mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n : ℕ
⊢ multichoose (-↑n) (n + 1) = 0

Tactic:
rw [← nsmul_right_inj (Nat.factorial_ne_zero _),
    factorial_nsmul_multichoose_eq_ascPochhammer, smeval_ascPochhammer_succ_neg, smul_zero]

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

Goal state before:
pp: case inl
⊢ (fun x ↦ 0) =O[atTop] fun x ↦ x ^ (re 0 - 1)

Tactic:
Asymptotics.isBigO_zero

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : E [×n]→L[𝕜] F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
⊢ (∑ x_1, f fun j ↦ if h : ∃ y, x_1 y = j then g (v (x_1.toEquivRange.symm ⟨j, ⋯⟩)) j else g x j) =
    ∑ i, f fun i_1 ↦ v (((Equiv.inv (Perm (Fin n))) ((embeddingEquivOfFinite (Fin n)) i)) i_1)

Tactic:
congr with σ

Goal state after:
case e_f
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : E [×n]→L[𝕜] F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i ↦ ContinuousLinearMap.id 𝕜 E
σ : Fin n ↪ Fin n
⊢ (f fun j ↦ if h : ∃ y, σ y = j then g (v (σ.toEquivRange.symm ⟨j, ⋯⟩)) j else g x j) =
    f fun i ↦ v (((Equiv.inv (Perm (Fin n))) ((embeddingEquivOfFinite (Fin n)) σ)) i)
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case e'_10
a : ℕ
z : ℂ
hz : z ≠ 0
this✝ :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1
H : 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ
this :
  HasFPowerSeriesOnBall (fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹)
    ((FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)) 0 ‖z‖ₑ
⊢ (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) =
    (z ^ (a + 1))⁻¹ •
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)).compContinuousLinearMap (z⁻¹ • 1)

Tactic:
ext n
    simp only [FormalMultilinearSeries.smul_apply, smul_apply,
      FormalMultilinearSeries.compContinuousLinearMap_apply]
    simp [add_assoc, pow_add _ _ (a + 1), mul_assoc]

Goal state after:
Module: Mathlib.Analysis.Analytic.RadiusLiminf

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
⊢ p.radius = liminf (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop

Tactic:
apply le_antisymm

Goal state after:
case a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
⊢ p.radius ≤ liminf (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop

case a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
⊢ liminf (fun n ↦ 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop ≤ p.radius
Module: Mathlib.Topology.MetricSpace.MetricSeparated

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : PseudoEMetricSpace X
s t s' : Set X
r : ℝ≥0∞
r0 : r ≠ 0
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
r' : ℝ≥0∞
r0' : r' ≠ 0
hr' : ∀ x ∈ s', ∀ y ∈ t, r' ≤ edist x y
⊢ min r r' ≠ 0

Tactic:
rw [← pos_iff_ne_zero] at r0 r0' ⊢

Goal state after:
case refine_1
X : Type u_1
inst✝ : PseudoEMetricSpace X
s t s' : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
r' : ℝ≥0∞
r0' : 0 < r'
hr' : ∀ x ∈ s', ∀ y ∈ t, r' ≤ edist x y
⊢ 0 < min r r'
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
r : ℝ≥0
hr : r ≤ D.packingRadius / 2
x y z : X
hx : x ∈ D
hy : y ∈ D
hxz : x ∈ ball z ↑r
hyz : y ∈ ball z ↑r
⊢ x = y

Tactic:
by_contra hne

Goal state after:
X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
r : ℝ≥0
hr : r ≤ D.packingRadius / 2
x y z : X
hx : x ∈ D
hy : y ∈ D
hxz : x ∈ ball z ↑r
hyz : y ∈ ball z ↑r
hne : ¬x = y
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Log.ERealExp

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

Tactic:
exp_eq_zero_iff

Goal state after:
case bot
y : EReal
h : ⊥ < y
⊢ ¬y = ⊥
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u v : ℕ → EReal
h : linearGrowthSup u ≠ ⊥ ∨ linearGrowthSup v ≠ ⊤
h' : linearGrowthSup u ≠ ⊤ ∨ linearGrowthSup v ≠ ⊥
⊢ linearGrowthSup (u + v) ≤ linearGrowthSup u + linearGrowthSup v

Tactic:
refine (limsup_add_le h h').trans_eq' (limsup_congr (Eventually.of_forall fun n ↦ ?_))

Goal state after:
u v : ℕ → EReal
h : linearGrowthSup u ≠ ⊥ ∨ linearGrowthSup v ≠ ⊤
h' : linearGrowthSup u ≠ ⊤ ∨ linearGrowthSup v ≠ ⊥
n : ℕ
⊢ ((fun n ↦ u n / ↑n) + fun n ↦ v n / ↑n) n = (u + v) n / ↑n
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✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hf : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
z : ℤ
⊢ ∀ (z_1 : ℤ), Tendsto (fun a ↦ k a ^ z_1 * (k a ^ z * f a)) l (𝓝 0)

Tactic:
refine fun z' => (hf <| z' + z).congr' ((hk.eventually_ne_atTop 0).mono fun x hx => ?_)

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
hf : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)
z z' : ℤ
x : α
hx : k x ≠ 0
⊢ k x ^ (z' + z) * f x = (fun a ↦ k a ^ z' * (k a ^ z * f a)) x
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
h : expGrowthSup u ≠ ⊥ ∨ expGrowthSup v ≠ ⊤
h' : expGrowthSup u ≠ ⊤ ∨ expGrowthSup v ≠ ⊥
⊢ expGrowthSup (u * v) ≤ expGrowthSup u + expGrowthSup v

Tactic:
refine (limsup_add_le h h').trans_eq' (limsup_congr (Eventually.of_forall fun n ↦ ?_))

Goal state after:
u v : ℕ → ℝ≥0∞
h : expGrowthSup u ≠ ⊥ ∨ expGrowthSup v ≠ ⊤
h' : expGrowthSup u ≠ ⊤ ∨ expGrowthSup v ≠ ⊥
n : ℕ
⊢ ((fun n ↦ (u n).log / ↑n) + fun n ↦ (v n).log / ↑n) n = ((u * v) n).log / ↑n
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
⊢ ∃ᶠ (x : ℕ) in atTop, ↑(v x) ≤ a' * ↑x → (u ∘ v) x ≤ a * b * ↑x

Tactic:
refine frequently_atTop.2 fun M ↦ ?_

Goal state after:
case neg.inr
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
M : ℕ
⊢ ∃ b_1, M ≤ b_1 ∧ (↑(v b_1) ≤ a' * ↑b_1 → (u ∘ v) b_1 ≤ a * b * ↑b_1)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
U : Set (Set ι)
UT : U ⊆ T
hU : IsChain (fun x1 x2 ↦ x1 ⊆ x2) U
a : ι
hat : a ∈ t
b : ι
u : Set ι
uU : u ∈ U
hbu : b ∈ u
hab : (B a ∩ B b).Nonempty
⊢ ∃ c, (∃ t ∈ U, c ∈ t) ∧ (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c

Tactic:
obtain ⟨c, cu, ac, hc⟩ : ∃ c, c ∈ u ∧ (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c :=
      (UT uU).2.2 a hat b hbu hab

Goal state after:
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
U : Set (Set ι)
UT : U ⊆ T
hU : IsChain (fun x1 x2 ↦ x1 ⊆ x2) U
a : ι
hat : a ∈ t
b : ι
u : Set ι
uU : u ∈ U
hbu : b ∈ u
hab : (B a ∩ B b).Nonempty
c : ι
cu : c ∈ u
ac : (B a ∩ B c).Nonempty
hc : δ a ≤ τ * δ c
⊢ ∃ c, (∃ t ∈ U, c ∈ t) ∧ (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
⊢ expGrowthSup (u + v) = expGrowthSup (u ⊔ v)

Tactic:
apply le_antisymm

Goal state after:
case a
u v : ℕ → ℝ≥0∞
⊢ expGrowthSup (u + v) ≤ expGrowthSup (u ⊔ v)

case a
u v : ℕ → ℝ≥0∞
⊢ expGrowthSup (u ⊔ v) ≤ expGrowthSup (u + v)
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
⊢ ∃ᶠ (x : ℕ) in atTop, a' * ↑x ≤ ↑(v x) → a * b * ↑x ≤ (u ∘ v) x

Tactic:
refine frequently_atTop.2 fun M ↦ ?_

Goal state after:
case neg
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
M : ℕ
⊢ ∃ b_1, M ≤ b_1 ∧ (a' * ↑b_1 ≤ ↑(v b_1) → a * b * ↑b_1 ≤ (u ∘ v) b_1)
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
h' : ∀ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ linearGrowthInf (u ∘ v) = a * linearGrowthInf u

Tactic:
apply le_antisymm

Goal state after:
case neg.a
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
h' : ∀ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ linearGrowthInf (u ∘ v) ≤ a * linearGrowthInf u

case a
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
h' : ∀ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ a * linearGrowthInf u ≤ linearGrowthInf (u ∘ v)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂(μ univ)⁻¹ • μ = ∫⁻ (x : α), f x ∂μ

Tactic:
measure_univ,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂1⁻¹ • μ = ∫⁻ (x : α), f x ∂μ
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
u_1 : ∃ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ linearGrowthSup (u ∘ v) = a * linearGrowthSup u

Tactic:
apply le_antisymm

Goal state after:
case neg.a
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
u_1 : ∃ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ linearGrowthSup (u ∘ v) ≤ a * linearGrowthSup u

case a
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
u_1 : ∃ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ a * linearGrowthSup u ≤ linearGrowthSup (u ∘ v)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → E
⊢ ∫ (x : α), f x ∂(μ univ)⁻¹ • μ = ∫ (x : α), f x ∂μ

Tactic:
measure_univ,

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → E
⊢ ∫ (x : α), f x ∂1⁻¹ • μ = ∫ (x : α), f x ∂μ
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 := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0

Tactic:
obtain ⟨u, ut', u_disj, hu⟩ : ∃ u ⊆ t',
      u.PairwiseDisjoint B ∧ ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b := by
    have A : ∀ a ∈ t', r a ≤ 1 := by
      intro a ha
      apply ha.2.trans (hR1 (c a))
    have A' : ∀ a ∈ t', (B a).Nonempty :=
      fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)
    refine exists_disjoint_subfamily_covering_enlargement
      B t' r 2 one_lt_two (fun a ha => ?_) 1 A A'
    exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))

Goal state after:
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 := {a | a ∈ t ∧ r a ≤ R (c a)}
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
⊢ ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
h : NeZero μ
c : E
⊢ (((μ univ)⁻¹ • μ) univ).toReal • c = c

Tactic:
measure_univ,

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
h : NeZero μ
c : E
⊢ ENNReal.toReal 1 • c = c
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hf : Integrable f μ
⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0

Tactic:
integral_sub hf (integrable_const _),

Goal state after:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hf : Integrable f μ
⊢ ∫ (a : α), f a ∂μ - ∫ (a : α), ⨍ (a : α), f a ∂μ ∂μ = 0
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case refine_2
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 < ∞)
⊢ IntegrableOn (fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) s μ

Tactic:
exact hf.sub (integrableOn_const hμ₁)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

Tactic:
← withDensity_apply f hs,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ (μ.withDensity f) s = ∫⁻ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

Tactic:
← withDensity_apply f hs,

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ (μ.withDensity f) s = ∫⁻ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ sInf {d | μ ≤ d + 0} = μ

Tactic:
apply le_antisymm

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ sInf {d | μ ≤ d + 0} ≤ μ

case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ ≤ sInf {d | μ ≤ d + 0}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ sInf {d | μ ≤ d + ν} = measure_sub

Tactic:
apply le_antisymm

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ sInf {d | μ ≤ d + ν} ≤ measure_sub

case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ measure_sub ≤ sInf {d | μ ≤ d + ν}
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍⁻ (x : α) in s, g x ∂μ ≤ g x
hx : x ∉ {x | g x = ∞}
⊢ x ∈ {a | ⨍⁻ (a : α) in s, g a ∂μ ≤ f a}

Tactic:
exact hfx.trans (hgf _)

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

Goal state before:
pp: 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
⊢ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν}) = sInf {d | μ.restrict s ≤ d + ν.restrict s}

Tactic:
apply le_antisymm

Goal state after:
case a
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
⊢ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν}) ≤ sInf {d | μ.restrict s ≤ d + ν.restrict s}

case a
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
⊢ sInf {d | μ.restrict s ≤ d + ν.restrict s} ≤ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν})
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a
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
t : Measure α
h_t_in : t ∈ {d | μ ≤ d + ν}
⊢ t.restrict s ∈ {d | μ.restrict s ≤ d + ν.restrict s}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case a
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
t : Measure α
h_t_in : t ∈ {d | μ ≤ d + ν}
⊢ μ.restrict s ≤ t.restrict s + ν.restrict s
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : μ.HaveLebesgueDecomposition ν
inst✝ : IsFiniteMeasure μ
⊢ μ - μ.singularPart ν = ν.withDensity (μ.rnDeriv ν)

Tactic:
nth_rw 1 [← rnDeriv_add_singularPart μ ν]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : μ.HaveLebesgueDecomposition ν
inst✝ : IsFiniteMeasure μ
⊢ ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν - μ.singularPart ν = ν.withDensity (μ.rnDeriv ν)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case refine_1.inl
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
C : ℝ≥0
h : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)
s : Set α
f : α → Set (Set α)
fsubset :
  ∀ x ∈ s,
    f x ⊆
      {a |
        IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧
          (interior a).Nonempty ∧ ∃ r, a ⊆ closedBall x r ∧ μ (closedBall x (3 * r)) ≤ ↑C * μ a}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
t : Set (ℝ × α × Set α) :=
  {p |
    p.2.2 ⊆ closedBall p.2.1 p.1 ∧
      μ (closedBall p.2.1 (3 * p.1)) ≤ ↑C * μ p.2.2 ∧
        (interior p.2.2).Nonempty ∧
          IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] p.2.2 ∧ p.2.2 ∈ f p.2.1 ∧ p.2.1 ∈ s}
x : α
xs : x ∈ s
ε : ℝ
εpos : ε > 0
a : Set α
ha : a ∈ f x
h'a : a ⊆ closedBall x ε
a_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
a_int : (interior a).Nonempty
r : ℝ
ar : a ⊆ closedBall x r
μr : μ (closedBall x (3 * r)) ≤ ↑C * μ a
h' : min r ε = r ∧ r ≤ ε
⊢ (min r ε, x, a).2.2 ⊆ closedBall (min r ε, x, a).2.1 (min r ε, x, a).1

Tactic:
rwa [h'.1]

Goal state after: