mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_4
β : Type u_5
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
f : β → α
s : Finset α
μ : Measure β
hf : AEMeasurable f μ
⊢ (∀ᵐ (b : β) ∂μ, f b ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a

Tactic:
by
  rw [← ae_map_iff hf (by measurability), ae_mem_finset_iff]
  simp [map_apply₀ hf]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
R : Type u_3
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
m : Measure (Measure α)
s : Set α
hs : MeasurableSet s
⊢ (c • m).join s = (c • m.join) s

Tactic:
simp [hs]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s : Set α
hs : MeasurableSet s
⊢ (join 0) s = 0 s

Tactic:
simp [hs]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
⊢ ∫⁻ (x : α), f x ∂μ ≤ C

Tactic:
have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α), f x ∂μ ≤ C
Module: Mathlib.MeasureTheory.Measure.Doubling

Goal state before:
pp: case succ
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ihn : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑C ^ n * μ (closedBall x ε)
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ (n + 1) * ε)) ≤ ↑C ^ (n + 1) * μ (closedBall x ε)

Tactic:
| succ n ihn =>

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

Goal state before:
pp: case inl
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
hx : 0 ≠ ∞
⊢ (0 • μ).OuterRegular

pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular

Tactic:
· rw [zero_smul]
    exact OuterRegular.zero

Goal state after:
case inr
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ : Measure α
inst✝ : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
h : IsComplement (↑H) T
⊢ Finite ↑T ↔ H.FiniteIndex

Tactic:
by
  rw [← h.rightQuotientEquiv.finite_iff,
    (QuotientGroup.quotientRightRelEquivQuotientLeftRel H).finite_iff]
  exact ⟨fun _ ↦ finiteIndex_of_finite_quotient, fun _ ↦ finite_quotient_of_finiteIndex⟩

Goal state after:
Module: Mathlib.MeasureTheory.Group.Action

Goal state before:
pp: G : Type u
α : Type w
m : MeasurableSpace α
inst✝² : Group G
inst✝¹ : MulAction G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
c : G
s : Set α
⊢ μ s ≤ μ ((fun x ↦ c • x) ⁻¹' s)

Tactic:
by
    simpa [preimage_preimage] using measure_preimage_smul_le μ c⁻¹ ((c • ·) ⁻¹' s)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝² : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝¹ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
y : ℕ → γ
hy : ∀ (n : ℕ), y n ∈ ↑(s n)
I : ∀ (m n : ℕ), (↑(s m) ∩ ↑(s n)).Nonempty
cauchy_y : CauchySeq y
this✝ : Nonempty γ
z : γ := atTop.limUnder y
y_lim : Tendsto y atTop (𝓝 z)
this : f z = x
⊢ f z ∈ range f

Tactic:
exact mem_range_self _

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
⊢ μ F = μ (⋃ i, F ∩ s i)

Tactic:
hBU

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
⊢ μ F = μ F
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
hr : r < μ s
B : ℕ → Set α := spanningSets μ
hBU : ⋃ n, s ∩ B n = s
⊢ μ s = μ (⋃ i, s ∩ spanningSets μ i)

Tactic:
hBU

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
hr : r < μ s
B : ℕ → Set α := spanningSets μ
hBU : ⋃ n, s ∩ B n = s
⊢ μ s = μ s
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
⊢ μ.prod 0 = 0

Tactic:
simp [Measure.prod]

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

Goal state before:
pp: G✝ : Type u_1
H✝ : Type u_2
inst✝⁷ : MeasurableSpace G✝
inst✝⁶ : MeasurableSpace H✝
inst✝⁵ : Mul G✝
μ✝ : Measure G✝
inst✝⁴ : MeasurableMul G✝
G : Type u_3
α : Type u_4
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
H : Subgroup G
y : ↥H
s : Set α
hs : MeasurableSet s
⊢ μ ((fun x ↦ y • x) ⁻¹' s) = μ s

Tactic:
by convert! SMulInvariantMeasure.measure_preimage_smul (μ := μ) (y : G) hs

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

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ map (Prod.map (fun x ↦ g * x) fun x ↦ h * x) (μ.prod ν) = μ.prod ν

Tactic:
rw [← map_prod_map _ _ (measurable_const_mul g) (measurable_const_mul h),
    map_mul_left_eq_self μ g, map_mul_left_eq_self ν h]

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

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulRightInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod (map (fun x ↦ x * h) ν) = μ.prod ν

Tactic:
map_mul_right_eq_self ν h

Goal state after:
G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulRightInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod ν = μ.prod ν
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ ∫⁻ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫⁻ (x : α) in s, ∫⁻ (y : β) in t, f (x, y) ∂ν ∂μ

Tactic:
rw [← Measure.prod_restrict, lintegral_prod _ (by rwa [Measure.prod_restrict])]

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

Goal state before:
pp: X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
r : ℝ≥0∞
hr : r < μ F
B : ℕ → Set X := compactCovering X
hBc : ∀ (n : ℕ), IsCompact (F ∩ B n)
hBU : ⋃ n, F ∩ B n = F
⊢ μ F = μ (⋃ i, F ∩ B i)

pp: X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
r : ℝ≥0∞
hr : r < μ F
B : ℕ → Set X := compactCovering X
hBc : ∀ (n : ℕ), IsCompact (F ∩ B n)
hBU : ⋃ n, F ∩ B n = F
⊢ Monotone fun n ↦ F ∩ B n

Tactic:
hBU

Goal state after:
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
r : ℝ≥0∞
hr : r < μ F
B : ℕ → Set X := compactCovering X
hBc : ∀ (n : ℕ), IsCompact (F ∩ B n)
hBU : ⋃ n, F ∩ B n = F
⊢ μ F = μ F

X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
r : ℝ≥0∞
hr : r < μ F
B : ℕ → Set X := compactCovering X
hBc : ∀ (n : ℕ), IsCompact (F ∩ B n)
hBU : ⋃ n, F ∩ B n = F
⊢ Monotone fun n ↦ F ∩ B n
Module: Mathlib.MeasureTheory.Function.Egorov

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : PseudoEMetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Nonempty ι
n : ℕ
hfg : μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)} = 0
⊢ μ (s ∩ ⋂ j, notConvergentSeq f g n j) ≤ μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)}

Tactic:
refine measure_mono fun x => ?_

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : PseudoEMetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Nonempty ι
n : ℕ
hfg : μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)} = 0
x : α
⊢ x ∈ s ∩ ⋂ j, notConvergentSeq f g n j → x ∈ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)}
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : f a = 0
ha_null✝ : a ∈ {x | f' x ≠ 0}ᶜ
ha_null : f' a = 0
⊢ ↑(f a) * g a = ↑0 * g' a

Tactic:
rw [ha]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : f a = 0
ha_null✝ : a ∈ {x | f' x ≠ 0}ᶜ
ha_null : f' a = 0
⊢ ↑0 * g a = ↑0 * g' a
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
c : β
⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ c = 0 ∨ IsFiniteMeasure μ

Tactic:
by
  simp [hasFiniteIntegral_const_iff_enorm enorm_ne_top]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
F_le_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Tactic:
← ae_all_iff

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
F_le_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ENNReal.ofReal ‖F i a‖ ≤ ENNReal.ofReal (bound a)
⊢ ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
F' : ℕ → α → ε
f' : α → ε
bound' : α → ℝ≥0∞
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F' n a‖ₑ ≤ bound' a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F' n a) atTop (𝓝 (f' a))
⊢ ∀ᵐ (a : α) ∂μ, ‖f' a‖ₑ ≤ bound' a

Tactic:
← ae_all_iff

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
F' : ℕ → α → ε
f' : α → ε
bound' : α → ℝ≥0∞
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ‖F' i a‖ₑ ≤ bound' a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F' n a) atTop (𝓝 (f' a))
⊢ ∀ᵐ (a : α) ∂μ, ‖f' a‖ₑ ≤ bound' a
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ √(x ^ 2) = x

Tactic:
by rw [sq, sqrt_mul_self h]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ ↑(NNReal.sqrt x.toNNReal⁻¹) = (√x)⁻¹

Tactic:
NNReal.sqrt_inv,

Goal state after:
x : ℝ
⊢ ↑(NNReal.sqrt x.toNNReal)⁻¹ = (√x)⁻¹
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ dist ((starRingEnd ℂ) z) z = 2 * |z.im|

Tactic:
dist_conj_self

Goal state after:
z : ℂ
⊢ 2 * |z.im| = 2 * |z.im|
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: m : ℤ
⊢ I ^ m = I ^ (m % 4)

Tactic:
conv_lhs => rw [← Int.mul_ediv_add_emod m 4]

Goal state after:
m : ℤ
⊢ I ^ (4 * (m / 4) + m % 4) = I ^ (m % 4)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
s✝ : Set α
hs : MeasurableSet s✝
⊢ MeasurableSet s✝

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
s✝ : Set α
hs : μ s✝ = 0
⊢ (Measure.map (fun z ↦ z.1) (μ.prod ν)) s✝ = 0

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
s✝ : Set β
hs : MeasurableSet s✝
⊢ MeasurableSet s✝

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
s✝ : Set β
hs : ν s✝ = 0
⊢ (Measure.map (fun z ↦ z.2) (μ.prod ν)) s✝ = 0

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
s✝ : Set β
hs : MeasurableSet s✝
⊢ MeasurableSet s✝

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
s✝ : Set β
hs : ν s✝ = 0
⊢ (Measure.map (fun z ↦ z.2) (μ.prod ν)) s✝ = 0

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
s✝ : Set α
hs : MeasurableSet s✝
⊢ MeasurableSet s✝

Tactic:
simp [hs]

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

Goal state before:
pp: case funProp.discharger
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
s✝ : Set α
hs : μ s✝ = 0
⊢ (Measure.map (fun z ↦ z.1) (μ.prod ν)) s✝ = 0

Tactic:
simp [hs]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Star

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : StarAddMonoid A
inst✝³ : Module R A
inst✝² : StarModule R A
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousStar A
x : A
⊢ (starL R).symm x = starAddEquiv.symm x

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Star

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : AddCommMonoid A
inst✝⁵ : StarAddMonoid A
inst✝⁴ : Module R A
inst✝³ : StarModule R A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousStar A
inst✝ : TrivialStar R
x : A
⊢ (starL' R).symm x = starAddEquiv.symm x

Tactic:
by
  simp

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: f : ℂ →ₐ[ℝ] ℂ
⊢ f = AlgHom.id ℝ ℂ ∨ f = ↑conjAe

Tactic:
refine
      (eq_or_eq_neg_of_sq_eq_sq (f I) I <| by rw [← map_pow, I_sq, map_neg, map_one]).imp ?_ ?_

Goal state after:
case refine_1
f : ℂ →ₐ[ℝ] ℂ
⊢ f I = I → f = AlgHom.id ℝ ℂ

case refine_2
f : ℂ →ₐ[ℝ] ℂ
⊢ f I = -I → f = ↑conjAe
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + ↑(re z) = 2 * ↑(re z)

Tactic:
two_mul

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + ↑(re z) = ↑(re z) + ↑(re z)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(im z) = I * ((starRingEnd K) z - z) / 2

Tactic:
← neg_inj,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ -↑(im z) = -(I * ((starRingEnd K) z - z) / 2)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ |re z| ≤ ‖z‖

Tactic:
rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
  apply re_sq_le_normSq

Goal state after:
Module: Mathlib.Tactic.NormNum.BigOperators

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
s : Finset α
h : a ∉ s
⊢ insert a s = Finset.cons a s h

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i j : ℕ
ji : j ≥ max n i
k : ℕ
hk : j = k + max n i
⊢ abv (∑ i ∈ range (k + max n i), f i - ∑ i ∈ range (max n i), f i) ≤
    ∑ k ∈ range (k + max n i), a k - ∑ k ∈ range (max n i), a k

Tactic:
clear hk ji j

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k : ℕ
⊢ abv (∑ i ∈ range (k + max n i), f i - ∑ i ∈ range (max n i), f i) ≤
    ∑ k ∈ range (k + max n i), a k - ∑ k ∈ range (max n i), a k
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: case succ
α : Type u_1
R : Type u_13
inst✝² : SeminormedRing R
S : Type u_17
inst✝¹ : NormedRing S
inst✝ : NormMulClass S
l : Filter α
f : α → R
g : α → S
h : f =o[l] g
n : ℕ
ihn : (fun x ↦ f x ^ n.succ) =o[l] fun x ↦ g x ^ n.succ
⊢ (fun x ↦ f x ^ (n + 1).succ) =o[l] fun x ↦ g x ^ (n + 1).succ

Tactic:
| succ n ihn =>

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

Goal state before:
pp: α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
s : Finset ι
B : ι → α → ℝ
hAB : ∀ i ∈ s, A i =o[l] B i
⊢ (fun H ↦ ∑ i ∈ s, A i H) =o[l] fun H ↦ ∑ i ∈ s, ‖B i H‖

Tactic:
by
  induction s using Finset.cons_induction with
  | empty => simp [isLittleO_zero]
  | cons i s his h =>
  simp_rw [Finset.sum_cons]
  calc (fun H => A i H + ∑ j ∈ s, A j H)
      =o[l] fun H => ‖B i H‖ + ‖∑ j ∈ s, ‖B j H‖‖ :=
          (hAB i (by simp)).add_add (h (fun j hj => hAB j (by simp [hj])))
    _ =ᶠ[l] fun H => ‖B i H‖ + ∑ j ∈ s, ‖B j H‖ := by
        refine Eventually.of_forall fun H ↦ congr_arg (‖B i H‖ + ·) ?_
        exact Real.norm_of_nonneg (Finset.sum_nonneg fun _ _ => norm_nonneg _)

Goal state after:
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case succ.succ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : j + 1 ≥ 1 → ‖∑ m ∈ range (j + 1), 0 ^ m / ↑m.factorial - 1‖ = 0
hj : j + 1 + 1 ≥ 1
⊢ ‖∑ m ∈ range (j + 1 + 1), 0 ^ m / ↑m.factorial - 1‖ = ‖∑ m ∈ range (j + 1), 0 ^ m / ↑m.factorial - 1‖

Tactic:
simp only [sum_range_succ, pow_succ]

Goal state after:
case succ.succ
ε : ℝ
ε0 : ε > 0
j : ℕ
ih : j + 1 ≥ 1 → ‖∑ m ∈ range (j + 1), 0 ^ m / ↑m.factorial - 1‖ = 0
hj : j + 1 + 1 ≥ 1
⊢ ‖∑ m ∈ range j, 0 ^ m / ↑m.factorial + 0 ^ j / ↑j.factorial + 0 ^ j * 0 / ↑(j + 1).factorial - 1‖ =
    ‖∑ m ∈ range j, 0 ^ m / ↑m.factorial + 0 ^ j / ↑j.factorial - 1‖