : ι), (μ i).rnDeriv ν)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := ⋯
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := ⋯
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋯
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
z_notmem_k : z ∉ k
⊢ z ∈ ball x (R x) \ k

Tactic:
exact (mem_sdiff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ (↑ε)⁻¹ * (ε • μ) (s ∩ o) ≤ (↑ε)⁻¹ * ρ (s ∩ o)

Tactic:
gcongr _ * ?_

Goal state after:
case h₂
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ (ε • μ) (s ∩ o) ≤ ρ (s ∩ o)
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: case h2
f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[<] x, x_1 + 1 * (x_1 - x) ^ 2 ∈ Iio x

Tactic:
· filter_upwards [Ioo_mem_nhdsLT <| show x - 1 < x by simp]
      rintro y ⟨hy : x - 1 < y, h'y : y < x⟩
      rw [mem_Iio]
      nlinarith

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

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
hx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y ↦ (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y ↦ (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hy : y < x
⊢ leftLim (↑f) y - ↑f x ≤ ↑f y - ↑f x

Tactic:
simpa only [sub_le_sub_iff_right] using f.mono.leftLim_le (le_refl y)

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

Goal state before:
pp: f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : ¬¬ContinuousAt f x
⊢ ↑hf.stieltjesFunction x = f x

Tactic:
rw [Classical.not_not, hf.continuousAt_iff_leftLim_eq_rightLim] at h'x

Goal state after:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ ↑hf.stieltjesFunction x = f x
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
b : α
bs : b ∈ s
a : α
as : a ∈ s
c : α
cs : c ∈ s
ac : a ≤ c
⊢ (fun a ↦ variationOnFromTo f s a b) c ≤ (fun a ↦ variationOnFromTo f s a b) a

Tactic:
dsimp only

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : LocallyBoundedVariationOn f s
b : α
bs : b ∈ s
a : α
as : a ∈ s
c : α
cs : c ∈ s
ac : a ≤ c
⊢ variationOnFromTo f s c b ≤ variationOnFromTo f s a b
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ∞
⊢ ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ

Tactic:
refine rnDeriv_smul_right' ν μ ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ∞
⊢ r.toNNReal ≠ 0
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
x : α
x✝ : x ∈ v.limRatioMeas hρ ⁻¹' {0}
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : μ o < ∞
s : Set α := ⋯
μs : μ s ≠ ∞
A : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s
B : Tendsto (fun q ↦ ↑q * μ s) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, ρ s ≤ ↑c * μ s

Tactic:
filter_upwards [self_mem_nhdsWithin] using A

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
I : Integrable (AEStronglyMeasurable.mk f ⋯) μ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
filter_upwards [v.ae_tendsto_lintegral_enorm_sub_div'_of_integrable I hf.1.stronglyMeasurable_mk,
    hf.1.ae_eq_mk] with x hx h'x

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
I : Integrable (AEStronglyMeasurable.mk f ⋯) μ
x : α
hx :
  Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖AEStronglyMeasurable.mk f ⋯ y - AEStronglyMeasurable.mk f ⋯ x‖ₑ ∂μ) / μ a)
    (v.filterAt x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f ⋯ x
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
h : L.NeBot
ε : ℝ≥0∞
εpos : ε > 0
y : α
y_mem : y ∈ s
hy : eVariationOn f (s ∩ Ici y) < ε
⊢ ∃ t ∈ map f L, ∀ x ∈ t, ∀ y ∈ t, edist x y < ε

Tactic:
refine ⟨f '' (s ∩ Ici y), ?_, ?_⟩

Goal state after:
case refine_1
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
h : L.NeBot
ε : ℝ≥0∞
εpos : ε > 0
y : α
y_mem : y ∈ s
hy : eVariationOn f (s ∩ Ici y) < ε
⊢ f '' (s ∩ Ici y) ∈ map f L

case refine_2
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
L : Filter α
hL : ∀ y ∈ s, s ∩ Ici y ∈ L
x₀ : α
hx₀ : x₀ ∈ s
h : L.NeBot
ε : ℝ≥0∞
εpos : ε > 0
y : α
y_mem : y ∈ s
hy : eVariationOn f (s ∩ Ici y) < ε
⊢ ∀ x ∈ f '' (s ∩ Ici y), ∀ y_1 ∈ f '' (s ∩ Ici y), edist x y_1 < ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
f_zero : f 0 = 0
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)

Tactic:
· rw [cfcₙ_apply f a, mkD_of_continuousOn f_cont f_zero]

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ cfcₙ f a = (cfcₙHom ha) (mkD ((σₙ R a).domRestrict f) 0)
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: n ν : ℕ
x : ↑I
⊢ 0 ≤ (bernstein n ν) x

Tactic:
simp only [bernstein_apply]
  have h₁ : (0 : ℝ) ≤ x := by unit_interval
  have h₂ : (0 : ℝ) ≤ 1 - x := by unit_interval
  positivity

Goal state after: